Eon-Kyung Lee’s research while affiliated with Sejong University and other places

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number [Formula: see text] of a knot [Formula: see text] is the minimum number of loops among all petal diagrams of [Formula: see text]. Let [Formula: see text] denote the [Formula: see text]-torus knot for relatively prime integers [Formula: see text]. Recently, Kim et al. proved that [Formula: see text] whenever [Formula: see text]. They conjectured that the inequality holds without the assumption [Formula: see text]. They also showed that [Formula: see text] whenever [Formula: see text] and [Formula: see text]. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that [Formula: see text] holds for any [Formula: see text]. We also show that [Formula: see text] holds for any [Formula: see text]. Our proofs construct petal grid diagrams for any torus knots.

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called [Formula: see text]-quasi-stabilizer of a pair of distant points is bounded above by a function of [Formula: see text]. The known upper bound of the cardinality is an exponential function of [Formula: see text]. In this paper we show that the [Formula: see text]-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of [Formula: see text]. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right-angled Artin group on its extension graph.

For the right-angled Artin group action on the extension graph, it is known that the minimal asymptotic translation length is bounded above by 2 provided that the defining graph has diameter at least 3. In this paper, we show that the same result holds without any assumption. This is done by exploring some graph theoretic properties of biconnected graphs, i.e. connected graphs whose complement is also connected.

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called r-quasi-stabilizer of a pair of distant points is bounded above by a function of r. The known upper bound of the cardinality is an exponential function of r. In this paper we show that the r-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of r. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right angled Artin group on its extension graph.

We introduce the notion of quasi-roots and study their uniqueness in right-angled Artin groups.

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for finite simplicial graphs $\Gamma_1$ and $\Gamma_2$ if $\Gamma_1$ embeds into $\Gamma_2^e$ as an induced subgraph then $A(\Gamma_1)$ embeds into $A(\Gamma_2)$. In this article we show that the converse of this theorem does not hold for the case $\Gamma_1$ is the complement of a tree and for the case $\Gamma_2$ is the complement of a path graph.

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for finite simplicial graphs $\Gamma_1$ and $\Gamma_2$ if $\Gamma_1$ embeds into $\Gamma_2^e$ as an induced subgraph then $A(\Gamma_1)$ embeds into $A(\Gamma_2)$. In this article we show that the converse of this theorem does not hold for the case $\Gamma_1$ is the complement of a tree and for the case $\Gamma_2$ is the complement of a path graph.

An element in Artin's braid group $B_n$ is called periodic if it has a power which lies in the center of $B_n$. The conjugacy problem for periodic braids can be reduced to the following: given a divisor $1\le d<n-1$ of $n-1$ and an element $\alpha$ in the super summit set of $\epsilon^d$, find $\gamma\in B_n$ such that $\gamma^{-1}\alpha\gamma=\epsilon^d$, where $\epsilon=(\sigma_{n-1}\cdots\sigma_1)\sigma_1$. In this article we characterize the elements in the super summit set of $\epsilon^d$ in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids. Our characterization directly provides a conjugating element $\gamma$. And it determines the size of the super summit set of $\epsilon^d$ by using the zeta polynomial of the noncrossing partition lattice.

An element in Artin's braid group $B_n$ is called periodic if it has a power which lies in the center of $B_n$. The conjugacy problem for periodic braids can be reduced to the following: given a divisor $1\le d<n-1$ of $n-1$ and an element $\alpha$ in the super summit set of $\epsilon^d$, find $\gamma\in B_n$ such that $\gamma^{-1}\alpha\gamma=\epsilon^d$, where $\epsilon=(\sigma_{n-1}\cdots\sigma_1)\sigma_1$. In this article we characterize the elements in the super summit set of $\epsilon^d$ in the dual Garside structure by studying the combinatorics of noncrossing partitions arising from periodic braids. Our characterization directly provides a conjugating element $\gamma$. And it determines the size of the super summit set of $\epsilon^d$ by using the zeta polynomial of the noncrossing partition lattice.

For a finite simplicial graph $\Gamma$, let $G(\Gamma)$ denote the right-angled Artin group on the complement graph of $\Gamma$. In this article, we introduce the notions of "induced path lifting property" and "semi-induced path lifting property" for immersions between graphs, and obtain graph theoretical criteria for the embedability between right-angled Artin groups. We recover the result of S.-h.{} Kim and T.{} Koberda that an arbitrary $G(\Gamma)$ admits a quasi-isometric group embedding into G(T) for some finite tree T. The upper bound on the number of vertices of T is improved from $2^{2^{(m-1)^2}}$ to $m2^{m-1}$, where m is the number of vertices of $\Gamma$. We also show that the upper bound on the number of vertices of T is at least $2^{m/4}$. Lastly, we show that $G(C_m)$ embeds in $G(P_n)$ for $n\geqslant 2m-2$, where $C_m$ and $P_n$ denote the cycle and path graphs on m and n vertices, respectively.