arXiv subjects

Hyungkee Yoo

Publications and source records attributed to Hyungkee Yoo.

13 recordsLinked to original sources

arXiv2026

Fox $p$-Colorings as Fixed Points of Braid Representations

Fox $p$-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid $β\in B_n$ we associate a representation $M : B_n \to \mathrm{GL}(n,\mathbb{F}_p)$ such that Fox $p$-colorings of the closed braid $\widehatβ$ correspond to fixed points of $M(β)$. As a consequence, $$ \mathrm{Col}_p(\widehatβ) = p^{\dim \ker(M(β)-I)}, $$ yielding an explicit formula for the number of Fox $p$-colorings of arbitrary knots and links. We apply the method to torus links and spiral links, obtaining explicit descriptions of their Fox 3-coloring spaces.

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arXiv2026

Four-page index and linear upper bounds for ribbonlength

We introduce the four-page index of a knot or link as a presentation invariant arising from embeddings in a four-page open book decomposition. Using spanning trees of the checkerboard graph of a reduced non-split diagram, we construct a Kauffman state consisting of a single state circle. The associated Eulerian tour of the underlying 4-valent plane graph determines a binding circle intersecting each edge exactly once, producing a four-page presentation with at most $2c(K)$ arcs. Hence $$ α_4(K) \le 2c(K), $$ with strict inequality in the non-alternating case. We further prove that ribbonlength is bounded above by the four-page index, and therefore obtain the linear bound $$ \mathrm{Rib}(K) \le 2c(K). $$ This improves the previously known general linear upper bound for ribbonlength and provides a diagrammatic method for estimating ribbonlength.

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arXiv2026

A diagrammatic approach to the three-page index

The three-page index $α_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that $α_3(L)$ admits a linear upper bound in terms of the crossing number, with equality realized by the Hopf link. In this paper, we investigate the equality case of this bound from a diagrammatic viewpoint. Starting from a reduced link diagram, we construct three-page presentations via binding circles arising as boundaries of suitable contractible subcomplexes of the induced cell decomposition of the $2$-sphere. This approach allows a refined control of the number of arcs in the resulting three-page presentation. As a consequence, we prove that for any non-split, nontrivial link $L$ other than the Hopf link, \[ α_3(L)\le 3c(L)-1, \] and hence characterize completely the links for which $α_3(L)=3c(L)$.

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arXiv2026

New Upper Bounds on the Ribbonlength of Alternating Links with Bipartite Dual Graphs

The ribbonlength of a link is a geometric invariant defined as the infimum of the ratio of the length to the width of a folded ribbon realization of the link. In this paper, we prove that if an alternating link admits an alternating diagram with a bipartite dual graph, then its ribbonlength satisfies $$ \mathrm{Rib}(L) \le \sqrt{3} \, c(L). $$ Using this result, we present improved upper bounds on the ribbonlength for several knots and links with small crossing numbers, and determines the exact ribbonlength of the Hopf link to be $2\sqrt{3}$.

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arXiv2025

Three-page indices of torus links

An arc presentation of a link is an embedding into the open book decomposition of $\mathbb{R}^3$ with a finite number of pages. An important rule of arc presentations is that different arcs must be placed on separate pages. In 1999, Dynnikov proposed a three-page presentation that bends this rule by restricting the total number of pages to three. Dynnikov showed that every link admits a three-page presentation. In this paper, we provide an alternative proof of this result. Also we define the three-page index $α_3(L)$ of a link $L$ that the minimum number of arcs needed to represent $L$ in a three-page presentation. We examine three-page presentations for torus links, leading to the determination of the exact three-page indices for several torus links.

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arXiv2024

Linear Upper Bounds on the Ribbonlength of Knots and Links

A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length required to tie the given knot type as a folded ribbon knot. The ribbonlength has been conjectured to grow linearly or sub-linearly with respect to a minimal crossing number. Several knot types provide evidence that this conjecture is true, but there is no proof for general cases. In this paper, we show that for any knot or link, the ribbonlength is bounded by a linear function of the crossing number. In more detail, $$ \text{Rib}(K) \leq 2.5 c(K)+1. $$ for a knot or link $K$. Our approach involves binary grid diagrams and bisected vertex leveling techniques.

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arXiv2024

Linking numbers of Montesinos links

The linking number of an oriented two-component link is an invariant indicating how intertwined the two components are. Tuler proved that the linking number of a two-component rational $\frac{p}{q}$-link is $$\sum^{\frac{|p|}{2}}_{k=1} (-1)^{\big\lfloor (2k-1) \frac{q}{p} \big\rfloor }.$$ In this paper, we provide a simple proof the above result, and introduce the numerical algorithm to find linking numbers of rational links. Using this result, we find linking numbers between any two components in a Montesinos link.

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arXiv2022

On the Laplacian spectrum of $k$-symmetric graphs

For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are integers.

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arXiv2022

Petal number of torus knots using superbridge indices

A petal projection of a knot $K$ is a projection of a knot which consists of a single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})=2s-1$$ when $1 < r < s$ and $r \equiv 1 \mod s-r$. Furthermore, we also find the upper bound of the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})\leq2s- 2\Big\lfloor \frac{s}{r} \Big\rfloor +1$$ when $s \equiv \pm 1 \mod r$.

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arXiv2022

Folded ribbonlength of 2-bridge knots

A ribbon is a two-dimensional object with one-dimensional properties which is related with geometry, robotics and molecular biology. A folded ribbon structure provides a complex structure through a series of folds. We focus on a folded ribbon with knotted core. The folded ribbonlength $Rib(K)$ of a knot $K$ is the infimum of the quotient of length by width among the ribbons representing a knot type of $K$. This quantity tells how efficiently the folded ribbon is realized. Kusner conjectured that folded ribbonlength is bounded by a linear function of the minimal crossing number $c(K)$. In this paper, we confirm that the folded ribbonlength of a 2-bridge knot $K$ is bounded above by $2c(K)+2$.

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arXiv2022

Ribbonlength of twisted torus knots

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent strands $s$ full twists. In this paper, we show that the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(\max \{ p, q, r \} +|s|r)$ where $p$ and $q$ are positive. Furthermore, if $r \leq p-q$, then the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(p+(|s|-1)r)$.

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arXiv2018

Bisected vertex leveling of plane graphs: braid index, arc index and delta diagrams

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $α(L)$ for any knot or non-split link $L$, which are $b(L) \leq \frac{1}{2} c(L) + 1$ and $α(L) \leq c(L) + 2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.

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arXiv2018

Lattice stick number of spatial graphs

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number $s_{L}(G)$ of spatial graphs $G$ with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number $c(G)$ $$ s_{L}(G) \leq 3c(G)+6e-4v-2s+3b+k, $$ where $G$ has $e$ edges, $v$ vertices, $s$ cut-components, $b$ bouquet cut-components, and $k$ knot components.

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