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Hwa Jeong Lee

Publications and source records attributed to Hwa Jeong Lee.

16 recordsLinked to original sources
arXiv2025

Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13, 14

There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.

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arXiv2024

Minimal grid diagrams of the prime alternating knots with 13 crossings

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half planes with a common boundary line such that each half plane contains a simple arc of the knot. The minimal number of such half planes among all arc presentations of a given knot is called the arc index of the knot. A knot is usually presented as a planar diagram with finitely many crossings of two strands where one of the strands goes over the other. A grid diagram is a planar diagram which is a non-simple rectilinear polygon such that vertical edges always cross over horizontal edges at all crossings. It is easily seen that an arc presentation gives rise to a grid diagram and vice versa. It is known that the arc index of an alternating knot is two plus its minimal crossing number. There are 4878 prime alternating knots with minimal crossing number 13. We obtained minimal arc presentations of them in the form of grid diagrams having 15 vertical segments. This is a continuation of the works on prime alternating knots of 11 crossings and 12 crossings.

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arXiv2024

Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13

We give a list of minimal grid diagrams of the 13 crossing prime nonalternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other nonalternating knots, 49, 399, 1,412 and 3,250 have arc index 10, 11, 12, and 13, respectively. We used the Dowker-Thistlethwaite code of the 3,250 knots provided by the program Knotscape to generate spanning trees of the corresponding knot diagrams to obtain minimal arc presentations in the form of grid diagrams.

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arXiv2017

Grid diagram for singular links

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto each singular object. This is an extension of the known result of Ng-Thurston for non-singular links and braids.

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arXiv2017

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.

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arXiv2017

Period and toroidal knot mosaics

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \! \times \! n$ matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.

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arXiv2016

Knot mosaic tabulation

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior of knot theory is assumed or required.

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arXiv2015

Exactly fourteen intrinsically knotted graphs have 21 edges

Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the thirteen graphs obtained from K7 by rY moves are intrinsically knotted graphs with 21 edges. We prove that these 14 graphs are the only intrinsically knotted graphs with 21 edges.

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arXiv2014

Quantum knots and the number of knot mosaics

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \times n$ matrix of mosaic tiles ($T_0$ through $T_{10}$ depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. $D^{(m,n)}$ is the total number of all knot (m,n)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. $D^{(m,n)}$ is already found for $m,n \leq 6$ by the authors. In this paper, we construct an algorithm producing the precise value of $D^{(m,n)}$ for $m,n \geq 2$ that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. $$ D^{(m,n)} = 2 \, \| (X_{m-2}+O_{m-2})^{n-2} \| $$ where $2^{m-2} \times 2^{m-2}$ matrices $X_{m-2}$ and $O_{m-2}$ are defined by $$ X_{k+1} = \begin{bmatrix} X_k & O_k \\ O_k & X_k \end{bmatrix} \ \mbox{and } \ O_{k+1} = \begin{bmatrix} O_k & X_k \\ X_k & 4 \, O_k \end{bmatrix} $$ for $k=0,1, \cdots, m-3$, with $1 \times 1$ matrices $X_0 = \begin{bmatrix} 1 \end{bmatrix}$ and $O_0 = \begin{bmatrix} 1 \end{bmatrix}$. Here $\|N\|$ denotes the sum of all entries of a matrix $N$. For $n=2$, $(X_{m-2}+O_{m-2})^0$ means the identity matrix of size $2^{m-2} \times 2^{m-2}$.

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arXiv2014

Mosaic number of knots

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number $m(K)$ of a knot $K$ is the smallest integer $n$ for which $K$ is representable as a knot $n$-mosaic. In this paper we establish an upper bound on the mosaic number of a knot or a link $K$ in terms of the crossing number $c(K)$. Let $K$ be a nontrivial knot or a non-split link except the Hopf link. Then $m(K) \leq c(K) + 1$. Moreover if $K$ is prime and non-alternating except $6^3_3$ link, then $m(K) \leq c(K) - 1$.

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arXiv2014

Arc index of pretzel knots of type $(-p,q,r)$

We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $α(K)$ equals the minimal crossing number $c(K)$. If $p\ge3$ and $q=3$, then $α(K)=c(K)-1$. If $p\ge5$ and $q=4$, then $α(K)=c(K)-2$.

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arXiv2014

Upper bound on the total number of knot $n$-mosaics

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot $n$-mosaic is an $n \times n$ matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. $D_n$ denotes the total number of all knot $n$-mosaics. Already known is that $D_1=1$, $D_2=2$, and $D_3=22$. In this paper we establish the lower and upper bounds on $D_n$ $$\frac{2}{275}(9 \cdot 6^{n-2} + 1)^2 \cdot 2^{(n-3)^2} \ \leq \ D_n \ \leq \ \frac{2}{275}(9 \cdot 6^{n-2} + 1)^2 \cdot (4.4)^{(n-3)^2}.$$ and find the exact number of $D_4 = 2594$.

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arXiv2014

Small knot mosaics and partition matrices

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $(m,n)$-mosaic is an $m \times n$ matrix of mosaic tiles which are $T_0$ through $T_{10}$ depicted as below, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot $(m,n)$-mosaics are there. $D_{m,n}$ denotes the total number of all knot $(m,n)$-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of $D_{m,n}$ for $4 \leq m \leq n \leq 6$ as below. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics. \begin{center} \begin{tabular}{|c|r|r|r|} \hline $D_{m,n}$ & $n=4$ & $n=5$ & $n=6$ \\ \hline $m=4$ & $2594$ & $54,226$ & $1,144,526$ \\ \hline $m=5$ & & $4,183,954$ & $331,745,962$ \\ \hline $m=6$ & & & $101,393,411,126$ \\ \hline \end{tabular} \end{center}

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arXiv2011

Prime knots whose arc index is smaller than the crossing number

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

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