Michael Willis

Search for another author

Author Description

Ask a Question About This Mathematician

All published works (29)

A Rasmussen invariant for links in ℝℙ³

2025-05-09

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> … 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.

Jones-Wenzl projectors and the Khovanov homotopy of the infinite twist

2024-02-15

Matthew Stoffregen , Michael Willis

We construct and study a lift of Jones-Wenzl projectors to the setting of Khovanov spectra, and provide a realization of such lifted projectors via a Cooper-Krushkal-like sequence of maps. We … We construct and study a lift of Jones-Wenzl projectors to the setting of Khovanov spectra, and provide a realization of such lifted projectors via a Cooper-Krushkal-like sequence of maps. We also give a polynomial action on the 3-strand spectral projector allowing a complete computation of the 3-colored Khovanov spectrum of the unknot, proving a conjecture of Lobb-Orson-Sch\"{u}tz. As a byproduct, we disprove a conjecture of Lawson-Lipshitz-Sarkar on the topological Hochschild homology of tangle spectra.

Chat View PDF

Khovanov homology and exotic $4$-manifolds

2024-02-15

Qiuyu Ren , Michael Willis

We show that the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna module distinguishes the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free … We show that the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna module distinguishes the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact $4$-manifolds. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen $s$-invariant, which are of independent interests. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna $s$-invariant, and a construction of induced maps on Khovanov homology for cobordisms in $k\mathbb{CP}^2$.

Chat View PDF

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.

Chat View PDF

A Rasmussen invariant for links in $\mathbb{RP}^3$

2023-01-01

Ciprian Manolescu , Michael Willis

Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in … Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in this setting. We show that the s-invariant gives constraints on the genera of link cobordisms in the cylinder $I \times \mathbb{RP}^3$. As an application, we give examples of freely 2-periodic knots in $S^3$ that are concordant but not standardly equivariantly concordant.

Chat View PDF

On the rank of knot homology theories and concordance

2023-01-01

Nathan M. Dunfield , Sherry Gong , Thomas Hockenhull +2 more

For a ribbon knot, it is a folk conjecture that the rank of its knot Floer homology must be 1 modulo 8, and another folk conjecture says the same about … For a ribbon knot, it is a folk conjecture that the rank of its knot Floer homology must be 1 modulo 8, and another folk conjecture says the same about reduced Khovanov homology. We give the first counter-examples to both of these folk conjectures, but at the same time present compelling evidence for new conjectures that either of these homologies must have rank congruent to 1 modulo 4 for any ribbon knot. We prove that each revised conjecture is equivalent to showing that taking the rank of the homology modulo 4 gives a homomorphism of the knot concordance group. We check the revised conjectures for 2.4 million ribbon knots, and also prove they hold for ribbon knots with fusion number 1.

Chat View PDF

Towards an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">sl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> action on the annular Khovanov spectrum

2022-07-25

Rostislav Akhmechet , Vyacheslav Krushkal , Michael Willis

Chat View PDF

Stable homotopy refinement of quantum annular homology

2021-04-01

Rostislav Akhmechet , Vyacheslav Krushkal , Michael Willis

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link … We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

Chat View PDF

Strands algebras and Ozsváth and Szabó’s Kauffman-states functor

2020-12-29

Andrew Manion , Marco Marengon , Michael Willis

We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for … We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'ath-Szab\'o's gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.

Chat View PDF

Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum

2020-11-23

Rostislav Akhmechet , Vyacheslav Krushkal , Michael Willis

Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider … Given a link in the thickened annulus, its annular Khovanov homology carries an action of the Lie algebra $\mathfrak{sl}_2$, which is natural with respect to annular link cobordisms. We consider the problem of lifting this action to the stable homotopy refinement of the annular homology. As part of this program, the actions of the standard generators of $\mathfrak{sl}_2$ are lifted to maps of spectra. In particular, it follows that the $\mathfrak{sl}_2$ action on homology commutes with the action of the Steenrod algebra. The main new technical ingredients developed in this paper, which may be of independent interest, concern certain types of cancellations in the cube of resolutions and the resulting more intricate structure of the moduli spaces in the framed flow category.

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.

