ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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The empty link, trivial circle, and annular core multicurves

Example

Assume the Axiom of Choice and let $R = Z [A^{\pm 1}]$ and $δ = - A^{2} - A^{- 2}$ . In $K_{A} (S^{3})$ , a zero-framed trivial circle equals $δ$ times the empty-link generator. In the solid-torus module, one and two parallel blackboard-framed core curves have classes $[z]$ and $[z^{2}]$ , which are distinct basis elements. If an annulus is specified as a boundary subsurface of the solid torus with a product collar, its closed-link skein module acts on the solid-torus module by gluing framed links into that collar. The schematic shows the one-core and two-core representatives; every curve between the annulus boundary circles carries its blackboard framing.

Example references

Axiom of Choice

Choice implies countable choice

Kauffman bracket skein module

Boundary-collar action

S³ skein-module basis

Solid-torus skein-module basis

Facts & Assumptions

Given: The Axiom of Choice, the S³ and solid-torus skein modules, and a chosen annular subsurface with a product collar in the solid-torus boundary.

[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 module definition imposes the zero-framed trivial-circle relation $[L ⊔ O] = δ [L]$ (Kauffman bracket skein module of a three-manifold).

[F2]

The empty link is a free basis of the S³ module and $[O] = δ [\emptyset]$ (The Kauffman bracket skein module of S³ is rank one).

[F3]

The parallel blackboard-framed core classes form a free basis of the solid-torus module (The solid-torus skein module has a core-multicurve basis).

[F4]

A specified product collar carries a unital closed-link skein action by gluing and disjoint union (Functoriality and boundary-collar action for the three-manifold skein quotient).

Proof

1.1A1A2A3F1F2algebra

Substitute the empty link for $L$ in the trivial-circle relation [F1] to get $[O] = δ [\emptyset]$ ; [F2] identifies $[\emptyset]$ as the generator of the free rank-one module.

2.1A1F3F4givenalgebra∎

By [F3], the core multicurves $[z]$ and $[z^{2}]$ are distinct basis vectors and are linearly independent over $R$ . The collar action in [F4] applies to the chosen annular boundary subsurface and is defined by placing a closed framed link in a thin collar layer and taking its union with the interior link; thus it adds the indicated operation without adding boundary endpoints.

Depends on

Used by

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Dependency tree · two levels

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Sources

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §2, Theorem 2.3(a), 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–§6.2, printed pp. 30–31 (standard reference, not scraped)