Algorithmic solutions to decision problems in the braid groups

Abstract

The subject of this thesis is the algorithmic solution to various decision problems in the braid groups. In chapter 1, we introduce the notion of braids and important decision problems of the braid groups. In chapter 2, we introduce the band-generator presentation and study the semigroup of positive words in this presentation. Then we give an algorithmic solution to the word problem using this new presentation. Our algorithm is faster than the classical algorithm using Artin``s presentation. In chapter 3, we solve the conjugacy problem by an algorithm which is parallel to the prior works of Garside, Thurston and Elrifai-Morton. Moreover we improve the cycling theorem, which is one of two main theorems for the conjguacy problem, for both of Artin``s presentation and the band generator presentation. In chapter 4, it is proved that there is a polynomial time algorithm for the conjugacy problem of the 4-braid group, after analyzing the structure of the reduced super summit set. In chapter 5, we solve the shortest word problem in the 4-braid group and show that the closure of a positive 4-braid bounds a Bennequin surface. And we give an example which shows that the Bennequin theorem cannot be generalized to all of the 4-braids. In chapter 6, we classify all conjugacy classes of 3-braids that are related by flype operations. Among them we determine which conjugacy classes have representatives that admit both (+) and (-) flypes as an effort to search for a potential example of a pair of transversal knots that are topologically isotopic and have the same Bennequin number but are not transversally isotopic.