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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Knot invariants from
symbolic dynamical systems


Authors: Daniel S. Silver and Susan G. Williams
Journal: Trans. Amer. Math. Soc. 351 (1999), 3243-3265
MSC (1991): Primary 57Q45; Secondary 54H20, 20E06, 20F05
Published electronically: April 7, 1999
MathSciNet review: 1466957
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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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Additional Information

Daniel S. Silver
Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
Email: silver@mathstat.usouthal.edu

Susan G. Williams
Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
Email: williams@mathstat.usouthal.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02167-4
PII: S 0002-9947(99)02167-4
Received by editor(s): June 27, 1996
Received by editor(s) in revised form: July 16, 1997
Published electronically: April 7, 1999
Article copyright: © Copyright 1999 American Mathematical Society