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A Rasmussen invariant for links in ℝℙ³

Authors

Ciprian Manolescu, Michael Willis
Type: Article
Publication Date: 2025-05-09
Citations: 0
DOI: https://doi.org/10.1090/btran/221

Abstract

Asaeda-Przytycki-Sikora, Manturov, and Gabrovšek extended Khovanov homology to links in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R double-struck upper P cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {RP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant in this setting. We show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant gives constraints on the genera of link cobordisms in the cylinder <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I times double-struck upper R double-struck upper P cubed"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">I \times \mathbb {RP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we give examples of freely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-periodic knots in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are concordant but not standardly equivariantly concordant.

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  • Transactions of the American Mathematical Society Series B
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Topological isotopy and Cochran’s derived invariants

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Null surgery on knots in L-spaces

2018-01-24
Yi Ni , Faramarz Vafaee
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Paul A. Sundheim
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2016-08-16
Takahiro Kitayama , Masanori Morishita , Ryoto Tange +1 more
We study the twisted knot module for the universal deformation of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="SL Subscript 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>SL</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\textrm {SL}_2</mml:annotation> … We study the twisted knot module for the universal deformation of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="SL Subscript 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>SL</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\textrm {SL}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-representation of a knot group and introduce an associated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function, which may be seen as an analogue of the algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we compute the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function by some concrete examples for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bridge knots.

A Khovanov stable homotopy type

2014-04-22
Robert Lipshitz , Sucharit Sarkar
Given a link diagram <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct spectra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K … Given a link diagram <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct spectra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that the Khovanov homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript i comma j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K^{i,j}(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to the (reduced) singular cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With tilde Superscript i Baseline left-parenthesis script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {H}^{i}(\mathcal {X}_{Kh}^j(L))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The construction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is combinatorial and explicit. We prove that the stable homotopy type of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends only on the isotopy class of the corresponding link.
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An approach to homotopy classification of links

1988-01-01
Jerome Levine
A reformulation and refinement of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu overbar"> <mml:semantics> <mml:mover> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline \mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariants of Milnor … A reformulation and refinement of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu overbar"> <mml:semantics> <mml:mover> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline \mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariants of Milnor are used to give a homotopy classification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> component links and suggest a possible general homotopy classification. The main idea is to use the (reduced) group of a link and its "geometric" automorphisms to define the precise indeterminacy of these invariants.
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The Gysin triangle via localization and 𝐴¹-homotopy invariance

2016-09-08
Gonçalo Tabuada , Michel Van den Bergh
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth scheme, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth scheme, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a smooth closed subscheme, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the open complement. Given any localizing and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariant of dg categories <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct an associated Gysin triangle relating the value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at the dg categories of perfect complexes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the particular case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homotopy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory, this Gysin triangle yields a new proof of Quillen’s localization theorem, which avoids the use of devissage. As a first application, we prove that the value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an étale descent result concerning noncommutative mixed motives with rational coefficients.

Stable-homotopy and homology invariants of boundary links

1992-01-01
Michael Färber
An <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n greater-than-or-equal-to 5 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> … An <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n greater-than-or-equal-to 5 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n \geq 5)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> link in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n plus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n + 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional sphere is <italic>stable</italic> if the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i"> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding="application/x-tex">i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th homotopy group of its complement <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to i less-than-or-equal-to left-parenthesis n plus 1 right-parenthesis slash 3"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>i</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2 \leq i \leq (n + 1)/3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _1}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is freely generated by meridians. In this paper a classification of stable links in terms of stable homotopy theory is given. For <italic>simple</italic> links this classification gives a complete algebraic description. We also study Poincaré duality in the space of the free covering of the complement of a boundary link. The explicit computation of the corresponding Ext-functors gives a construction of new homology pairings, generalizing the Blanchfield and the torsion pairings for knots.
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Burnside groups and 𝑛-moves for links

