Understanding the Kauffman bracket skein module

Doug Bullock, Charles Frohman, Joanna Kania-Bartoszyńska

Research output: Contribution to journalArticlepeer-review

50 Scopus citations

Abstract

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the SL2(ℂ)-characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine SL2(ℂ)-characters of π1(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.

Original languageEnglish
Pages (from-to)265-277
Number of pages13
JournalJournal of Knot Theory and its Ramifications
Volume8
Issue number3
DOIs
StatePublished - May 1999

Keywords

  • 3-manifold
  • Character theory
  • Knot
  • Link
  • Skein module

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