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

MathSciNet review:
1351491

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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]**Kenneth I. Appel,*On the conjugacy problem for knot groups*, Math. Z.**138**(1974), 273–294. MR**0357622****[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**0294460**, 10.1090/S0002-9939-1972-0294460-X**[AS2]**K. I. Appel and P. E. Schupp,*Artin groups and infinite Coxeter groups*, Invent. Math.**72**(1983), no. 2, 201–220. MR**700768**, 10.1007/BF01389320**[D]**Max Dehn,*Papers on group theory and topology*, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell; With an appendix by Otto Schreier. MR**881797****[GeS1]**S. M. Gersten and H. B. Short,*Small cancellation theory and automatic groups*, Invent. Math.**102**(1990), no. 2, 305–334. MR**1074477**, 10.1007/BF01233430**[GeS2]**S. M. Gersten and H. Short,*Small cancellation theory and automatic groups. II*, Invent. Math.**105**(1991), no. 3, 641–662. MR**1117155**, 10.1007/BF01232283**[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]**Roger C. Lyndon and Paul E. Schupp,*Combinatorial group theory*, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR**0577064****[M]**W. Menasco,*Closed incompressible surfaces in alternating knot and link complements*, Topology**23**(1984), no. 1, 37–44. MR**721450**, 10.1016/0040-9383(84)90023-5**[P]**C. D. Papakyriakopoulos,*On Dehn’s lemma and the asphericity of knots*, Ann. of Math. (2)**66**(1957), 1–26. MR**0090053****[Scb]**Horst Schubert,*Die eindeutige Zerlegbarkeit eines Knotens in Primknoten*, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl.**1949**(1949), no. 3, 57–104 (German). MR**0031733****[Scp]**Paul E. Schupp,*On Dehn’s algorithm and the conjugacy problem*, Math. Ann.**178**(1968), 119–130. MR**0237620****[Se]**Z. Sela,*The conjugacy problem for knot groups*, Topology**32**(1993), no. 2, 363–369. MR**1217075**, 10.1016/0040-9383(93)90026-R**[Wa]**Friedhelm Waldhausen,*The word problem in fundamental groups of sufficiently large irreducible 3-manifolds*, Ann. of Math. (2)**88**(1968), 272–280. MR**0240822****[We]**C. M. Weinbaum,*The word and conjugacy problems for the knot group of any tame, prime, alternating knot*, Proc. Amer. Math. Soc.**30**(1971), 22–26. MR**0279169**, 10.1090/S0002-9939-1971-0279169-X

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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