Khovanov–Rozansky homology for infinite multicolored braids

2020-06-08

Michael Willis

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. … We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids.

Chat View PDF

GENERATORS, RELATIONS, AND HOMOLOGY FOR OZSVÁTH–SZABÓ’S KAUFFMAN-STATES ALGEBRAS

2020-04-17

Andrew Manion , Marco Marengon , Michael Willis

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras … We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

Chat View PDF

Towards an $\mathfrak{sl}_2$ action on the annular Khovanov spectrum

2020-01-01

Rostislav Akhmechet , Vyacheslav Krushkal , Michael Willis

Chat View PDF

Colored Khovanov–Rozansky homology for infinite braids

2019-10-20

Michael Abel , Michael Willis

We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors … We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp.

Chat View PDF

Generators, relations, and homology for Ozsv\'ath-Szab\'o's Kauffman-states algebras

2019-03-13

Andrew Manion , Marco Marengon , Michael Willis

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsvath and Szabo for the computation of knot Floer homology. We also compute the homology of these algebras … We give a generators-and-relations description of differential graded algebras recently introduced by Ozsvath and Szabo for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

Chat View PDF

Strands algebras and Ozsváth-Szabó's Kauffman-states functor

2019-03-01

Andrew Manion , Marco Marengon , Michael Willis

A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds

2019-01-01

Ciprian Manolescu , Marco Marengon , Sucharit Sarkar +1 more

We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links … We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 \times S^2$, 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 four-manifolds: $B^2 \times S^2$, $S^1 \times B^3$, $\mathbb{CP}^2$, 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 $B^4$ by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard.

Chat View PDF

Generators, relations, and homology for Ozsváth-Szabó's Kauffman-states algebras

2019-01-01

A. Manion , Marco Marengon , Michael Willis

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras … We give a generators-and-relations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

Chat View PDF

Khovanov homology for links in $\#^r(S^2\times S^1)$

2018-12-17

Michael Willis

We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded … We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ 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 $\mathcal{S}(M^r)$ of Hoste and Przytycki when $L$ is null-homologous in $M^r$. The construction also allows for a clear path towards 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 $S^3$ 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.

The Khovanov homology of infinite braids

2018-07-09

Gabriel Islambouli , Michael Willis

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones–Wenzl projector. This result extends Lev Rozansky’s categorification of the Jones–Wenzl projectors using the limiting … We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones–Wenzl projector. This result extends Lev Rozansky’s categorification of the Jones–Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz–Sarkar–Khovanov stable homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.

Chat View PDF

A colored Khovanov spectrum and its tail for B–adequate links

2018-04-03

Michael Willis

We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ colored links and quantum spin networks and derive some of its basic properties. In the case of $n$-colored B-adequate links, we show … We define a Khovanov homotopy type for $sl_2(\mathbb{C})$ colored links and quantum spin networks and derive some of its basic properties. In the case of $n$-colored B-adequate links, we show a stabilization of the homotopy types as the coloring $n\rightarrow\infty$, generalizing the tail behavior of the colored Jones polynomial. Finally, we also provide an alternative, simpler stabilization in the case of the colored unknot.

Chat View PDF

Khovanov homology for links in $\#^r(S^2\times S^1)$

2018-01-01

Michael Willis

Chat View PDF

The Khovanov homology of infinite braids

2016-10-14

Gabriel Islambouli , Michael Willis

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting … We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.

Chat View PDF

Stabilization of the Khovanov Homotopy Type of Torus Links

2016-06-27

Michael Willis

The structure of the Khovanov homology of |$(n,m)$| torus links has been studied extensively. In particular, Marko Stošić proved that the homology groups stabilize as |$m\rightarrow\infty$|⁠. We show that the … The structure of the Khovanov homology of |$(n,m)$| torus links has been studied extensively. In particular, Marko Stošić proved that the homology groups stabilize as |$m\rightarrow\infty$|⁠. We show that the Khovanov homotopy types of |$(n,m)$| torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as |$m\rightarrow\infty$|⁠. We provide an explicit bound on values of |$m$| beyond which the stabilization begins. As an application, we give new examples of torus links with non-trivial |$Sq^2$| action.

