Speaker: Aidan McCue (University of Arkansas)

Title: Part 2 - Braids with large Artin and Birman-Ko-Lee super summit sets

Abstract: Garside theory provides us with two parallel but distinct solutions to the word problem in the braid group, one coming from the standard Artin presentation, and another coming from the band generator presentation due to Birman, Ko, and Lee. Extending these algorithms provides us with several solutions to the conjugacy problem, each involving computing certain invariant subsets of the conjugacy classes of the input braids. In this talk, we review the band generator presentation and the solutions to the word problem, present several of the aforementioned subsets, and review some calculus that is useful in computations in the braid group.

The Super Summit Set is one of the most well-known of the subsets mentioned. Caruso produced a family of 5-stranded braids for which the Artin super summit sets grow exponentially in the word length of the braids, but left the Birman-Ko-Lee SSS unanalyzed. I will present a family of 5-stranded braids for which exponential growth is exhibited in the intersection of the Artin and Birman-Ko-Lee super summit sets, demonstrating the inefficiency of the super summit set algorithms.