Algorithmic Problems in the Braid Group

TL;DR

This paper reviews braid groups, discusses known decision problem solutions, and introduces new algorithms for the word and conjugacy problems in braid groups and their cyclic amalgamations, with implications for cryptography.

Contribution

It provides new algorithms for the generalized word problem and conjugacy problem in cyclic amalgamations of braid groups, along with complexity analysis and applications to cryptography.

Findings

Quick solution to the generalized word problem in cyclic subgroups

Word problem solution for cyclic amalgamation of two braid groups

New theorem on conjugacy problem in cyclic amalgamations

Abstract

We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution and its complexity using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups. Again using a multitape Turing machine, we solve the word problem for the cyclic amalgamation of two braid groups. We analyze its complexity as well. We then turn to a more general study of the conjugacy problem in cyclic amalgamations. We revise and prove some theorems of Lipschutz[L1] and show their application to cyclic amalgamations of braid groups. We generalize this application to prove a new theorem regarding the conjugacy problem in cyclic amalgamations. We then discuss some application of braid groups, culminating in a section devoted to the discussion of braid group cryptography. We conclude with a discussion of some open questions that we would like to pursue in future research.