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.References
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Additional Information
- Daniel S. Silver
- Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
- MR Author ID: 162170
- Email: silver@mathstat.usouthal.edu
- Susan G. Williams
- Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
- MR Author ID: 201838
- Email: williams@mathstat.usouthal.edu
- Received by editor(s): June 27, 1996
- Received by editor(s) in revised form: July 16, 1997
- Published electronically: April 7, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 3243-3265
- MSC (1991): Primary 57Q45; Secondary 54H20, 20E06, 20F05
- DOI: https://doi.org/10.1090/S0002-9947-99-02167-4
- MathSciNet review: 1466957