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Knots, representation shifts and TQFT
by
Susan Williams
University of South Alabama
Coauthors: Daniel Silver
We discuss applications of symbolic dynamics to knot theory. Let k be an oriented knot in S3, and K the commutator subgroup of the knot group. For any finite group \Sigma, the set \Phi\Sigma of homomorphisms of K into \Sigma, under a natural Z action, has the structure of a shift of finite type which we call the representation shift of k in \Sigma. The topological conjugacy class of \Phi\Sigma is a knot invariant for each \Sigma. The representation shift is obtained algorithmically from a knot diagram. Points of period r in \Phi\Sigma correspond to representations of the fundamental group of the r-fold branched cyclic cover of k.
Recently, Patrick Gilmer has shown that the representation shift is closely related to the Finite Total Homotopy TQFT studied by Frank Quinn. In fact, this TQFT is readily obtained from a quotient of \Phi\Sigma.
Date received: February 12, 1998
Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-65.