Chat View PDF

A Colored Khovanov Homotopy Type For Links, And Its Tail For The Unknot

2016-02-11

Michael Willis

In a previous paper, the author showed that the Khovanov homotopy types of the torus links $T(n,m)$ stabilize as $m\rightarrow\infty$. In this sequel, we use similar techniques to extend this … In a previous paper, the author showed that the Khovanov homotopy types of the torus links $T(n,m)$ stabilize as $m\rightarrow\infty$. In this sequel, we use similar techniques to extend this result in two directions. First, we construct a stable colored version of the Khovanov homotopy type whose reduced cohomology recovers the colored Khovanov homology of the link. Second, in the case of the $T(n,\infty)$ as above, we show a further stabilization as $n\rightarrow\infty$.

Chat View PDF

The Khovanov homology of infinite braids

2016-01-01

Gabriel Islambouli , Michael Willis

Chat View PDF

Stabilization of the Khovanov Homotopy Type of Torus Links

2015-11-09

Michael Willis

The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as $m\rightarrow\infty$. We show that the … The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as $m\rightarrow\infty$. We show that the Khovanov homotopy types of $(n,m)$ torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as $m\rightarrow\infty$. We provide an explicit bound on values of $m$ beyond which the stabilization begins. As an application, we give new examples of torus links with non-trivial $Sq^2$ action.

Chat View PDF

Stabilization of the Khovanov Homotopy Type of Torus Links

2015-01-01

Michael Willis

Chat View PDF

Common Coauthors

Coauthor	Papers Together
Marco Marengon	8
Rostislav Akhmechet	4
Andrew Manion	4
Vyacheslav Krushkal	4
Gabriel Islambouli	3
Sucharit Sarkar	2
Ciprian Manolescu	2
Ciprian Manolescu	2
A. Manion	1
Nathan M. Dunfield	1
Thomas Hockenhull	1
Sherry Gong	1
Qiuyu Ren	1
Michael Abel	1
Matthew Stoffregen	1

Commonly Cited References

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.

Chat View PDF

Khovanov’s homology for tangles and cobordisms

2005-08-08

Dror Bar-Natan

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots.By staying within a world of … We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots.By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological.And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

Chat View PDF

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.

Chat View PDF

Khovanov homology and the slice genus

2010-09-06

Jacob Rasmussen

Chat View PDF

A refinement of Rasmussen’s S-invariant

2014-03-26

Robert Lipshitz , Sucharit Sarkar

In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations … In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq2 our refinement is strictly stronger than s.

Chat View PDF

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.

Chat View PDF

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.

Chat View PDF

Categorification of the Jones–Wenzl projectors

2012-03-03

Benjamin Cooper , Vyacheslav Krushkal

The Jones–Wenzl projectors p_n play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes P_n , … The Jones–Wenzl projectors p_n play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes P_n , whose graded Euler characteristic is the “classical” projector p_n in the Temperley–Lieb algebra. We show that the {P}_n are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov’s categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of \mathrm{U}_q\mathfrak{sl}(2) and a categorification of quantum spin networks. We introduce 6j-symbols in this context.

Chat View PDF

An infinite torus braid yields a categorified Jones–Wenzl projector

2014-01-01

Lev Rozansky

A sequence of Temperley–Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones–Wenzl projector. We show that a sequence of categorification complexes of these braids … A sequence of Temperley–Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones–Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may

Chat View PDF

Holomorphic disks and knot invariants

2003-11-15

Peter Ozsváth , Zoltán Szabó

Chat View PDF

A functor-valued invariant of tangles

2002-09-06

Mikhail Khovanov

We construct a family of rings.To a plane diagram of a tangle we associate a complex of bimodules over these rings.Chain homotopy equivalence class of this complex is an invariant … We construct a family of rings.To a plane diagram of a tangle we associate a complex of bimodules over these rings.Chain homotopy equivalence class of this complex is an invariant of the tangle.On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle.When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.

Chat View PDF

The Khovanov homology of infinite braids

2018-07-09

Gabriel Islambouli , Michael Willis

Chat View PDF

The cube and the Burnside category

2017-01-01

Tyler Lawson , Robert Lipshitz , Sucharit Sarkar

In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between … In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.

