The 3^rd Mini Workshop on Knot theory

Low dimensional topology

June 16–17, 2017

Korea University, Seoul, Korea

sponsered by

Speaker: Sangbum Cho (Hanyang University)
Title: Disk complexes, mapping class groups, and 2-bridge knots
Abstract: We describe the combinatorial structure of the disk complex of a genus-2 handlebody. In particular, when the handlebody is one of the handlebodies of a reducible genus-2 Heegaard splitting, the disk complex admits an interesting subcomplex, called the primitive disk complex. As applications of a study of the primitive disk complexes, first we provide a finite presentation of the mapping class group of each of the reducible genus-2 Heegaard splittings, and next we give an alternative proof of a result of Kobayashi and Saeki that every (1, 1)-position of a non-trivial 2-bridge knot is a stabilization of its 2-bridge position. This is a joint work with Yuya Koda.

Speaker: Seonhwa Kim (IBS-CGP)
Title: Partially abelian PSL(2,C) representations of knot groups
Abstract: Octahedral developings of knot complement are inspired by Volume conjecture. These are parametrized by several ways using complex variables related to cross-ratios, in particular segment variables and region variables. We will see a condition if there is a missing representation in a solution set of gluing equation and introduce the notion of partially abelian representation, which is also related to a PSL(2,C) representation of virtually knot group.

Speaker: Bo-hyun Kwon (Korea University)
Title: Introduction to trisection and bridge trisection
Abstract: In this talk, we introduce the Trisection of closed, smooth 4-manifolds which was developed by Gay and Kirby and the Bridge Trisection of knotted surfaces in $S^4$ which is introduced by Meier and Zupan. Also, I would give some interesting open problems about these topics.

Speaker: Kyungbae Park (KIAS)
Title: Heegaard Floer correction terms, definite 4-manifolds, and Dehn surgery
Abstract: Heegaard Floer correction term is a rational-valued invariant for closed, oriented 3-manifolds equipped with torsion $spin^c$ structures, introduced by Ozsváth and Szabó using the absolute grading of Heegaard Floer homology. In particular, it is known that the invariant gives constraints on definite smooth 4-manifolds bounded by a give 3-manifold. In this talk, we discuss the construction and the formal properties of the correction term, and introduce two applications of it. First, we present infinitely many examples of closed, oriented, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $\pi_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This is a joint work with Matt Hedden and Min Hoon Kim. Secondly, we show if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are only finitely many intersection forms of negative definite smooth 4-manifolds bounded by $Y$. This is a joint work with Dong Heon Choe.

Speaker: Hyunshik Shin (KAIST)
Title: Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex
Abstract: Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(R_n, \psi_n)$ of fibers contained in the fibered cone of $(S,\psi)$ such that the asymptotic translation length of $\psi_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like $1/|\chi(R_n)|^2$. As an application, we can reprove the previous result by Gadre--Tsai that the minimal asymptotic translation lengths of a closed surface of genus $g$ are bounded below and above by $C/g^2$ and $D/g^2$ for some positive constants $C$ and $D$, respectively. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.