2018-12-05
Haruko A. Miyazawa , Kodai Wada , Akira Yasuhara
M. K. Da̧bkowski and J. H. Przytycki introduced the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>th Burnside group of a link, which is an invariant preserved by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-moves. Using this … M. K. Da̧bkowski and J. H. Przytycki introduced the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>th Burnside group of a link, which is an invariant preserved by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-moves. Using this invariant, for an odd prime<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>, they proved that there are links which cannot be reduced to trivial links via<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-moves. It is generally difficult to determine if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>th Burnside groups associated to a link and the corresponding trivial link are isomorphic. In this paper, we give a necessary condition for the existence of such an isomorphism. Using this condition we give a simple proof for their results that concern<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-move reducibility of links.
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Dehn surgery on knots—tracing the evolution of research

2023-04-20
Kimihiko Motegi
Every closed orientable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold is obtained from the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> … Every closed orientable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold is obtained from the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Dehn surgery on a link, and thus Dehn surgery is a useful way to construct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds. In this survey article we restrict our attention to Dehn surgery on knots in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and take a quick look at the evolution of study on this field along developments of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional topology.

A deformation of Robert-Wagner foam evaluation and link homology

2024-01-01
Mikhail Khovanov , Nitu Kitchloo
We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state … We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state spaces for planar <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> MOY graphs (Murakami-Ohtsuki-Yamada graphs). Skein relations for the deformation are worked out in details in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> case. These skein relations deform <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> foam relations of Beliakova, Hogancamp, Putyra and Wehrli. We establish the Reidemeister move invariance of the resulting chain complexes assigned to link diagrams, giving us a link homology theory.
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A stroll in equivariant 𝐾-theory

2024-01-01
Chi-Kwong Fok
Equivariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory is a generalized equivariant cohomology theory that is a hybrid of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Equivariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory is a generalized equivariant cohomology theory that is a hybrid of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory of a topological space and the representation theory of a group acting on it. In this article, we review the basics of equivariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory and focus on the localization theorem of Atiyah and Segal, which have become important tools in equivariant topology nowadays. We then discuss the application of equivariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory to equivariant formality, and briefly mention some recent developments.

𝔫-homology of generic representations for 𝔊𝔏(𝔑)

1999-01-01
Jen-Tseh Chang , James Cogdell
We compute the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homology for a class of representations of <inline-formula content-type="math/mathml"> <mml:math … We compute the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homology for a class of representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis upper N comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(N,\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis upper N comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(N,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which admit a Whittaker model. They are all completely reducible.
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A topological interpretation for the bias invariant

1986-01-01
Micheal N. Dyer
The bias invariant has been used to distinguish between the homotopy types of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complexes. In this note we show … The bias invariant has been used to distinguish between the homotopy types of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complexes. In this note we show that two finite, connected <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complexes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with isomorphic fundamental groups and the same Euler characteristic have the same bias invariant if and only if there is a map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:X \to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is a homology equivalence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _1}f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 2 f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_2}f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphisms).
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Slice links in 𝑆⁴

1984-01-01
Tim D. Cochran
We produce necessary and sufficient conditions of a homotopy-theoretic nature for a link of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres in <inline-formula content-type="math/mathml"> <mml:math … We produce necessary and sufficient conditions of a homotopy-theoretic nature for a link of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 4"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{S^4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be slice (i.e., cobordant to the unlink). We give algebraic conditions on the link <italic>group</italic> sufficient to guarantee sliceness, generalizing the known results for boundary links. The notion of a "stable link" is introduced and shown to be useful in constructing cobordisms in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Suspension spectra and homology equivalences

