arXiv subjects

Hyoungjun Kim

Publications and source records attributed to Hyoungjun Kim.

17 recordsLinked to original sources

arXiv2026

Four triangle-free intrinsically knotted graphs with 22 edges

An intrinsically knotted graph is one for which every spatial embedding contains a nontrivially knotted cycle. Classifying such graphs is a central problem in spatial graph theory. It is known that every intrinsically knotted graph has at least 21 edges, and the case of 21 edges has been completely resolved. For 22 edges, however, the classification remains incomplete. In particular, exactly eight triangle-free examples with a vertex of degree at least 5 are known, leaving only the case in which all vertices have degree 3 or 4. In this paper, we introduce a method for detecting intrinsic knottedness based on induced subgraphs obtained by deleting pairs of vertices. Using this method, we classify all triangle-free intrinsically knotted graphs with 22 edges having eight vertices of degree~4 and four of degree~3. We prove that there are exactly four: Cousins 43, 105, and 109 in the $E_9\!+\!e$ family and the graph $H_{12}\! +\! e$ in the $H_9\!+\!e$ family.

math.CO↗

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.

math.GT↗

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.

math.GT↗

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 $s \equiv \pm 1 \mod r$.

math.GT↗

arXiv2022

Chirality for simple graphs of size up to 12

Chirality is one of the important assymmetrical property in wide area of natural science, which has been studied to predict molecular behavior. One of good methods to analyze molecules with complex structures is representing them as graphs embedded in 3-dimensional space. So it is important to study the chirality of spatial graphs to understand structure of chiral molecules. Moreover, Robertson and Seymour's graph minor theorem implies that a set of minor minimal graphs with respect to intrinsic properties is finite. So it is also important to find a complete set of minor minimal graphs for intrinsic properties. In this paper, we classify minor minimal intrinsically chiral graphs among simple graphs of size up to twelve.

math.GT↗

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 $2c(K)+2$.

math.GT↗

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 $2(p+(|s|-1)r)$.

math.GT↗

arXiv2022

Dips at small sizes for topological graph obstruction sets

The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs.

math.GT↗

arXiv2022

Bipartite intrinsically knotted graphs with 23 edges

A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the $E_9+e$ family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex has degree 3 or more. Four among them contain the Heawood graph and the other two contain Cousin 110 of the $E_9+e$ family. Consequently, there is no minor minimal intrinsically knotted graph with 23 edges that is bipartite.

math.CO↗

arXiv2020

The minor minimal intrinsically chiral graphs

Molecular chirality is actively researched in a variety of areas of biology, including biochemistry, physiology, pharmacology, etc., and today many chiral compounds are widely known to exhibit biological properties. The molecular structure is represented by a graph structure. Therefore, the study of the mirror symmetry of a graph is important in the natural sciences. A graph $G$ is said to be intrinsically chiral if no embedding of G is ambient isotopic to its mirror image. In this paper, we find two minor minimal intrinsically chiral graphs $\Gamma_7$ and $\Gamma_8$. Furthermore, we classify all intrinsically chiral graphs with at most eleven edges.

math.GT↗

arXiv2018

Circuit presentation and lattice stick number with exactly 4 $z$-sticks

The lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $L$ in the cubic lattice. Hong, No and Oh found a general upper bound $s_L(K) \leq 3 c(K) +2$. A rational link can be represented by a lattice presentation with exactly 4 $z$-sticks. An $n$-circuit is the disjoint union of $n$ arcs in the lattice plane $\mathbb{Z}^2$. An $n$-circuit presentation is an embedding obtained from the $n$-circuit by connecting each $n$ pair of vertices with one line segment above the circuit. By using a 2-circuit presentation, we can easily find the lattice presentation with exactly 4 $z$-sticks. In this paper, we show that an upper bound for the lattice stick number of rational $\dfrac{p}{q}$-links realized with exactly 4 $z$-sticks is $2p+6$. Furthermore it is $2p+5$ if $L$ is a 2-component link.

math.GT↗

arXiv2017

More intrinsically knotted graphs with 22 edges and the restoring method

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper we show that there are exactly five triangle-free intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the $K_{3,3,1,1}$ family, Cousins 97 and 99 of the $E_9+e$ family and two others that were previously unknown.

math.GT↗

arXiv2014

Bipartite intrinsically knotted graphs with 22 edges

A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are the only minor minimal intrinsically knotted graphs with 21 edges. This set includes exactly one bipartite graph, the Heawood graph. In this paper we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the $K_{3,3,1,1}$ and $E_9+e$ families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the $E_9+e$ family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.

math.GT↗

arXiv2014

Minimum lattice length and ropelength of knots

Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows: $\mbox{Len}(K) \leq \min \left\{ \frac{3}{4}c(K)^2 + 5c(K) + \frac{17}{4}, \, \frac{5}{8}c(K)^2 + \frac{15}{2}c(K) + \frac{71}{8} \right\}.$ The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the minimum ropelength $\mbox{Rop}(K)$ which is close to twice $\mbox{Len}(K)$: $\mbox{Rop}(K) \leq \min \left\{ 1.5 c(K)^2 + 9.15 c(K) + 6.79, 1.25 c(K)^2 + 14.58 c(K) + 16.90 \right\}.$

math.GT↗

arXiv2014

Minimum lattice length and ropelength of 2-bridge knots and links

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot $K$ indicates the minimum length necessary to construct $K$ in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of knot energies measuring the complexity of knot conformation. The minimum ropelength $\mbox{Rop}(K)$ is the minimum length of an ideally flexible rope necessary to tie a given knot $K$. Much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number $c(K)$ of the knot. It is known that $\mbox{Len}(K)$ and $\mbox{Rop}(K)$ lie between $\mbox{O}(c(K)^{\frac{3}{4}})$ and $\mbox{O}(c(K) [\ln (c(K))]^5)$, but unknown yet whether any family of knots has superlinear growth. In this paper, we focus on 2-bridge knots and links. Linear growth upper bounds on the minimum lattice length and minimum ropelength for nontrivial 2-bridge knots or links are presented: $\mbox{Len}(K) \leq 8 c(K) + 2$. $\mbox{Rop}(K) \leq 11.39 c(K) + 12.37$.

math.GT↗

arXiv2014

A new intrinsically knotted graph with 22 edges

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. In this paper we present the following results: there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Two are the cousins 94 and 110 of the $E_9+e$ family and the third is a previously unknown graph named $M_{11}$. These graphs are shown in Figure 3 and 4. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5.

math.GT↗

arXiv2014

Equilateral stick number of knots

An equilateral stick number $s_{=}(K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct a polygonal knot of $K$ which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a nontrivial knot K in terms of the minimal crossing number $c(K)$ which is $s_{=}(K) \le 2c(K) + 2$. Moreover if $K$ is a non-alternating prime knot, then $s_{=}(K) \le 2c(K) - 2$. Furthermore we find another upper bound on the equilateral stick number for composite knots which is $s_{=}(K_1 \# K_2) \le 2c(K_1) + 2c(K_2)$.

math.GT↗