Solutions to the conjugacy search and decision problems in the braid group using finite conjugacy class invariants - Erasmus, Sane´ (Department of Mathematics and Applied Mathematics, 2023)
For two braids, A, B ∈ Bn, the conjugacy decision problem asks whether another braid X ∈ Bn exists such that X−1 A X = B. If we know A, B ∈ Bn are indeed conjugate, the conjugacy search problem asks us to find a braid Y ∈ Bn such that Y −1 A Y = B. In this dissertation we investigate a number of solutions to the conjugacy search problem and conjugacy decision problem in the braid group, all of which use finite invariant subsets of the conjugacy class. In particular, we study the summit set, the super summit set, the improved super summit set algorithm which utilises minimal simple elements, the ultra summit set, improvements to the ultra summit set solution using graph theory, and lastly the set of sliding circuits. As part of this investigation, we also study normal forms of braids, partial orders on the braid group, and the Garside group which generalises the braid group.