SU\G(𝔸)/K·U
Sign Up for free Log In

Author Description

Login to generate an author description

Ask a Question About This Mathematician

Complementary Regions of Knot and Link Diagrams

2011-10-01
Colin Adams , Reiko Shinjo , Kokoro Tanaka
View PDF Chat

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory

2006-05-18
Kokoro Tanaka
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain … Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the <italic>Khovanov-Jacobsson number</italic>, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan’s theory, and prove that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T squared"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">T^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-knot has trivial Khovanov-Jacobsson number.

Homology for quandles with partial group operations

2017-02-06
John Carter , Atsushi Ishii , Masahico Saito +1 more
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to … A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group and quandle homology theories. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle (MCQ, defined rigorously within). In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. A homology theory defined here for MCQs take into consideration both group and quandle operations, as well as their compatibility. The first homology group is characterized, and the notion of extensions by $2$-cocycles is provided. Degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes and group inverses. Cocycle invariants are also defined for handlebody-links.
View PDF Chat

Charts, signatures, and stabilizations of Lefschetz fibrations

2015-12-29
Hisaaki Endo , Isao Hasegawa , Seiichi Kamada +1 more
We employ a certain labeled finite graph, called a chart, in a closed oriented surface to describe the monodromy of a(n achiral) Lefschetz fibration over the surface.Applying charts and their … We employ a certain labeled finite graph, called a chart, in a closed oriented surface to describe the monodromy of a(n achiral) Lefschetz fibration over the surface.Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface.We then prove two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.
View PDF Chat

The braid index of surface-knots and quandle colorings

2005-04-01
Kokoro Tanaka
The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there … The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there exists a surface-knot with braid index $k$ for any positive integer $k$. To prove it, we use colorings of surface-knots by quandles and give lower bounds of the braid index of surface-knots.
View PDF Chat

Inequivalent surface-knots with the same knot quandle

2007-05-28
Kokoro Tanaka
View PDF Chat

ON SURFACE-LINKS REPRESENTED BY DIAGRAMS WITH TWO OR THREE TRIPLE POINTS

2005-12-01
Kokoro Tanaka
We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can … We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can take.

Independence of Roseman moves including triple points

2016-09-12
Kengo Kawamura , Kanako Oshiro , Kokoro Tanaka
The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space.In this paper, we focus on Roseman moves involving … The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space.In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence.For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the number of triple points of the two diagrams are the same).Moreover, we find a pair of diagrams of an S 2 -knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.
View PDF Chat

Regular-equivalence of 2-knot diagrams and sphere eversions

2016-05-05
Masamichi Takase , Kokoro Tanaka
For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ … For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in $4$-space.
View PDF Chat

Crossing changes for pseudo-ribbon surface-knots

2004-12-01
Kokoro Tanaka

The shadow nature of positive and twisted quandle cocycle invariants of knots

2014-01-01
Seiichi Kamada , Victoria Lebed , Kokoro Tanaka
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We … Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.
View PDF Chat

Complementary Regions of Knot and Link Diagrams

2008-01-01
Colin Adams , Reiko Shinjo , Kokoro Tanaka
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of … An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. This paper is an investigation into which sequences, either finite or infinite, are universal. We also consider how to minimize the number of odd-sided faces for projections of knots and links with n components.

On rack colorings for surface-knot diagrams without branch points

2015-05-29
Kanako Oshiro , Kokoro Tanaka
View PDF Chat

The shadow nature of positive and twisted quandle invariants of knots

2015-08-14
Seiichi Kamada , Victoria Lebed , Kokoro Tanaka
Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret … Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

The bridge number of surface links and kei colorings

2022-04-21
Kouki Sato , Kokoro Tanaka
Meier and Zupan introduced bridge trisections of surface links in S 4 $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface … Meier and Zupan introduced bridge trisections of surface links in S 4 $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number n $n$ for any integer n ⩾ 4 $n \geqslant 4$ . To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.
View PDF Chat

Charts, signatures, and stabilizations of Lefschetz fibrations

2014-01-01
Hisaaki Endo , Isao Hasegawa , Seiichi Kamada +1 more
We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and … We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.

A note on the local relations in geometric formalism of Khovanov homology theory

2015-11-23
Kokoro Tanaka
In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point … In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point out that the invariance and the functoriality can be proved without using the torus relation.

