Department of Mathematics

Geometry and Topology

  •  Charlie Frohman, U Iowa
  •  A Geometric Kauffman Bracket
  •  04/06/2021
  •  2:50 PM - 3:40 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Honghao Gao (gaohongh@msu.edu)

This is joint work with Joanna Kania-Bartoszynska and Thang Le $\\$ I will discuss the representation theory of the Kauffman bracket skein algebra of a finite type surface at a root of unity whose order is not divisible by 4. $\\$ Specifically, the Kauffman bracket skein algebra is an algebra with trace in the sense of De Concini, Procesi, Reshetikhin and Rosso, so it has a well defined character variety of trace preserving representations, which can be identified with a branched cover of the SL(2,C)-character variety of the fundamental group of the underlying surface. $\\$ In the case of a closed surface the branched cover is trivial so its just the character variety of the fundamental group of the surface. $\\$ The skein algebra is also a Poisson order, so the character variety representations of the Kauffman bracket skein algebra of a closed surface decomposes into representations corresponding to irreducible, abelian and central representations of the fundamental group of the underlying surface. The irreducible representations of the fundamental group of the surface correspond to irreducible representations of the skein algebra. $\\$ We then use this as basic data to define an invariant of framed links in a three-manifold equipped with an irreducible representation of its fundamental group. The invariant satisfies the Kauffman bracket skein relations. $\\$ Such a representation could be the lift of the holonomy of a hyperbolic structure on the three-manifold, hence the title : A Geometric Kauffman Bracket. $\\$

 

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