2013 Fiscal Year Final Research Report

A study on knots and transverse knots using braid theory and Floer theory

Research Project

Project/Area Number	23740053
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Multi-year Fund
Research Field	Geometry
Research Institution	Yamagata University
Principal Investigator	MATSUDA Hiroshi 山形大学, 理学部, 准教授 (70372703)
Project Period (FY)	2011 – 2013
Keywords	横断的結び目 / 組み紐
Research Abstract	I constructed an example of a pair of closed 4-braids with the following properties; (1) they are related by a Hopf-flype, (2) they are distinct as transverse knots, (3) they have the same self-linking number. I also constructed a similar example of a pair of a closed 3-braid and a closed 7-braid. I determined 2-bridge numbers of torus knots of type (p, q), where p and q are integers. I also determined 2-bridge numbers of knots that had alternating diagrams of closed braids. An invariant of a mapping class group of a surface (fixing its boundary) is defined in bordered Floer theory. When a surface has one boundary component and is of genus 2, I calculated this invariant for elements in Torelli group. Torelli group is a subgroup of a mapping class group of a surface that acts trivially on its first homology group.

(6 results)