Quandle colorings vs. biquandle colorings

2019-01-01
Katsumi Ishikawa , Kokoro Tanaka
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those … Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for classical/surface links so far. In this paper, we give an explicit one-to-one correspondence between biquandle colorings and quandle colorings for classical/surface links. We also show that biquandle homotopy invariants and quandle homotopy invariants are equivalent. As a byproduct, we can interpret biquandle cocycle invariants in terms of shadow quandle cocycle invariants.
View PDF Chat

Shifting chain maps in quandle homology and cocycle invariants

2022-03-30
Yu Hashimoto , Kokoro Tanaka
Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a <italic>shifting chain map</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a <italic>shifting chain map</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on each quandle chain complex that lowers the dimensions by one. By using its pull-back <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯"> <mml:semantics> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^\sharp</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each <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>-cocycle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives us 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>-cocycle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented classical links in 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>-space, we explore relation between their quandle <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their shadow <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented surface links in the <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>-space, we explore how powerful their quandle <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.
View PDF Chat

Inequivalent surface-knots with the same knot quandle

2005-01-01
Kokoro Tanaka
We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is … We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical knot theory, and prove the following: There exist arbitrarily many inequivalent surface-knots of genus $g$ with the same knot quandle, and there exist two inequivalent surface-knots of genus $g$ with the same knot quandle and with the same fundamental class.

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

2005-01-01
Kokoro Tanaka
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain … Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invariant of a surface-knot which is a generalization of the Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any $T^2$-knot has the trivial Khovanov-Jacobsson number.

On rack colorings for surface-knot diagrams without branch points

2014-01-01
Kanako Oshiro , Kokoro Tanaka
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then … Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rack colorings are invariants of $S^2$-knots. We also prove that rack colorings for $S^2$-knots can be interpreted in terms of quandles, and discuss a relationship with regular-equivalences of surface-knot diagrams.
View PDF Chat

The bridge number of surface links and kei colorings

2020-01-01
Kouki Sato , Kokoro Tanaka
Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called … Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number $n$ for any integer $n \geq 4$. To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.
View PDF Chat

Shifting chain maps in quandle homology and cocycle invariants

2020-01-01
Yu Hashimoto , Kokoro Tanaka
Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\sigma$ on each quandle … Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\sigma$ on each quandle chain complex that lowers the dimensions by one. By using its pull-back $\sigma^\#$, each $2$-cocycle $\phi$ gives us the $3$-cocycle $\sigma^\# \phi$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $\phi$ and their shadow $3$-cocycle invariants associated with $\sigma^\# \phi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $\sigma^\# \phi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.
View PDF Chat

$2$-knots with the same knot group but different knot quandles

2023-01-01
Kokoro Tanaka , Yuta Taniguchi
We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, … We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot quandles.
View PDF Chat

Any link has a diagram with only triangles and quadrilaterals

2023-01-01
Reiko Shinjo , Kokoro Tanaka
A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has … A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has a diagram with only triangles and quadrilaterals. This extends previous results shown by the authors and C. Adams.
View PDF Chat

The second quandle homology group of the knot $n$-quandle

2023-01-01
Kokoro Tanaka , Yuta Taniguchi
The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed … The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed that it characterizes the unknot. In this paper, we compute the second quandle homology group of the knot $n$-quandle completely, where the knot $n$-quandle is a certain quotient of the knot quandle for each integer $n$ greater than one. Although the knot $n$-quandle is weaker than the knot quandle, the second quandle homology group of the former is found to have more information than that of the latter. As one of the consequences, it follows that the second quandle homology group of the knot $3$-quandle characterizes the unknot, the trefoil and the cinquefoil.
View PDF Chat

Quandle colorings vs. biquandle colorings

2024-01-23
Katsumi Ishikawa , Kokoro Tanaka
View PDF Chat

Any link has a diagram with only triangles and quadrilaterals

2025-03-05
Reiko Shinjo , Kokoro Tanaka

Any link has a diagram with only triangles and quadrilaterals

2025-03-05
Reiko Shinjo , Kokoro Tanaka

Quandle colorings vs. biquandle colorings

2024-01-23
Katsumi Ishikawa , Kokoro Tanaka
View PDF Chat

$2$-knots with the same knot group but different knot quandles

2023-01-01
Kokoro Tanaka , Yuta Taniguchi
We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, … We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot quandles.
View PDF Chat