1984-01-01
Nicholas J. Kuhn
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon normal upper Sigma Superscript normal infinity Baseline upper X right-arrow normal upper Sigma Superscript normal infinity Baseline upper Y"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon normal upper Sigma Superscript normal infinity Baseline upper X right-arrow normal upper Sigma Superscript normal infinity Baseline upper Y"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:{\Sigma ^\infty }X \to {\Sigma ^\infty }Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a stable map between two connected spaces, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_{\ast }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a generalized homology theory. We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript asterisk Baseline left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_{\ast }}(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an isomorphism then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript asterisk Baseline left-parenthesis normal upper Omega Superscript normal infinity Baseline f right-parenthesis colon upper E Subscript asterisk Baseline left-parenthesis upper Q upper X right-parenthesis right-arrow upper E Subscript asterisk Baseline left-parenthesis upper Q upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_{\ast }}({\Omega ^\infty }f):{E_{\ast }}(QX) \to {E_{\ast }}(QY)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a monomorphism, but possibly not an epimorphism. Applications of this theorem include results of Miller and Snaith concerning the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory of the Kahn-Priddy map.
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Nil 𝐾-theory maps to cyclic homology

1987-01-01
Charles A. Weibel
Algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory breaks into two pieces: nil <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> … Algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory breaks into two pieces: nil <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory and Karoubi-Villamayor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory. Karoubi has constructed Chern classes from the latter groups into cyclic homology. We construct maps from nil <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory to cyclic homology which are compatible with Karoubi’s maps, but with a degree shift. Several recent results show that in characteristic zero our map is often an isomorphism.
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Property G and the 4-genus

2024-01-12
Yi Ni
We say a null-homologous knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> … We say a null-homologous knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Property G, if the Thurston norm and fiberedness of the complement of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is preserved under the zero surgery on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we will show that, if the smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-genus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K times StartSet 0 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K\times \{0\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in a certain homology class) in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper Y times left-bracket 0 comma 1 right-bracket right-parenthesis number-sign upper N ModifyingAbove double-struck upper C upper P squared With bar"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="normal">#<!-- # --></mml:mi> <mml:mi>N</mml:mi> <mml:mover> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">(Y\times [0,1])\#N\overline {\mathbb CP^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a rational homology sphere, is smaller than the Seifert genus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Property G. When the smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-genus is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be taken to be any closed, oriented <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold.
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Witten-Helffer-Sjöstrand theory for 𝑆¹-equivariant cohomology

1999-02-24
Hon-kit Wai
Given an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant Morse function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation … Given an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant Morse function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant Riemannian metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a family of finite dimensional subcomplexes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ModifyingAbove normal upper Omega With tilde Subscript i n v comma s m Superscript asterisk Baseline left-parenthesis upper M comma t right-parenthesis comma upper D left-parenthesis t right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\widetilde \Omega ^*_{inv,sm}(M,t), D(t))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">t\in [0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of the Witten deformation of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Delta With tilde Superscript k Baseline left-parenthesis t right-parenthesis equals upper D Subscript k Superscript asterisk Baseline left-parenthesis t right-parenthesis upper D Subscript k Baseline left-parenthesis t right-parenthesis plus upper D Subscript k minus 1 Baseline left-parenthesis t right-parenthesis upper D Subscript k minus 1 Superscript asterisk Baseline left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>k</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In fact the spectrum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Delta With tilde Superscript k Baseline left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde \Delta ^k(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ModifyingAbove normal upper Omega With tilde Subscript i n v comma s m Superscript asterisk Baseline left-parenthesis upper M comma t right-parenthesis comma upper D left-parenthesis t right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">( \widetilde \Omega ^*_{inv,sm}(M,t),D(t))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the complex of eigenforms corresponding to the small eigenvalues of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Delta With tilde left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde \Delta (t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This permits us to verify the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant Morse inequalities. Moreover suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is self-indexing and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(f,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a geometric complex which is induced by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(f,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and calculates the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant cohomology of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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References (24)

Floer homology and knot complements

2003-01-01
Jacob Rasmussen
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call … We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology. This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in math.GT/0209056.
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A Khovanov stable homotopy type

