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Framing transport and compatible collar actions under oriented diffeomorphisms

Statement

Assume the Axiom of Choice. An orientation-preserving diffeomorphism between smooth oriented $3$ -manifolds transports the full framing of every link and induces an isomorphism of their Kauffman bracket skein modules. If the diffeomorphism carries a specified product collar of a compact oriented boundary surface productwise to another specified collar, the induced isomorphism intertwines the two closed-link collar actions. A positive or negative full framing twist on one link component multiplies its class by $- A^{3}$ or $- A^{- 3}$ , respectively. In $K_{A} (S^{3})$ , a positive twist of a zero-framed trivial circle changes its class from $δ [\emptyset]$ to $- A^{3} δ [\emptyset]$ , with $δ [\emptyset] \neq - A^{3} δ [\emptyset]$ . In the solid-torus module, a positive twist of the blackboard-framed core changes $[z]$ to $- A^{3} [z]$ , with $[z] \neq - A^{3} [z]$ .

Statement references

Axiom of Choice

Choice implies countable choice

Functoriality and boundary-collar action

Full framing twists

S³ skein-module basis

Solid-torus skein-module basis

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules (1998), §2, Proposition 2.2(i), and §3, Theorem 3.1

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

Facts & Assumptions

Given: The Axiom of Choice, oriented smooth three-manifolds, an orientation-preserving diffeomorphism, and matching specified product collars when a boundary action is considered.

[A1]

The Axiom of Choice, written $A C$ (The Axiom of Choice).

[A2]

Countable choice, written ${A C}_{ω}$ , means every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A3]

In ZF, $A C$ implies ${A C}_{ω}$ (AC implies DC implies countable choice).

[F1]

A diffeomorphism induces a skein-module isomorphism by transporting framed links, and a matching product collar carries the closed-link action by gluing and disjoint union; this action is natural under collar-preserving diffeomorphisms (Functoriality and boundary-collar action for the three-manifold skein quotient).

[F2]

A positive or negative full framing twist on one component multiplies its skein class by $- A^{3}$ or $- A^{- 3}$ (Full framing twists in the Kauffman bracket skein module).

[F3]

The empty class is a free $R$ -basis of $K_{A} (S^{3})$ and a zero-framed trivial circle has class $δ [\emptyset]$ (The Kauffman bracket skein module of S³ is rank one).

[F4]

The blackboard-framed core classes $[z^{n}]$ , $n \geq 0$ , are an $R$ -basis of the solid-torus module (The solid-torus skein module has a core-multicurve basis).

[F5]

Here $R = Z [A^{\pm 1}]$ is the Laurent-polynomial ring over the integral domain $Z$ (Kauffman bracket smoothing and loop value).

Proof

1.1F1F2givenconstruct

The quotient isomorphism and collar-action naturality are exactly [F1]. Under a diffeomorphism, the framing annulus is carried by the derivative along with its core, so this map preserves the full link framing rather than only the unframed core. A single full twist is changed according to [F2].

1.2A1A2A3F2F3F5algebra

Apply [F2] to the zero-framed trivial circle $O$ in $S^{3}$ . By [F3], $[O] = δ [\emptyset]$ , so its positive twist represents $- A^{3} δ [\emptyset]$ . The difference from $[O]$ is $(- A^{3} - 1) δ [\emptyset]$ . In $R$ , $δ = - (A^{2} + A^{- 2}) \neq 0$ and $- A^{3} - 1 \neq 0$ ; the product is nonzero because the product of two nonzero Laurent polynomials over the integral domain $Z$ is nonzero (its highest-exponent coefficient is the product of their nonzero highest-exponent coefficients). Since $[\emptyset]$ is a basis element by [F3], the two classes are unequal.

2.1A1F2F4F5algebra∎

For the solid-torus core, [F2] gives the twisted class $- A^{3} [z]$ . Its difference from $[z]$ is $(- A^{3} - 1) [z]$ , which is nonzero because $- A^{3} - 1 \neq 0$ in $R$ and $[z]$ is one of the free basis elements in [F4]. This proves the second displayed framing change is nontrivial.

Depends on

Used by

Forgetting framing identifies distinct skein classes Counterexample
The empty link, trivial circle, and annular core multicurves Example

Dependency tree · two levels

22 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, Proposition 2.2(i), and §3, Theorem 3.1 (standard reference, not scraped)
Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §3.1, pp. 11–12 (standard reference, not scraped)
Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.1, printed pp. 30–31 (standard reference, not scraped)