Any link has a diagram with only triangles and quadrilaterals

2023-01-01
Reiko Shinjo , Kokoro Tanaka
A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has … A link diagram can be considered as a $4$-valent graph embedded in the $2$-sphere and divides the sphere into complementary regions. In this paper, we show that any link has a diagram with only triangles and quadrilaterals. This extends previous results shown by the authors and C. Adams.
View PDF Chat

The second quandle homology group of the knot $n$-quandle

2023-01-01
Kokoro Tanaka , Yuta Taniguchi
The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed … The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed that it characterizes the unknot. In this paper, we compute the second quandle homology group of the knot $n$-quandle completely, where the knot $n$-quandle is a certain quotient of the knot quandle for each integer $n$ greater than one. Although the knot $n$-quandle is weaker than the knot quandle, the second quandle homology group of the former is found to have more information than that of the latter. As one of the consequences, it follows that the second quandle homology group of the knot $3$-quandle characterizes the unknot, the trefoil and the cinquefoil.
View PDF Chat

The bridge number of surface links and kei colorings

2022-04-21
Kouki Sato , Kokoro Tanaka
Meier and Zupan introduced bridge trisections of surface links in S 4 $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface … Meier and Zupan introduced bridge trisections of surface links in S 4 $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number n $n$ for any integer n ⩾ 4 $n \geqslant 4$ . To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.
View PDF Chat

Shifting chain maps in quandle homology and cocycle invariants

2022-03-30
Yu Hashimoto , Kokoro Tanaka
Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a <italic>shifting chain map</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a <italic>shifting chain map</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on each quandle chain complex that lowers the dimensions by one. By using its pull-back <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯"> <mml:semantics> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\sigma ^\sharp</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each <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>-cocycle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> gives us 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>-cocycle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented classical links in 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>-space, we explore relation between their quandle <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and their shadow <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For oriented surface links in the <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>-space, we explore how powerful their quandle <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>-cocycle invariants associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript normal ♯ Baseline phi"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ</mml:mi> <mml:mi mathvariant="normal">♯</mml:mi> </mml:msup> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma ^\sharp \phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.
View PDF Chat

The bridge number of surface links and kei colorings

2020-01-01
Kouki Sato , Kokoro Tanaka
Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called … Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links called the bridge number. We prove that there exist infinitely many surface knots with bridge number $n$ for any integer $n \geq 4$. To prove it, we use colorings of surface links by keis and give lower bounds for the bridge number of surface links.
View PDF Chat

Shifting chain maps in quandle homology and cocycle invariants

2020-01-01
Yu Hashimoto , Kokoro Tanaka
Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\sigma$ on each quandle … Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\sigma$ on each quandle chain complex that lowers the dimensions by one. By using its pull-back $\sigma^\#$, each $2$-cocycle $\phi$ gives us the $3$-cocycle $\sigma^\# \phi$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $\phi$ and their shadow $3$-cocycle invariants associated with $\sigma^\# \phi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $\sigma^\# \phi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.
View PDF Chat

Quandle colorings vs. biquandle colorings

2019-01-01
Katsumi Ishikawa , Kokoro Tanaka
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those … Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for classical/surface links so far. In this paper, we give an explicit one-to-one correspondence between biquandle colorings and quandle colorings for classical/surface links. We also show that biquandle homotopy invariants and quandle homotopy invariants are equivalent. As a byproduct, we can interpret biquandle cocycle invariants in terms of shadow quandle cocycle invariants.
View PDF Chat

Homology for quandles with partial group operations

2017-02-06
John Carter , Atsushi Ishii , Masahico Saito +1 more
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to … A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group and quandle homology theories. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle (MCQ, defined rigorously within). In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. A homology theory defined here for MCQs take into consideration both group and quandle operations, as well as their compatibility. The first homology group is characterized, and the notion of extensions by $2$-cocycles is provided. Degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes and group inverses. Cocycle invariants are also defined for handlebody-links.
View PDF Chat

Independence of Roseman moves including triple points

2016-09-12
Kengo Kawamura , Kanako Oshiro , Kokoro Tanaka
The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space.In this paper, we focus on Roseman moves involving … The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space.In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence.For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the number of triple points of the two diagrams are the same).Moreover, we find a pair of diagrams of an S 2 -knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.
View PDF Chat

