The 5th Mini Workshop on Knot theory

July 19–23, 2020

Maison Glad Jeju, Jeju, Korea

sponsered by

Department of Mathematics Education, Kyungpook National University

Department of Mathematics Education, Dongguk University-Gyeongju

and

National Research Foundation of Korea

(중견연구 2019R1A2C1005506, 2020R1A2C1A01003201)

Knot

Minicourse Lecturers

  • Youngjin Bae (KIAS)
  • Seungsang Oh (Korea University)

Invited Speakers

  • YoungJin Cho (KAIST)
  • Hyoungjun Kim (Ewha Womans University)
  • Jung Hoon Lee (Jeonbuk National University)
  • Sungjong No (Kyonggi University)
  • Minkyoung Song (IBS-CGP)
  • Hyungkee Yoo (Korea University)

Confirmed Participants

  • Byung Hee An (Kyungpook National University)
  • Youngjin Bae (KIAS)
  • Youngjin Cho (KAIST)
  • Hyungjun Kim (Ewha Womans University)
  • Seonjung Kim (Korea University)
  • Hwa Jeong Lee (Dongguk University, Gyeongju)
  • Jung Hoon Lee (Jeonbuk National University)
  • Sangyop Lee (Chung-Ang University)
  • Sungjong No (Kyonggi University)
  • Sangrok O (KAIST)
  • Seungsang Oh (Korea University)
  • Hyowon Park (Handong University)
  • Juno Seo (KAIST)
  • Minkyoung Song (IBS-CGP)
  • Seungyeop Yang (Kyungpook National University)
  • Byungyong Yoo (Korea University)
  • Hyungkee Yoo (Korea University)

Organizers

  • Byung Hee An (Kyungpook National University)
  • Hwa Jeong Lee (Dongguk University, Gyeongju)

Venue

Maison Glad Jeju, Jeju, Korea

Room rate

Standard Double or Twin
KRW 120,000

* The above rate is as of July, 2020 (VAT included)

** The rate may vary if you make your own reservation.

Talk Schedule (tentative)

July 19 (Sun) July 20 (Mon) July 21 (Tue) July 22 (Wed) July 23 (Thu)
10:00 – 10:45 Bae Bae Bae Song
10:45 – 11:15 Break Closing Remark
11:15 – 12:00 Oh Oh Oh Departure
12:00 – 14:00 Lunch
14:00 – 14:45 Arrival
and
Registration		 Kim Cho Free Discussion
&
Question Session
14:45 – 15:15 Break
15:15 – 16:00 No Lee
16:00 – 16:15 Opening Remark Free Discussion
16:15 – 17:00 Yoo

* The schedule may vary.

Title & Abstract (TBA)

Minicourses

Speaker
Youngjin Bae (KIAS)
Title
Introduction to Legendrian knot theory
Abstract
In this series of lectures, I will introduce the Legendrian knot theory and its various invariants. After introducing basic concepts of Legendrian knots in a contact 3-manifold, I will discuss classical invariants of Legendrian knots including Thurston-Bennequin number and the rotation number. As high-tech invariants of Legendrian knots, I will introduce Chekanov-Eliashberg algebra and the ruling invariant. We will compute the above various invariants for certain Legendrian knots. No preliminary background is required.

Lecture 1: Contact manifolds, Legendrian submanifolds, and their classical invariants.

Lecture 2: Non-classical invariant of Legendrian knots and their computations I (differential algebra type invariant).

Lecture 3: Non-classical invariant of Legendrian knots and their computations II (polynomial type invariant).

Speaker
Seungsang Oh (Korea University)
Title
Mathematics in Blockchain
Abstract
블록체인은 거대한 분산 공개 장부이며, 그 장부 안에 포함된 개별 거래는 모두 디지털 서명이 붙어 있어서 은행이나 다른 제 3자의 개입이 없어도 진본임을 보증할 수 있다. 거래 당사자간의 신뢰 확보를 위해 중앙 기관을 필요로 하지 않는 탈중앙화를 달성한 최초의 스프트웨어 기술이다. 여기에는 작업 증명이라는 수학적 계산 작업과 경제 관점에서의 논리를 통해 위,변조가 사실상 불가능한 구조를 갖게 되어, 그 안에 기록된 거래들은 은행같은 중앙의 보증 기관이 없이도 신뢰할 수 있는 거래로서 확정될 수 있다. 이번 강의를 통해서 우리는 블록체인의 핵심 개념인 분산 공개 장부, 해시함수, 전자서명, 작업증명(PoW), 채굴(보상), 블록의 생성 및 전파, 블룸 필터, 이중지불 문제, 완료된 거래정보의 변경 불가 등에 대해서 배운다.