Chat View PDF

A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S2 x S1

2010-01-01

Lev Rozansky

The WRT invariant of a link L in S2xS1 at sufficiently high values of the level r can be expresses as an evaluation of a special polynomial invariant of L … The WRT invariant of a link L in S2xS1 at sufficiently high values of the level r can be expresses as an evaluation of a special polynomial invariant of L at 2r-th root of unity. We categorify this polynomial invariant by associating to L a bigraded homology whose graded Euler characteristic is equal to this polynomial. If L is presented as a closure of a tangle in S2xS1, then the homology of L is defined as the Hochschild homology of the H_n-bimodule associated to the tangle by M. Khovanov. This homology can also be expressed as a stable limit of Khovanov homology of the circular closure of the tangle in S3 through the torus braid with high twist.

Chat View PDF

FAST KHOVANOV HOMOLOGY COMPUTATIONS

2007-03-01

Dror Bar-Natan

We introduce a local algorithm for Khovanov homology computations — that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, … We introduce a local algorithm for Khovanov homology computations — that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational efficiency. Thus our program can rapidly compute certain Khovanov homology groups that otherwise would have taken centuries to evaluate.

Chat View PDF

Khovanov homotopy type, Burnside category and products

2020-09-23

Tyler Lawson , Robert Lipshitz , Sucharit Sarkar

In this paper, we give a new construction of a Khovanov stable homotopy type, or spectrum.We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra … In this paper, we give a new construction of a Khovanov stable homotopy type, or spectrum.We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed in [LS14a] and [HKK16] and, as a corollary, that those two constructions give equivalent spectra.We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a].Finally, combining these results with computations from [LS14c] and the refined s-invariant from [LS14b] we obtain new results about the slice genera of certain knots.

Chat View PDF

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

2020-07-09

Michael Willis

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

2017-06-15

Andrew Manion

Chat View PDF

Holomorphic disks and three-manifolds invariants: Properties and applications

2004-05-01

Peter Ozsváth , Zoltán Szabó

In [27], we introduced Floer homology theories HF -(Y, s), HF ∞ (Y, s), HF + (Y, t), HF (Y, s),and HF red (Y, s) associated to closed, oriented three-manifolds … In [27], we introduced Floer homology theories HF -(Y, s), HF ∞ (Y, s), HF + (Y, t), HF (Y, s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y ).In the present paper, we give calculations and study the properties of these invariants.The calculations suggest a conjectured relationship with Seiberg-Witten theory.The properties include a relationship between the Euler characteristics of HF ± and Turaev's torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences.We also include some applications of these techniques to three-manifold topology.

Chat View PDF

Temperley-Lieb recoupling theory and invariants of 3-manifolds

1994-01-01

Louis H. Kauffman , Sóstenes Lins

This is a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for … This is a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose.

An invariant of tangle cobordisms via subquotients of arc rings

2014-01-01

Yanfeng Chen , Mikhail Khovanov

We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum ${\rm sl}(2)$. We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum ${\rm sl}(2)$.

Chat View PDF

Chen–Khovanov Spectra for Tangles

2021-05-14

Tyler Lawson , Robert Lipshitz , Sucharit Sarkar

We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of Chen–Khovanov and Stroppel's platform algebras and tangle invariants, and discuss the topological Hochschild … We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of Chen–Khovanov and Stroppel's platform algebras and tangle invariants, and discuss the topological Hochschild homology of these refinements.

Chat View PDF

Homological thickness and stability of torus knots

2007-03-29

Marko Stošić

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce … In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in \cite{dgr}.

Chat View PDF

Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134)

1994-12-31

Louis H. Kauffman , Sóstenes Lins

This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model … This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Annular Khovanov homology and knotted Schur–Weyl representations

2017-11-28

J. Elisenda Grigsby , Anthony Licata , Stephan M. Wehrli

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$ , the exterior current algebra … Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$ , the exterior current algebra of $\mathfrak{sl}_{2}$ . When $\mathbb{L}$ is an $m$ -framed $n$ -cable of a knot $K\subset S^{3}$ , its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$ . One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

Chat View PDF

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

2018-10-31

Andrew Manion

We relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsv\'ath- Szab\'o's algebra $\mathcal{B}(n,\mathcal{S})$, and … We relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsv\'ath- Szab\'o's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $\mathcal{U}_q(\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsv\'ath-Szab\'o's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsv\'ath-Szab\'o's theory and Viro's quantum relative $\mathcal{A}^1$ of the Reshetikhin-Turaev functor based on $\mathcal{U}_q(\mathfrak{gl}(1|1))$.