Regular-equivalence of 2-knot diagrams and sphere eversions

2016-05-05
Masamichi Takase , Kokoro Tanaka
For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ … For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in $4$-space.
View PDF Chat

Charts, signatures, and stabilizations of Lefschetz fibrations

2015-12-29
Hisaaki Endo , Isao Hasegawa , Seiichi Kamada +1 more
We employ a certain labeled finite graph, called a chart, in a closed oriented surface to describe the monodromy of a(n achiral) Lefschetz fibration over the surface.Applying charts and their … We employ a certain labeled finite graph, called a chart, in a closed oriented surface to describe the monodromy of a(n achiral) Lefschetz fibration over the surface.Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface.We then prove two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.
View PDF Chat

A note on the local relations in geometric formalism of Khovanov homology theory

2015-11-23
Kokoro Tanaka
In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point … In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point out that the invariance and the functoriality can be proved without using the torus relation.

The shadow nature of positive and twisted quandle invariants of knots

2015-08-14
Seiichi Kamada , Victoria Lebed , Kokoro Tanaka
Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret … Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

On rack colorings for surface-knot diagrams without branch points

2015-05-29
Kanako Oshiro , Kokoro Tanaka
View PDF Chat

The shadow nature of positive and twisted quandle cocycle invariants of knots

2014-01-01
Seiichi Kamada , Victoria Lebed , Kokoro Tanaka
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We … Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.
View PDF Chat

Charts, signatures, and stabilizations of Lefschetz fibrations

2014-01-01
Hisaaki Endo , Isao Hasegawa , Seiichi Kamada +1 more
We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and … We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.

On rack colorings for surface-knot diagrams without branch points

2014-01-01
Kanako Oshiro , Kokoro Tanaka
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then … Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rack colorings are invariants of $S^2$-knots. We also prove that rack colorings for $S^2$-knots can be interpreted in terms of quandles, and discuss a relationship with regular-equivalences of surface-knot diagrams.
View PDF Chat

Complementary Regions of Knot and Link Diagrams

2011-10-01
Colin Adams , Reiko Shinjo , Kokoro Tanaka
View PDF Chat

Complementary Regions of Knot and Link Diagrams

2008-01-01
Colin Adams , Reiko Shinjo , Kokoro Tanaka
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of … An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. This paper is an investigation into which sequences, either finite or infinite, are universal. We also consider how to minimize the number of odd-sided faces for projections of knots and links with n components.

Inequivalent surface-knots with the same knot quandle

2007-05-28
Kokoro Tanaka
View PDF Chat

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory

2006-05-18
Kokoro Tanaka
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain … Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov’s theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the <italic>Khovanov-Jacobsson number</italic>, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan’s theory, and prove that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T squared"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">T^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-knot has trivial Khovanov-Jacobsson number.

ON SURFACE-LINKS REPRESENTED BY DIAGRAMS WITH TWO OR THREE TRIPLE POINTS

2005-12-01
Kokoro Tanaka
We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can … We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can take.

The braid index of surface-knots and quandle colorings

2005-04-01
Kokoro Tanaka
The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there … The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there exists a surface-knot with braid index $k$ for any positive integer $k$. To prove it, we use colorings of surface-knots by quandles and give lower bounds of the braid index of surface-knots.
View PDF Chat

Inequivalent surface-knots with the same knot quandle

2005-01-01
Kokoro Tanaka
We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is … We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical knot theory, and prove the following: There exist arbitrarily many inequivalent surface-knots of genus $g$ with the same knot quandle, and there exist two inequivalent surface-knots of genus $g$ with the same knot quandle and with the same fundamental class.

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

2005-01-01
Kokoro Tanaka
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain … Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invariant of a surface-knot which is a generalization of the Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any $T^2$-knot has the trivial Khovanov-Jacobsson number.

Crossing changes for pseudo-ribbon surface-knots

2004-12-01
Kokoro Tanaka
Coauthor Papers Together
Reiko Shinjo 4
Seiichi Kamada 4
Kanako Oshiro 3
Hisaaki Endo 2
Katsumi Ishikawa 2
Colin Adams 2
Victoria Lebed 2
Yuta Taniguchi 2
Kouki Sato 2
Isao Hasegawa 2
Yu Hashimoto 1
Masamichi Takase 1
Kengo Kawamura 1
Yu Hashimoto 1
Masahico Saito 1
Atsushi Ishii 1
John Carter 1