arXiv subjects
Publications and source records attributed to Kokoro Tanaka.
Jabłonowski proved that the knot quandles of Suciu's $n$-knots, which share isomorphic knot groups, are mutually non-isomorphic, and Yasuda later gave a different proof. In this paper, we present yet another proof of this result by analyzing the conjugacy classes of certain automorphisms of the free group of rank two.
We study the structure of a series of groups $Γ_n^4$ defined by Kim and Manturov. We show that the groups are finite for all $n \ge 6$ and in fact they are 2-step nilpotent $2$-groups.
We consider a series of groups defined by Kim and Manturov. These groups have their background in triangulations of a surface and configurations of points, lines or circles on the surface. They are expected to have relationships to many geometric objects. In this paper, we give a minimal generating set of the group and determine the abelianization. We also introduce some related groups which might be helpful to understand the structure of the original groups.
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.
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.
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.
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 $σ$ on each quandle chain complex that lowers the dimensions by one. By using its pull-back $σ^\#$, each $2$-cocycle $ϕ$ gives us the $3$-cocycle $σ^\# ϕ$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $ϕ$ and their shadow $3$-cocycle invariants associated with $σ^\# ϕ$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $σ^\# ϕ$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.
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.
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.
Roseman moves are seven types of local modification 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 numbers of triple points of the two diagrams are the same). Moreover, we can find a pair of two diagrams of an $S^2$-knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.
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.
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.
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.
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.
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.
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.
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 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.