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Eri Matsudo

Publications and source records attributed to Eri Matsudo.

11 recordsLinked to original sources
arXiv2026

The minimum number of Dehn $\mathbb Z$-colors of any nonsplittable $\mathbb Z$-colorable link is three

Our previous papers [8, 9] are the first and second to discuss minimum numbers of ``region'' colors, while minimum numbers of arc colors such as Fox colors are well-studied. As the third installment, in this paper, we investigate the minimum number of Dehn $\mathbb{Z}$-colors. In particular, we show that the minimum number of Dehn $\mathbb{Z}$-colors of a nonsplittable $\mathbb{Z}$-colorable link is three.

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arXiv2025

Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs

In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number $p$ and any Dehn $p$-colorable knot $K$, the minimum number of colors for $K$ is at least $\lfloor \log_2 p \rfloor +2$. Moreover, we will define the $\R$-palette graph for a set of colors. The $\R$-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn $p$-colored diagram. In Appendix, we also prove that for Dehn $5$-colorable knot, the minimum number of colors is $4$.

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arXiv2024

Two-tone colorings and surjective dihedral representations for links

It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In this paper, we introduce a two-tone coloring on a link diagram, and give a condition for links so that the link groups admit surjective representations to the dihedral groups. In particular, it is shown that the link group of any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.

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arXiv2022

Constructing Goeritz matrix from Dehn coloring matrix

Associated to a knot diagram, Goeritz introduced an integral matrix, which is now called a Goeritz matrix. It was shown by Traldi that the solution space of the equations with Goeritz matrix (precisely, unreduced Goeritz matrix called in his paper) as a coefficient matrix is isomorphic to the linear space consisting of the Dehn colorings for a knot. In this paper, we give a construction of a Goeritz matrix from a Dehn coloring matrix, from which Dehn colorings are induced. Moreover, if the knot diagram is prime, we give a purely algebraic construction of a Goeritz matrix from a Dehn coloring matrix.

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arXiv2022

Coloring links by the symmetric group of degree three

We consider the number of colors for the colorings of links by the symmetric group $S_3$ of degree $3$. For knots, such a coloring corresponds to a Fox 3-coloring, and thus the number of colors must be 1 or 3. However, for links, there are colorings by $S_3$ with 4 or 5 colors. In this paper, we show that if a 2-bridge link admits a coloring by $S_3$ with 5 colors, then the link also admits such a coloring with only 4 colors.

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arXiv2019

Minimal coloring numbers on minimal diagrams of torus links

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorings on link diagrams.

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arXiv2017

Minimal coloring number on minimal diagrams for $\mathbb{Z}$-colorable links

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on minimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positive integer $N$, there exists a minimal diagram of a $\mathbb{Z}$-colorable link such that any $\mathbb{Z}$-coloring on the diagram has at least $N$ colors. On the other hand, it is shown that certain $\mathbb{Z}$-colorable torus links have minimal diagrams admitting $\mathbb{Z}$-colorings with only four colors.

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arXiv2017

Minimal coloring number for $\mathbb{Z}$-colorable links II

The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable $\mathbb{Z}$-colorable links is four. As an example, we consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. We show that an even parallel of a link is $\mathbb{Z}$-colorable except for the case of 2 parallels with non-zero linking number. We then give a simple way to obtain a diagram which attains the minimal coloring number for such even parallels of links.

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arXiv2016

Minimal coloring number for Z-colorable links

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link. We give sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.

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