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Shin Satoh

Publications and source records attributed to Shin Satoh.

At least 19 recordsLinked to original sources
arXiv2026

A note on Fox colorings of virtual tangles

We study Fox colorings of tangle diagrams by $R=\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$, where $p\geq3$ is an odd integer. For an $R$-colored $m$-string tangle diagram, the colors at the $2m$ boundary points form a vector $v\in R^{2m}$. We show that for classical tangle diagrams, such vectors are completely characterized by the alternating sum condition $\Delta(v)=0$. We then investigate how this restriction changes in the virtual setting. For $R=\mathbb{Z}$, the realizability of $v$ is determined by a divisibility condition on $\Delta(v)$. For $R=\mathbb{Z}/p\mathbb{Z}$, every vector is realizable by a virtual tangle diagram.

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arXiv2026

The $V_1$- and $V_2$-polynomials of a long virtual knot

We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $\alpha_3$-invariant.

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arXiv2025

The intersection polynomials of a long virtual knot I: Definitions and properties

We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed surface, considering the order of over- and under-crossings. We study their fundamental properties including behavior under symmetries, crossing changes, and concatenation products. All are finite-type invariants of degree two under crossing changes, but not under virtualizations, and we examine their relation to the closure and the values at $t=1$ of their derivatives.

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arXiv2025

The intersection polynomials of a long virtual knot II: Two supporting genera and characterizations

We develop the study of the twelve intersection polynomials of long virtual knots, previously introduced in our preceding paper. We define two geometric invariants, the $1$- and $2$-supporting genera, using two distinct surface realizations. These genera yield a natural filtration of the set of long virtual knots, and we analyze the behavior of the intersection polynomials for long virtual knots with small supporting genera. Moreover, we investigate virtual $2$-string tangles, analyzing how their sums with long virtual knots affect the intersection polynomials through right closures. As an application, we provide complete realizability criteria for all twelve intersection polynomials.

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arXiv2024

Hurwitz equivalence in the universal dihedral quandle

We investigate the Hurwitz action of the $m$-braid group on the $m$-fold Cartesian product of the universal dihedral quandle. We introduce three computable invariants and prove that they give a complete classification of the orbits under this action. As a consequence, we describe an explicit complete system of orbit representatives. We further obtain analogous classifications for the corresponding Hurwitz actions of the pure $m$-braid group, the virtual $m$-braid group, and the virtual pure $m$-braid group.

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arXiv2024

On wen knots

We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror reflections. This allow us to characterise completely extended welded knots by the parity of their number of wens, that we can always reduce to 0 or 1.

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arXiv2024

Virtualized Delta, sharp, and pass moves for oriented virtual knots and links

We study virtualized Delta, sharp, and pass moves for oriented virtual links, and give necessary and sufficient conditions for two oriented virtual links to be related by the local moves. In particular, they are unknotting operations for oriented virtual knots. We provide lower bounds for the unknotting numbers and prove that they are best possible.

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arXiv2024

Virtualized Delta moves for virtual knots and links

We introduce a local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized $\Delta$-moves.

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arXiv2021

The intersection polynomials of a virtual knot I: Definitions and calculations

We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. The calculations of intersection polynomials are given up to crossing number four. We also study several properties of intersection polynomials.

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arXiv2020

Classification of $2$-component virtual links up to $\Xi$-moves

The $\Xi$-move is a local move generated by forbidden moves in virtual knot theory. This move was introduced by Taniguchi and the second author, who showed that it characterizes the odd writhe of virtual knots, which is a fundamental invariant defined by Kauffman. In this paper, we extend this result by classifying $2$-component virtual links up to $\Xi$-moves, using refinements of the odd writhe and linking numbers.

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arXiv2019

Writhe polynomials and shell moves for virtual knots and links

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented $2$-component virtual links up to shell moves by using several invariants of virtual links.

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arXiv2018

A note on coverings of virtual knots

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.

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arXiv2015

11-colored knot diagram with five colors

We prove that any $11$-colorable knot is presented by an $11$-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially $11$-colored diagrams of the knot. We also prove a similar result for any $11$-colorable ribbon $2$-knot.

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arXiv2004

Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers

We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an unknotting operation, such as ribbon 2-knots and twist-spun knots, have trivial Khovanov-Jacobsson number.

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arXiv2003

Ribbon concordance of surface-knots via quandle cocycle invariants

We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.

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arXiv2001

A Theorem of Sanderson on Link Bordisms in Dimension 4

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we geometrically represent every element of the bordism group uniquely by a certain standard form of a surface-link, a generalization of a Hopf link. The standard forms give rise to an inverse of Sanderson's geometrically defined invariant.

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