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Kauffman bracket skein module of a three-manifold

Definition

Let $M$ be a smooth oriented $3$ -manifold, and put $R = Z [A^{\pm 1}]$ and $δ = - A^{2} - A^{- 2}$ . A framed unoriented link in $M$ is a finite disjoint union of smoothly embedded annuli $S^{1} \times [- 1, 1]$ in the interior of $M$ , considered up to ambient isotopy through such framed annuli; the empty union is allowed. Let $L_{f r} (M)$ be the set of these isotopy classes and let $F_{R} (M)$ be the free $R$ -module with basis $L_{f r} (M)$ .

For the local crossing relation, choose an orientation-preserving chart from a closed $3$ -ball in $M$ to a standard ball in $R^{3}$ , and project to the first two coordinates. In the displayed chart the over-strand runs from the southwest endpoint to the northeast endpoint, and the under-strand runs from the northwest endpoint to the southeast endpoint. The $A$ -smoothing pairs the two left endpoints and the two right endpoints, and the $B$ -smoothing pairs the two top endpoints and the two bottom endpoints. All three local tangles, including the crossing strands, carry the blackboard framing of the displayed projection, with no additional framing twists. Their framed strips coincide in a collar of the four endpoints.

For each such local triple, the three framed links agree outside the chart. Let $R_{A} (M)$ be the submodule of $F_{R} (M)$ generated by $[L_{\times}] - A [L_{A}] - A^{- 1} [L_{B}]$ for all crossing triples, and by $[L ⊔ O] - δ [L]$ for every framed link $L$ , where $O$ is a zero-framed trivial circle bounding a disk in a ball disjoint from $L$ . The Kauffman bracket skein module of $M$ is

K_{A} (M) : = F_{R} (M) / R_{A} (M) .

The relations are imposed on framed link classes; no unframed type-I identification is added.

Definition references

Kauffman bracket smoothing and loop value

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules (1998), §2, Definition 2.1

Sean Pohorence, 3-manifold Invariants and TQFTs (2016), §3.1, Definition 3.1.1

Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.1

Depends on

Kauffman bracket smoothing and loop value

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §2, Definition 2.1 (standard reference, not scraped)
Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §3.1, Definition 3.1.1, pp. 11–12 (standard reference, not scraped)
Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.1, printed p. 30 (standard reference, not scraped)