2014-04-22
Robert Lipshitz , Sucharit Sarkar
Given a link diagram <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct spectra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K … Given a link diagram <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct spectra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that the Khovanov homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript i comma j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K^{i,j}(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to the (reduced) singular cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper H With tilde Superscript i Baseline left-parenthesis script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {H}^{i}(\mathcal {X}_{Kh}^j(L))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The construction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is combinatorial and explicit. We prove that the stable homotopy type of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Subscript upper K h Superscript j Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>K</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}_{Kh}^j(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends only on the isotopy class of the corresponding link.
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Khovanov homology and the slice genus

2010-09-06
Jacob Rasmussen
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Categorification of the Kauffman bracket skein module of<i>I</i>–bundles over surfaces

2004-12-15
Marta Asaeda , Józef H. Przytycki , Adam S. Sikora
Khovanov defined graded homology groups for links L ⊂ R 3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in … Khovanov defined graded homology groups for links L ⊂ R 3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I -bundles M over surfaces F = D 2 (except for the homology with Z/2 coefficients only).Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients.After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a "categorification" of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups.Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.
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KHOVANOV HOMOLOGY FOR VIRTUAL KNOTS WITH ARBITRARY COEFFICIENTS

2007-03-01
Vassily Olegovich Manturov
In the present paper, we construct Khovanov homology theory with arbitrary coefficients for arbitrary virtual knots. We give a definition of the complex, which is homotopy equivalent to the initial … In the present paper, we construct Khovanov homology theory with arbitrary coefficients for arbitrary virtual knots. We give a definition of the complex, which is homotopy equivalent to the initial Khovanov complex in the classical case; our definition works in the virtual case as well. The method used in this work allows us to construct a Khovanov homology theory for "twisted virtual knots" in the sense of Bourgoin and Viro [4, 27] (in particular, for knots in RP 3 ). We also generalize some results of the Khovanov homology for virtual knots with arbitrary atoms (Wehrli and Kofman–Champanerkar spanning tree, minimality problems, Frobenius extensions) and orientable ones (Rasmussen's invariant).
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THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF

2013-04-11
Boštjan Gabrovšek
Abstract Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $ , is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler … Abstract Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $ , is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al . [‘Categorification of the Kauffman bracket skein module of $I$ -bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$ -bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $ , where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $ . This paper categorifies the missing case of the twisted $I$ -bundle over $ \mathbb{R} {\mathrm{P} }^{2} $ , $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $ , by redefining the differential in the Khovanov chain complex in a suitable manner.
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LINK COBORDISM IN RATIONAL HOMOLOGY 3-SPHERES

1993-09-01
Patrick M. Gilmer
We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form … We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form on non-oriented spanning surfaces. We develop their cobordism properties from this point of view. We give a good way to index these invariants. We also define d-signatures and d-nullities for links which are null-homologous modulo d in a rational homology sphere from the point of view of branched covers. We index d-signatures and d-nullities and develop their cobordism properties. Finally we define Arf invariants (when possible) in a general closed 3-manifold using spin structures.

Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

2008-12-01
Anna Beliakova , Stephan M. Wehrli
Abstract We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphismclasses of these brackets are invariants of framed colored links. The Bar-Natan … Abstract We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphismclasses of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.
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An endomorphism of the Khovanov invariant

2004-12-12
Eun Soo Lee
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A categorification of the Jones polynomial

2000-02-15
Mikhail Khovanov
Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial. Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
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τ–invariants for knots in rational homology spheres

2020-07-20
Katherine Raoux
Ozsváth and Szabó used the knot filtration on CF (S 3 ) to define the τ -invariant for knots in the 3-sphere.In this article, we generalize their construction and define … Ozsváth and Szabó used the knot filtration on CF (S 3 ) to define the τ -invariant for knots in the 3-sphere.In this article, we generalize their construction and define a collection of τ -invariants associated to a knot K in a rational homology sphere Y .We then show that some of these invariants provide lower bounds for the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y .
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A 2-category of chronological cobordisms and odd Khovanov homology

2014-01-01
Krzysztof K. Putyra
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram … We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain $2$-cat
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Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

2016-12-23
Heather A. Dye , Aaron Kaestner , Louis H. Kauffman
The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well … The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in [Formula: see text] in the sense of Turaev [42].
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Annular Khovanov–Lee homology, braids, and cobordisms

