Knot invariants from symbolic dynamical systems

Silver, Daniel; Williams, Susan

doi:10.1090/S0002-9947-99-02167-4

Knot invariants from symbolic dynamical systems
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by Daniel S. Silver and Susan G. Williams PDF

Trans. Amer. Math. Soc. 351 (1999), 3243-3265 Request permission

Abstract:

If $G$ is the group of an oriented knot $k$, then the set $\operatorname {Hom} (K, \Sigma )$ of representations of the commutator subgroup $K = [G,G]$ 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 $k$. Similar techniques are applied to oriented links.

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