한양대학교 저차원위상수학 겨울 Workshop

이정훈 (전북대학교)

On unperturbed weakly reducible non-minimal bridge positions

A bridge position of a knot is said to be perturbed if it is obtained from a lower index bridge position by introducing a new local maximum and adjacent local minimum. Motivated by the example of a knot admitting an unperturbed strongly irreducible non-minimal bridge position due to Ozawa and Takao, we derive examples of unperturbed weakly reducible non-minimal bridge positions. Also, a bridge version of Gordon's Conjecture is proposed: the connected sum of unperturbed bridge positions is unperturbed.

김형준 (국민대학교)

Intrinsic chirality of spatial graphs with 12 edges

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 embedded in S 3 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 talk, we classify all simple intrinsically chiral graphs with at most 12 edges. This work is collaborated with Howon Choi and Sungjong No.

유형기 (한양대학교)

knot projection with multi-crossings

An n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. In particular, the multi-crossing knot diagram that satisfies a specific condition with single multi-crossing is called petal diagram. The least number of loops in any petal projection of K is called petal number p(K) of K. In this talk, I will introduce the development process and results of the multi-crossing diagram. And I will calculate the exact petal number of several torus knots.

박경배 (강원대학교)

Recent Progress on Intersection Forms of 4-manifolds

The intersection pairing of the second homology group of 4-manifolds not only provides a strong invariant for topological 4-manifolds, but it also portrays the difference between topological and smooth categories of 4-manifolds. For example, a question sometimes called geography problem asks which symmetric bilinear forms can be realized as the intersection form of smooth 4-manifolds. In this talk, we survey classical results and open problems relating to the question and introduce some recent progress.

노성종 (경기대학교)

The folded ribbon length of twisted torus knots and 2-bridge knots

The folded ribbon length $Rib(K)$ of a knot $K$ is the infimum value of the ratio of the length of any ribbon with core to its width. In this talk, we introduce the upper bound of ribbon length of twisted torus knots and 2-bridge knots.

김세구 (경희대학교)

Knot trace and Rasmussen invariant

The knot trace of a knot K is the four-dimensional manifold obtained by attaching a 0-framed 2-handle to the standard 4-ball with attaching sphere K. If two knots K and K' have diffeomorphic knot traces and if K is not slice, then K' is not slice. Piccirillo used this fact and Rasmussen invariant to prove that the Conway knot is not slice. We discuss this result.

오승상 (고려대학교)

Lattice Brunnian Theta-curves

A theta-curve is a graph embedded in R3 which consists of three edges with share endpoints at two vertices. If we cannot continuously transform a theta-curve into a plane without intersecting its strand during the deformation, then it is said to be nontrivial. A Brunnian theta-curve is a nontrivial theta-curve that becomes a trivial knot if any one edge is removed.

In this talk, I introduce qualitative results of these theta-curves, using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve type in the cubic lattice. I present lower bounds of the lattice stick number for nontrivial theta-curves by 14, and Brunnian theta-curves by 16.

강성모 (전남대학교)

About Seifert-fibered surgeries on tunnel-number-one hyperbolic knots.

There are two conjectures about Seifert-fibered surgeries on hyperbolic knots: (1) Any Seifert-fibered surgeries on hyperbolic knots in the 3-sphere are integral, and (2) any Seifert-fibered surgeries on hyperbolic tunnel-number-one knots can be realized by a primitive/Seifert position whose surface slope corresponds to the surgery slope. In this talk, I will present Seifert-fibered surgeries under some circumstances on hyperbolic tunnel-number-one knots are integral and are realized by primitive/Seifert positions.

김민훈 (전남대학교)

Cappell-Shaneson homotopy 4-spheres

In 1976, Cappell and Shaneson constructed a family of homotopy 4-spheres parametrized by a certain 3 by 3 integer matrix and a Z/2-framing. It is still unknown whether all of these homotopy 4-spheres are diffeomorphic to the standard 4-sphere or not. In this talk, I will give a survey on Cappell-Shaneson homotopy 4-spheres.

이화정 (동국대학교)

Arc presentations of Montesinos links

Let $L$ be a Montesinos link $M(-p,q,r)$ with positive rational numbers $p, q,$ and $r$, each less than 1, and $c(L)$ the minimal crossing number of $L$. In this talk, we construct arc presentations of $L$ with $c(L)$, $c(L)-1$, and $c(L)-2$ arcs under some conditions for $p$, $q$, and $r$. Furthermore, we determine the arc index of infinitely many Montesinos links.