2017-01-01
J. Elisenda Grigsby , Anthony Licata , Stephan M. Wehrli
We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type … We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.
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Doubled Khovanov Homology

2017-11-27
William Rushworth
Abstract We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology . We … Abstract We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology . We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case, and we define a doubled Rasmussen invariant . This invariant is used to obtain various cobordism obstructions; in particular, it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot and use this to show that knots with non-zero odd writhe are not slice.
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Lift in the 3-sphere of knots and links in lens spaces

2014-04-01
Enrico Manfredi
An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage [Formula: see text] of … An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage [Formula: see text] of L under the universal covering of L(p, q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p, q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift [Formula: see text] in S 3 . Using this construction, we are able to find different knots and links in L(p, q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.
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Invariants of 4–manifolds from Khovanov–Rozansky link homology

2022-12-31
Scott Morrison , Kevin Walker , Paul Wedrich
We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is … We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.
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A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds

2023-01-06
Ciprian Manolescu , Marco Marengon , Sucharit Sarkar +1 more
We extend the definition of Khovanov–Lee homology to links in connected sums of S1×S2's and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S1×S2, … We extend the definition of Khovanov–Lee homology to links in connected sums of S1×S2's and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S1×S2, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following 4-manifolds: B2×S2, S1×B3, CP2, and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from B4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are nonstandard.
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The Conway knot is not slice

2020-02-13
Lisa Piccirillo
A knot is said to be slice if it bounds a smooth properly embedded disk in $B^4$. We demonstrate that the Conway knot is not slice. This completes the classification … A knot is said to be slice if it bounds a smooth properly embedded disk in $B^4$. We demonstrate that the Conway knot is not slice. This completes the classification of slice knots under $13$ crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Khovanov Homology for Links in #r(S2×S1)

2020-07-09
Michael Willis
We revisit Rozansky's construction of Khovanov homology for links in S2×S1, extending it to define the Khovanov homology Kh(L) for links L in Mr=#r(S2×S1) for any r. The graded Euler … We revisit Rozansky's construction of Khovanov homology for links in S2×S1, extending it to define the Khovanov homology Kh(L) for links L in Mr=#r(S2×S1) for any r. The graded Euler characteristic of Kh(L) can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module S(Mr) of Hoste and Przytycki when L is null-homologous in Mr. The construction also allows for a clear path toward defining a Lee's homology Kh′(L) and associated s-invariant for such L, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in S3 and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

Knot Floer homology and relative adjunction inequalities

2022-11-12
Matthew Hedden , Katherine Raoux
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Equivariant cobordisms between freely periodic knots

2022-06-22
Keegan Boyle , Jeffrey Musyt
Abstract We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the … Abstract We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.
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An estimate of the Rasmussen invariant for links and the determination for certain links

2015-05-28
Tomomi Kawamura

Geometry of knots in real projective 3-space

2023-08-28
Rama K. Mishra , Visakh Narayanan
This paper discusses some geometric ideas associated with knots in real projective 3-space [Formula: see text]. These ideas are borrowed from classical knot theory. Since knots in [Formula: see text] … This paper discusses some geometric ideas associated with knots in real projective 3-space [Formula: see text]. These ideas are borrowed from classical knot theory. Since knots in [Formula: see text] are classified into three disjoint classes: affine, class-[Formula: see text] non-affine and class-[Formula: see text] knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in [Formula: see text]. We then study the notion of companionship of knots in [Formula: see text] and using it we provide geometric criteria for a knot to be affine. We also define a notion of “genus” for knots in [Formula: see text] and study some of its properties. We prove that this genus detects knottedness in [Formula: see text] and gives some criteria for a knot to be affine and of class-[Formula: see text]. We also prove a “non-cancellation” theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-[Formula: see text] non-affine knots with genus [Formula: see text]. Thus we highlight that, [Formula: see text] admits a knot theory with a truly different flavor than that of [Formula: see text] or [Formula: see text].
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