arXiv subjects

Sungjong No

Publications and source records attributed to Sungjong No.

17 recordsLinked to original sources

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

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

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.

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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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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 $Γ_7$ and $Γ_8$. Furthermore, we classify all intrinsically chiral graphs with at most eleven edges.

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arXiv2018

Stick number of spatial graphs

For a nontrivial knot $K$, Negami found an upper bound on the stick number $s(K)$ in terms of its crossing number $c(K)$ which is $s(K) \leq 2 c(K)$. Later, Huh and Oh utilized the arc index $α(K)$ to present a more precise upper bound $s(K) \leq \frac{3}{2} c(K) + \frac{3}{2}$. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number $s_{=}(K)$ as follows; $s_{=}(K) \leq 2 c(K) +2$. As a sequel to this research program, we similarly define the stick number $s(G)$ and the equilateral stick number $s_{=}(G)$ of a spatial graph $G$, and present their upper bounds as follows; $$ s(G) \leq \frac{3}{2} c(G) + 2e + \frac{3b}{2} -\frac{v}{2}, $$ $$ s_{=}(G) \leq 2 c(G) + 2e + 2b - k, $$ where $e$ and $v$ are the number of edges and vertices of $G$, respectively, $b$ is the number of bouquet cut-components, and $k$ is the number of non-splittable components.

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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

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.

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arXiv2017

Arc index of spatial graphs

Bae and Park found an upper bound on the arc index of prime links in terms of the minimal crossing number. In this paper, we extend the definition of the arc presentation to spatial graphs and find an upper bound on the arc index $α(G)$ of any spatial graph $G$ as $$α(G) \leq c(G)+e+b,$$ where $c(G)$ is the minimal crossing number of $G$, $e$ is the number of edges, and $b$ is the number of bouquet cut-components. This upper bound is lowest possible.

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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 $\mbox{Len}(K)$: $\mbox{Rop}(K) \leq \min \left\{ 1.5 c(K)^2 + 9.15 c(K) + 6.79, 1.25 c(K)^2 + 14.58 c(K) + 16.90 \right\}.$

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arXiv2014

Upper bounds on the minimal length of cubic lattice knots

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we consider every knot in the cubic lattice. Especially the minimal length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used "minimal edge number" instead) and proved that the minimal length of the trefoil knot $3_1$ is $24$. Also the minimal lengths of the knots $4_1$ and $5_1$ are known to be $30$ and $34$, respectively. In the article we find a general upper bound of the minimal length of a nontrivial knot $K$, except the trefoil knot, in terms of the minimal crossing number $c(K)$. The upper bound is $\frac{3}{2}c(K)^2 + 2c(K) + \frac{1}{2}$. Moreover if $K$ is a non-alternating prime knot, then the upper bound is $\frac{3}{2}c(K)^2 - 4c(K) + \frac{5}{2}$. Furthermore if $K$ is $(n+1,n)$-torus knot, then the upper bound is $6 c(K) + 2 \sqrt{c(K)+1} +6$.

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arXiv2014

Stick numbers of $2$-bridge knots and links

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or link, except in the case of the unlink and the Hopf link. In this paper we construct any $2$-bridge knot or link $K$ of at least six crossings by using only $c(K)+2$ straight sticks. This gives a new upper bound on stick numbers of $2$-bridge knots and links in terms of crossing numbers.

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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: $\mbox{Len}(K) \leq 8 c(K) + 2$. $\mbox{Rop}(K) \leq 11.39 c(K) + 12.37$.

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arXiv2014

Links with small lattice stick numbers

Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of sticks required to construct a polygonal representation of the link in the cubic lattice. Huh and Oh found all knots whose lattice stick numbers are at most 14. They proved that only the trefoil knot $3_1$ and the figure-8 knot $4_1$ have lattice stick numbers 12 and 14, respectively. In this paper we find all links with more than one component whose lattice stick numbers are at most 14. Indeed we prove combinatorically that $s_L(2^2_1)=8$, $s_L(2^2_1 \sharp 2^2_1)=s_L(6^3_2)=s_L(6^3_3)=12$, $s_L(4^2_1)=13$, $s_L(5^2_1)=14$ and any other non-split links have stick numbers at least 15.

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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 $s_{=}(K_1 \# K_2) \le 2c(K_1) + 2c(K_2)$.

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