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Journal of Undergraduate Research

Keywords

braid group, symmetric group, symmetric group

College

Life Sciences

Department

Microbiology and Molecular Biology

Abstract

My project was to determine which elements of the symmetric group, or permutations on a finite set of objects, generate group representations for the braid group. I started out by finding group representations for the three-strand braid group. A three-strand braid is a set of three disjoint strings in three-dimensional space, whose height functions increase monotonically and which end at points vertically above where they start. The set of braids can be made into a group by letting one braid “multiply,” or perform an operation on, another braid by being placed above it, with their two sets of ends joined. The braid group on three strings is generated by two very simple braids x and y, which satisfy the relation xyx = yxy. In fact, this is the only relation needed in this braid group in the sense that all other relations follow from it. A group permutation representation of the braid group in the symmetric group Sn is a subgroup of Sn generated by two permutations x and y likewise satisfying xyx = yxy.

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