Chat View PDF

A colored Khovanov spectrum and its tail for B–adequate links

2018-04-03

Michael Willis

Chat View PDF

On knot Floer homology in double branched covers

2013-03-12

Lawrence P. Roberts

We define a link surgery spectral sequence for each knot Floer homology group for a knot, K , in a three manifold, Y .When K arises as the double cover … We define a link surgery spectral sequence for each knot Floer homology group for a knot, K , in a three manifold, Y .When K arises as the double cover of an unknot in S 3 , and Y is the double cover of S 3 branched over a link, we relate the E 2 -page to a version of Khovanov homology for links in an annulus defined by Asaeda, Przytycki and Sikora.Finally we examine the specific cases when the branch locus is a braid, and when it is alternating. 57M27; 57R58B L Let †.L/ be the double cover of S 3 branched over L, and let z B be the pre-image of B in †.L/.Then z B is a null-homologous knot in †.L/ so we can try to compute its knot Floer homology groups (Ozsváth and Szabó [11])

Chat View PDF

An odd Khovanov homotopy type

2020-03-19

Sucharit Sarkar , Christopher Scaduto , Matthew Stoffregen

Chat View PDF

A Steenrod square on Khovanov homology

2014-03-24

Robert Lipshitz , Sucharit Sarkar

In a previous paper, we defined a space-level version X Kh ( L ) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this … In a previous paper, we defined a space-level version X Kh ( L ) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq 2 : Kh i , j ( L ) → Kh i + 2 , j ( L ) . We compute this operation for all links up to 11 crossings; this, in turn, determines the stable homotopy type of X Kh ( L ) for all such links.

Chat View PDF

ENHANCED TRIANGULATED CATEGORIES

1991-02-28

Alexey Bondal , Mikhail Kapranov

A solution is given to the problem of describing a triangulated category generated by a finite number of objects. It requires the notion of "enhancement" of a triangulated category, by … A solution is given to the problem of describing a triangulated category generated by a finite number of objects. It requires the notion of "enhancement" of a triangulated category, by means of the complexes RHom.

Quantum link homology via trace functor I

2019-01-03

Anna Beliakova , Krzysztof K. Putyra , Stephan M. Wehrli

Chat View PDF

On Khovanov’s categorification of the Jones polynomial

2002-05-21

Dror Bar-Natan

The working mathematician fears complicated words but loves pictures and diagrams.We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of the Jones polynomial".For the … The working mathematician fears complicated words but loves pictures and diagrams.We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of the Jones polynomial".For the same low cost we also provide some computations, including one that shows that Khovanov's invariant is strictly stronger than the Jones polynomial and including a table of the values of Khovanov's invariant for all prime knots with up to 11 crossings.

Chat View PDF

Categorified Young symmetrizers and stable homology of torus links

2018-06-01

Michael Abel , Matthew Hogancamp

We show that the triply graded Khovanov-Rozansky homology of the torus link T n;k stabilizes as k ! 1.We explicitly compute the stable homology, as a ring, which proves a … We show that the triply graded Khovanov-Rozansky homology of the torus link T n;k stabilizes as k ! 1.We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky, Oblomkov, Rasmussen and Shende.To accomplish this, we construct complexes P n of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that P n is a stable limit of Rouquier complexes.A certain derived endomorphism ring of P n computes the aforementioned stable homology of torus links.18G60, 57M27 1. Introduction 2943 2. A categorified Young symmetrizer 2955 3. Derived categories and triply graded homology 2971 4. Structure of the projector 2979

