Understanding the Kauffman bracket skein module

Details Understanding the Kauffman bracket skein module Understanding the Kauffman bracket skein module

1996-04-20

arxiv.org/abs/q-alg/9604013v1

Mathematics

Quantum Algebra

LaTeX2e v1.2, customized document class jktr.cls (included), requires packages epsfig and amstex, 13 pages, 26 figures inserte

Scientific paper

The Kauffman bracket skein module $K(M)$ of a 3-manifold $M$ is defined over formal power series in the variable $h$ by letting $A=e^{h/4}$. For a compact oriented surface $F$, it is shown that $K(F \times I)$ is a quantization of the $\g$-characters of the fundamental group of $F$, corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined. Topologically free Kauffman bracket modules are shown to generate finite type invariants. It is shown 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 finitely generated.

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