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Full framing twists in the Kauffman bracket skein module

Statement

Let $M$ be a smooth oriented $3$ -manifold and let $L$ be a framed unoriented link in its interior with a specified component. A positive full framing twist is the local blackboard curl whose $A$ -smoothing is the straight strip with one additional zero-framed trivial circle; a negative full framing twist is the curl whose $B$ -smoothing is the straight strip with one additional zero-framed trivial circle. Write $L^{+}$ and $L^{-}$ for the links obtained by inserting these twists on one component. With $R = Z [A^{\pm 1}]$ and $δ = - A^{2} - A^{- 2}$ , the module classes satisfy

[L^{+}] = - A^{3} [L], [L^{-}] = - A^{- 3} [L] .

Statement references

Kauffman bracket skein module

Kauffman bracket smoothing and loop value

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

Facts & Assumptions

Given: A smooth oriented $3$ -manifold $M$ , a framed unoriented link $L$ in its interior, and the two local full-twist insertions from the Statement.

[F1]

A crossing triple in $K_{A} (M)$ satisfies $[L_{\times}] = A [L_{A}] + A^{- 1} [L_{B}]$ , and adjoining a zero-framed trivial circle multiplies a class by $δ$ (Kauffman bracket skein module of a three-manifold).

[F2]

The loop value is $δ = - A^{2} - A^{- 2}$ and the coefficients of the two smoothings are $A$ and $A^{- 1}$ (Kauffman bracket smoothing and loop value).

Proof

1.1F1F2givenalgebra

In the positive curl from the Statement, the $A$ -smoothing is $L$ with one zero-framed trivial circle and the $B$ -smoothing is $L$ . By [F1] and [F2], $[L^{+}] = A δ [L] + A^{- 1} [L] = (A (- A^{2} - A^{- 2}) + A^{- 1}) [L] = - A^{3} [L] .$

2.1F1F2step 1.1algebra∎

In the negative curl, the $A$ -smoothing is $L$ and the $B$ -smoothing is $L$ with one zero-framed trivial circle. Hence $[L^{-}] = A [L] + A^{- 1} δ [L] = (A + A^{- 1} (- A^{2} - A^{- 2})) [L] = - A^{- 3} [L] .$ These are equalities in the framed-link quotient and use no ambient-isotopy identification that forgets the framing.

Depends on

Used by

Dependency tree · two levels

4 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, framing relation following Definition 2.1 (standard reference, not scraped)
Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.1 (standard reference, not scraped)