Research talks

Speaker
Youngjin Cho (KAIST)
Title
The structure of automorphism groups of CLTTF Artin groups
Abstract
Beginning with defining Artin groups by labeled graphs, we compare generators of automorphism groups of some (CLTTF) Artin groups and a right angled Artin groups(RAAG’s) and talk about the difference and why CLTTF cases are more tricky. Furthermore we describe the relations of generators of an automorphism groups of CLTTF Artin groups and give you some examples.
Speaker
Hyungjun Kim (Ewha Womans University)
Title
Bipartite intrinsically knotted graphs with 23 edges
Abstract
A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. It is already known that intrinsically knotted graphs have at least 21 edges, and K7 and the 13 graphs obtained from K7 by ∇-Y moves are the only intrinsically knotted graphs with 21 edges. There are exactly two minor minimal bipartite intrinsically knotted graphs with at most 22 edges. Moreover, there are no minimal intrinsically knotted graphs with 22 edges that are bipartite. The goal of this talk is to show that there are no minor minimal intrinsically knotted graphs with 23 edges that are bipartite.
Speaker
Jung Hoon Lee (Jeonbuk National University)
Title
Primitive disks and intersection pattern
Abstract
For two intersecting properly embedded disks in a $3$-manifold, the intersection pattern is not unique. Concerning the intersection pattern of primitive disks, we give an example of primitive disks $D, E_1, E_2$ for a genus-$g$ ($g \ge 4$) Heegaard splitting of $S^3$ satisfying the following conditions:
  • $E_1$ and $E_2$ are isotopic.
  • Every surgery on $E_1$ along $D$ yields primitive disks.
  • Every surgery on $E_2$ along $D$ yields non-primitive disks.
Speaker
Sungjong No (Kyonggi University)
Title
Stick numbers of Montesinos knot
Abstract
Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$: $s(K) \leq 2c(K)$. Huh and Oh found an improved upper bound: $s(K) \leq \frac{3}{2}(c(K)+1)$. Huh, No and Oh proved that $s(K) \leq c(K)+2$ for a $2$-bridge knot or link $K$ with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let $K$ be a knot or link which admits a reduced Montesinos diagram with $c(K)$ crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then $s(K)\leq c(K)+3$. Furthermore, if $K$ is alternating, then we can additionally reduce the upper bound by $2$.
Speaker
Minkyoung Song (IBS-CGP)
Title
Lower central series and homology cylinders
Abstract
All of Johnson homomorphisms of a mapping class group of a surface, Milnor invariants and Orr invariants of links are related to lower central series of a free group. Moreover, it is known that they are closely connected. In this talk, we consider extension of those invariants to homology cylinders and a filtration via their kernels. A homology cylinder is a kind of 3-manifold, which is a generalization of both a string link and a mapping class group. We determine the images of the filtration under the invariants and get relations of quotients of the filtration to automorphism groups of free nilpotent groups, and free Lie algebras. We also obtain the numbers of linearly independent invariants.
Speaker
Hyungkee Yoo (Korea University)
Title
Minimally knotted spatial cubic graphs with two vertices
Abstract
A spatial graph is called minimally knotted if it is nontrivial, but every proper subgraph is trivial. Clearly, the minimum degree of any minimally knotted spatial graph is at least two. If every degree of vertex is two, then it becomes a Brunnian link. Therefore, we consider the simplest case, the minimally knotted spatial cubic graph with two vertices. In this talk, we observe the properties of a minimally knotted spatial cubic graph with two vertices. Using these properties, we find exact values of lattice stick numbers for several spatial graphs.

Contact

  • Byung Hee An (anbyhee_at_knu.ac.kr)
  • Hwa Jeong Lee (hjwith_at_dongguk.ac.kr)

Previous Editions

Sponsered by

KNU
NRF