Download PDF - Estimating the states of the Kauffman bracket skein moduleAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Estimating the states of the Kauffman bracket skein module

Authors 1

Affiliations

  1. Department of Mathematics, Boise State University, Boise, Idaho 83725, U.S.A.

Abstract

The states of the title are a set of knot types which suffice to create a generating set for the Kauffman bracket skein module of a manifold. The minimum number of states is a topological invariant, but quite difficult to compute. In this paper we show that a set of states determines a generating set for the ring of SL2(C) characters of the fundamental group, which in turn provides estimates of the invariant.

Bibliography

  1. G. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, Contemporary Mathematics 187 (1995).
  2. D. Bullock, The (2,∞)-skein module of the complement of a (2,2p+1) torus knot, J. Knot Theory Ramifications 4 no. 4 (1995) 619-632.
  3. D. Bullock, On the Kauffman bracket skein module of surgery on a trefoil, Pacific J. Math., to appear.
  4. D. Bullock, A finite set of generators for the Kauffman bracket skein algebra, preprint.
  5. D. Bullock, Estimating a skein module with SL2(C) characters, Proc. Amer. Math. Soc., to appear.
  6. M. Culler and P. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. Math. 117 (1983) 109-146.
  7. R. Fricke and F. Klein, Vorlesungen über die Theorie der automorphen Functionen, Vol. 1, B. G. Teubner, Leipzig 1897.
  8. W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math. 54 no. 2 (1984) 200-225.
  9. R. Horowitz, Characters of free groups represented in the two dimensional linear group, Comm. Pure Appl. Math. 25 (1972) 635-649.
  10. J. Hoste and J. H. Przytycki, The (2,∞)-skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theory Ramifications 2 no. 3 (1993) 321-333.
  11. J. Hoste and J. H. Przytycki, The Kauffman bracket skein module of S1×S2, Math Z. 220 (1995) 65-73.
  12. W. Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), 91-103.
  13. H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Supér. III. Sér. 6 (1889), 3-72.
Banach Center Publications
1998/42/1
Export to PBN, Opens in new tab
Pages:
23-28
Main language of publication
English
Published
1998
Exact and natural sciences