The conjugacy problem for groups of alternating prime tame links is polynomial-time

Author:
Karin Johnsgard

Journal:
Trans. Amer. Math. Soc. **349** (1997), 857-901

MSC (1991):
Primary 20F10, 03D15, 57M25

DOI:
https://doi.org/10.1090/S0002-9947-97-01617-6

MathSciNet review:
1351491

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An alternating projection of a prime link can to used to produce a group presentation (of the link group under free product with the infinite cyclic group) with some useful peculiarities, including small cancellation conditions. In this presentation, conjugacy diagrams are shown to have the form of a tiling of squares in the Euclidean plane in one of a limited number of shapes. An argument based on the shape of the link projection is used to show that the tiling requires no more square tiles than a linear function of word length (with constant multiple based on the number of link crossings). It follows that the computation time for testing conjugacy of two group elements (previously known to be no worse than exponential) is bounded by a cubic polynomial. This bounds complexity in the original link group.

**[A]**K. I. Appel,*On the Conjugacy Problem for Knot Groups*, Math. Z.**138**(1974), 273-294. MR**50:10090****[AS1]**K. I. Appel and P. E. Schupp,*The Conjugacy Problem for the Group of any Tame Alternating Knot is Solvable*, Proc. Amer. Math. Soc.**33**(1972), 329-336. MR**45:3530****[AS2]**-,*Artin Groups and Infinite Coxeter Groups*, Invent. Math.**72**(1983), 201-220. MR**84h:20028****[D]**M. Dehn,*Papers on Group Theory and Topology*, Springer-Verlag, Berlin, 1987. MR**88d:01041****[GeS1]**S. M. Gersten and H. Short,*Small Cancellation Theory and Automatic Groups*, Invent. Math.**102**(1990), 305-334. MR**92c:20058****[GeS2]**-,*Small Cancellation Theory and Automatic Groups: Part II*, Invent. Math.**105**(1991), 641-662. MR**92j:20030****[J1]**K. Johnsgard,*The Structure of the Cayley Complex and a Cubic-time Algorithm for Solving the Conjugacy Problem for Groups of Prime Alternating Knots*, Univ. Illinois-Urbana-Champaign: Ph. D. thesis, 1993.**[J2]**-,*Geometric Tilings for Equal Geodesic Words in Group Presentations*, 1993, Preprint.**[K]**I. Kapovitch,*Small Cancellation Groups and Translation Numbers*, 1993, Preprint.**[LS]**R. C. Lyndon and P. E. Schupp,*Combinatorial Group Theory*, Springer-Verlag, Berlin, 1977. MR**58:28182****[M]**W. Menasco,*Closed Incompressible Surfaces in Alternating Knot and Link Complements*, Topology**23**(1984), 37-44. MR**86b:57004****[P]**C. D. Papakyriakopoulos,*On Dehn's Lemma and the Asphericity of Knots*, Ann. of Math.**66**(1957), 1-26. MR**19:761a****[Scb]**H. Schubert,*Die eindeutige Zerlegbarkeit eines Knotens in Primknoten*, Sitzungsber. Heidelberger Akad. Wiss. Math.-Natur. Kl.**1949**(3), 57-104. MR**11:196f****[Scp]**P. E. Schupp,*On Dehn's Algorithm and the Conjugacy Problem*, Math. Ann.**178**(1968), 119-130. MR**38:5901****[Se]**Z. Sela,*The Conjugacy Problem for Knot Groups*, Topology**32**(1993), 363-369. MR**94g:57012****[Wa]**F. Waldhausen,*The Word Problem in Fundamental Groups of Sufficiently Large Irreducible 3-Manifolds*, Ann. of Math.**88**(1968), 272-280. MR**39:2167****[We]**C. Weinbaum,*The Word and Conjugacy Problem for the Knot Group of any Tame Prime Alternating Knot*, Proc. Amer. Math. Soc.**22**(1971), 22-26. MR**43:4895**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20F10,
03D15,
57M25

Retrieve articles in all journals with MSC (1991): 20F10, 03D15, 57M25

Additional Information

**Karin Johnsgard**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
karinj@math.cornell.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01617-6

Received by editor(s):
May 26, 1994

Received by editor(s) in revised form:
September 11, 1995

Additional Notes:
The author was supported in part by Alfred P. Sloan Dissertation Fellowship Grant #DD-414.

Article copyright:
© Copyright 1997
American Mathematical Society