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Knot invariants from symbolic dynamical systems

Authors:	Daniel S Silver Susan G Williams

Institution:	Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688 ; Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688

Abstract:	If is the group of an oriented knot , then the set $\operatorname{Hom} (K, \Sigma )$ of representations of the commutator subgroup into any finite group $\Sigma$ has the structure of a shift of finite type $\Phi _{\Sigma }$ , a special type of dynamical system completely described by a finite directed graph. Invariants of $\Phi _{\Sigma }$ , such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When $\Sigma$ is abelian, $\Phi _{\Sigma }$ gives information about the infinite cyclic cover and the various branched cyclic covers of . Similar techniques are applied to oriented links.

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