Chat View PDF

Man and machine thinking about the smooth 4-dimensional Poincaré conjecture

2010-06-22

Michael Freedman , Robert E. Gompf , Scott Morrison +1 more

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these … While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen’s s-invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots K for which nonzero s(K) would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those K, with the computations showing that s was 0, when a landmark posting of Akbulut [3] altered the terrain. His posting, appearing only six days after our initial posting, proved that the family of “Cappell–Shaneson” homotopy spheres that we had geared up to study were in fact all standard. The method we describe remains viable but will have to be applied to other examples. Akbulut’s work makes SPC4 seem more plausible, and in another section of this paper we explain that SPC4 is equivalent to an appropriate generalization of Property R (“in S3, only an unknot can yield S1 × S2 under surgery”). We hope that this observation, and the rich relations between Property R and ideas such as taut foliations, contact geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to look at SPC4.

Chat View PDF

Khovanov homology for links in $\#^r(S^2\times S^1)$

2018-12-17

Michael Willis

A categorification of quantum $\mathfrak{sl}_3$ projectors and the $\mathfrak{sl}_3$ Reshetikhin–Turaev invariant of tangles

2014-03-03

David E. V. Rose

We construct a categorification of the quantum \mathfrak{sl}_3 projectors, the \mathfrak{sl}_3 analog of the Jones–Wenzl projectors, as the stable limit of the complexes assigned to k -twist torus braids (as … We construct a categorification of the quantum \mathfrak{sl}_3 projectors, the \mathfrak{sl}_3 analog of the Jones–Wenzl projectors, as the stable limit of the complexes assigned to k -twist torus braids (as k \to \infty ) in a suitably shifted version of Morrison and Nieh’s geometric formulation of \mathfrak{sl}_3 link homology [14] We use these projectors to give a categorification of the \mathfrak{sl}_3 Reshetikhin–Turaev invariant of framed tangles.

Chat View PDF

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

2012-03-03

Igor Frenkel , Catharina Stroppel , Joshua Sussan

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted … We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan–Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones–Wenzl projector, \Theta -networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev–Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

Chat View PDF

Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

2015-04-01

Denis Auroux , J. Elisenda Grigsby , Stephan M. Wehrli

In 2002, Khovanov-Seidel constructed a faithful action of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m+1)</mml:annotation> … In 2002, Khovanov-Seidel constructed a faithful action of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–strand braid group, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper B Subscript m plus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">B</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {B}_{m+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, on the derived category of left modules over a quiver algebra, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript m"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the <italic>sutured Khovanov homology</italic> of the annular braid closure.

Chat View PDF

Formality of DG algebras (after Kaledin)

2009-12-07

Valery A. Lunts

Chat View PDF

A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles

2011-01-01

David E. V. Rose

We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k … We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754). We use these projectors to give a categorification of the sl_3 Reshetikhin-Turaev invariant of framed tangles.

Chat View PDF

Stabilization of the Khovanov Homotopy Type of Torus Links

2016-06-27

Michael Willis

Chat View PDF

Colored Khovanov–Rozansky homology for infinite braids

2019-10-20

Michael Abel , Michael Willis

Chat View PDF

A categorification of quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="normal">sl</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

2010-06-21

Aaron D. Lauda

Chat View PDF

A survey of skein modules of 3-manifolds

2014-06-10

Jim Hoste , Józef H. Przytycki

Categorical diagonalization of full twists

2018-01-01

Ben Elias , Matthew Hogancamp

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced … We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced earlier by the authors. We prove our conjecture in type $A$, and as a result we obtain a categorification of the Young idempotents.

Chat View PDF

A colored sl(N) homology for links in S<sup>3</sup>

2014-01-01

Hao Wu

Fix an integer $N\geq 2$. To each diagram of a link colored by $1,\dots,N$ we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of … Fix an integer $N\geq 2$. To each diagram of a link colored by $1,\dots,N$ we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of t

Chat View PDF

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.

Chat View PDF

Matrix factorizations and link homology

2008-01-01

Mikhail Khovanov , Lev Rozansky

For each positive integer $n$ the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum $sl(n)$. For each such $n$ … For each positive integer $n$ the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum $sl(n)$. For each such $n$ we build a doubly-graded homology theory of links with this

Chat View PDF

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.

Chat View PDF