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Forgetting framing identifies distinct skein classes

Statement refuted

Assume the Axiom of Choice and let $R = Z [A^{\pm 1}]$ . The assertion that the Kauffman bracket skein module can be formed from unframed link-isotopy classes without changing its classes is false. In $S^{3}$ , a zero-framed trivial circle and the same circle with one positive full framing twist have distinct skein classes. In a solid torus, a zero-framed core and the same core with one positive full framing twist also have distinct skein classes. In each pair the underlying unframed embedded link is identical.

Statement references

Axiom of Choice

Choice implies countable choice

Framing changes are nontrivial

S³ skein-module basis

Solid-torus skein-module basis

Facts & Assumptions

Given: The Axiom of Choice, the framed Kauffman bracket skein modules of $S^{3}$ and a solid torus, and the two pairs of framed links in the statement.

[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]

The positive full twist multiplies any link-component class by $- A^{3}$ ; on the two basis classes in the examples the resulting changes are nonzero (Framing transport and compatible collar actions under oriented diffeomorphisms).

[F2]

The S³ empty class is a free basis and a zero-framed trivial circle represents $δ [\emptyset]$ (The Kauffman bracket skein module of S³ is rank one).

[F3]

The solid-torus core classes form a free $R$ -basis, including $[z]$ (The solid-torus skein module has a core-multicurve basis).

Proof

1.1A1A2A3F1F2givenalgebra

In $K_{A} (S^{3})$ the zero-framed trivial circle has class $δ [\emptyset]$ by [F2]. Its positive framing twist has class $- A^{3} δ [\emptyset]$ by [F1]. These are unequal by the nonzero calculation in the framing-and-boundary lemma and the basis assertion [F2].

2.1A1F1F3step 1.1algebra∎

In the solid torus, the zero-framed core has class $[z]$ and its positive full framing twist has class $- A^{3} [z]$ by [F1]. Since $[z]$ is an element of a free basis by [F3] and $- A^{3} - 1$ is a nonzero Laurent polynomial, the two skein classes differ. Their underlying embedded core circle is unchanged when the framing decoration is forgotten. The S³ pair has the same underlying unframed circle, so forgetting the framing identifies both pairs while the framed skein classes remain unequal.

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Sources

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §2, framing relation and Theorem 2.3(a) (standard reference, not scraped)
Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §3.1, pp. 11–12 (standard reference, not scraped)