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

DOI:
https://doi.org/10.1090/S0002-9947-99-02167-4

Published electronically:
April 7, 1999

MathSciNet review:
1466957

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**[Al]**J.W. Alexander, ``Topological invariants of knots and links,'' Trans. Amer. Math. Soc.**30**(1928), 275-306.**[Ba1]**G. Baumslag, ``Wreath products and finitely presented groups,'' Math. Z.**75**(1960/61), 22-28. MR**22:11026****[Ba2]**G. Baumslag, Topics in Combinatorial Group Theory, Birkhäuser Verlag, Basel, 1993. MR**94j:20034****[BuZi]**G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics**5**, de Gruyter, Berlin, 1985. MR**87b:57004****[CrFo]**R.H. Crowell and R.H. Fox, An Introduction to Knot Theory, Ginn and Co., 1963. MR**26:4348****[CrTr]**R.H. Crowell and H.F. Trotter, ``A class of pretzel knots,'' Duke Math. J.**30**(1963), 373-377. MR**27:2977****[De]**M.R. Dellomo, ``On the inverse limit of the branched cyclic covers associated with a knot,'' J. Pure Appl. Algebra**40**(1986), 15-26. MR**87h:57008****[Do]**A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York, 1972. MR**54:3685****[Fo]**R.H. Fox, ``A quick trip through knot theory,'' in Topology of 3-Manifolds and Related Topics (edited by M.K. Fort), Prentice-Hall, N.J. (1961), 120-167. MR**25:3522****[Hal]**P. Hall, ``Finiteness conditions for soluble groups,'' Proc. LondonMath. Soc.**4**(1954), 419-436. MR**17:344c****[Har]**R. Hartley, ``Metabelian representations of knot groups,'' PacificJ. Math.**82**(1979), 93-104. MR**81a:57007****[HauKe]**J.C. Hausmann and M. Kervaire, ``Sous-groupes dérivés des groupes de noeuds,'' L'Enseign. Math.**24**(1978), 111-123. MR**58:7643****[Hil]**J.A. Hillman, ``A remark on branched cyclic covers,'' J. Pure Appl.Algebra**87**(1993), 237-240. MR**94f:57001****[Hir]**F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, New York, 1966. MR**34:2573****[KimRou]**K.H. Kim and F. Roush, email correspondence.**[Kit]**B.P. Kitchens, ``Expansive dynamics on zero-dimensional groups,'' Ergodic Theory and Dynamical Systems**7**(1987), 249-261. MR**88i:28039****[LiMa]**D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. MR**97a:58050****[Lo]**J. Los, ``Knots, braid index and dynamical type,'' Topology**33**(1994), 257-270. MR**95h:57007****[LySc]**R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, New York, 1977. MR**58:28182****[Mi]**J.W. Milnor, ``Infinite cyclic coverings,'' in Conference on the Topology of Manifolds (edited by J.G. Hocking), Prindle, Weber, and Schmidt, Boston, 1968, 115-133. MR**39:3497****[Pr]**J.H. Przytycki, ``-coloring and other elementary invariants of knots,'' Proceedings of the 1995 Conference in Knot Theory at the Banach Center in Warsaw, 275-295. CMP**98:16****[Ra]**E.S. Rapaport, ``Knot-like groups,'' in Annals of Math. Studies**84**, Princeton Univ. Press, Princeton, 1975, 119-133. MR**55:13406****[Ri]**R. Riley, ``Homomorphisms of knot groups on finite groups,'' Math. Comp.**25**(1971), 603-619. MR**45:4399****[Rol]**D. Rolfsen, ``Knots and Links,'' Mathematics Lecture Series**7**, Publish or Perish, Inc., Berkeley, 1976. MR**58:24236**; MR**95c:57018**(corrected reprint)**[Se]**H. Seifert, ``Über das Geschlect von Knoten,'' Math. Ann.**110**(1934), 571-592.**[Si1]**D.S. Silver, ``Augmented group systems and -knots,'' Math. Ann.**296**(1993), 585-593. MR**94i:57039****[Si2]**D.S. Silver, ``Knot invariants from topological entropy,'' Topology Appl.**61**, (1995), 159-177. MR**95m:57020****[SiWi1]**D.S. Silver and S.G. Williams, `` Augmented group systems and shifts of finite type,'' Israel J. Math.**95**, (1996), 231-251. MR**98b:20045****[SiWi2]**D.S. Silver and S.G. Williams, ``Generalized -colorings of links,'' Proceedings of the 1995 Conference in Knot Theory at the Banach Center in Warsaw, 381-394. CMP**9:16****[St]**W.H. Stevens, ``Periodicity for the -homology of cyclic covers of knots and**Z**-homology circles,'' J. Pure Appl. Algebra (in press).**[Vi]**J.W. Vick, Homology Theory, Second Edition, Springer-Verlag, New York, 1994. MR**94i:55002****[Zh]**D. Zheng, ``Symbolic dynamics applied to combinatorial group theory: a toolkit,'' Master of Science thesis in Computer Science, University of South Alabama, 1996.

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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:
https://doi.org/10.1090/S0002-9947-99-02167-4

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