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The solid-torus skein module has a core-multicurve basis

Statement

Assume the Axiom of Choice and let $R = Z [A^{\pm 1}]$ . Identify the solid torus with $A \times [- 1, 1]$ , where $A$ is a planar annulus. A simple closed curve is called essential if it does not bound a disk in $A$ . Every framed link in this product admits a regular annular diagram and an integer framing correction from its blackboard framing. Let $z$ be one blackboard-framed annular core curve, and for $n \geq 1$ let $z^{n}$ denote $n$ parallel copies of that curve, with $z^{0}$ the empty link. In the polynomial ring $R [z]$ , $z$ is also the formal variable sent to the core class $[z]$ . Then the classes $[z^{n}]$ , $n \geq 0$ , form an $R$ -basis of $K_{A} (A \times [- 1, 1])$ . Equivalently, the map $J : R [z] \to K_{A} (A \times [- 1, 1])$ sending $z^{n}$ to $[z^{n}]$ is an $R$ -module isomorphism.

Statement references

Axiom of Choice

Framed annular diagram calculus

Full framing twists

Kauffman bracket skein module

Annular state expansion

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules (1998), §3, Theorem 3.1

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

Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.2

Facts & Assumptions

Given: The Axiom of Choice, the annulus-product model of the solid torus, and its framed-link skein module.

[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 quotient is generated by framed links and imposes the crossing and zero-framed trivial-circle relations (Kauffman bracket skein module of a three-manifold).

[F2]

Every framed link in $A \times (- 1, 1)$ admits a regular annular diagram, with its framing correction recorded by signed full-twist marks (Framed link diagrams and local Reidemeister moves in plane products).

[F3]

The state expansion $Ψ$ is well-defined on the skein module and sends a diagram with twist correction $m$ to $(- A^{3})^{m} \sum_{S} A^{a (S) - b (S)} δ^{u (S)} z^{e (S)}$ (Annular state expansion for the Kauffman bracket skein module).

[F4]

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

Proof

1.1A1F3givenconstructalgebra

Define $J$ by the stated $R$ -linear assignment. On $z^{n}$ take the crossingless annular diagram of $n$ parallel essential circles with blackboard framing. It has $m = 0$ , $a = b = u = 0$ , and $e = n$ , so [F3] gives $Ψ (J (z^{n})) = z^{n}$ . By linearity, $Ψ \circ J$ is the identity on $R [z]$ , and hence $J$ is injective.

2.1A1A2A3F1F2F3F4step 1.1inductioncasesalgebra∎

Let $L$ be any framed link in the solid torus. By [F2], choose a regular annular diagram $D$ and write $m_{j}$ for the framing correction on its $j$ th component, with $m = \sum_{j} m_{j}$ . Apply [F4] componentwise: each positive twist contributes $- A^{3}$ , each negative twist contributes $- A^{- 3} = (- A^{3})^{- 1}$ , so $[L] = (- A^{3})^{m} [D]$ , where $D$ carries its blackboard framing. Induct on the number of crossings. At one crossing, [F1] replaces $D$ by its two smoothing diagrams with coefficients $A$ and $A^{- 1}$ , and each smoothing has one fewer crossing. Applying the induction hypothesis to each term yields $[D] = \sum_{S} A^{a (S) - b (S)} [D_{S}],$ where each $D_{S}$ is a crossingless annular state with blackboard framing. Every embedded state circle in the annulus either bounds a disk or is essential and cuts the annulus into two annuli, so it is isotopic to a core circle. For finitely many disjoint essential circles, cutting along them gives annular regions between consecutive circles; thus they are parallel copies of the core. Among the contractible circles choose one whose disk contains no other state circle; its disk has a small thickening disjoint from the remaining link, so the trivial-circle relation in [F1] removes it with factor $δ$ . Repeating removes all contractible circles. Therefore $[L] = (- A^{3})^{m} \sum_{S} A^{a (S) - b (S)} δ^{u (S)} [z^{e (S)}] = J (Ψ ([L])),$ where the last equality is [F3]. Thus $J \circ Ψ$ is the identity on every link generator, and by $R$ -linearity on the whole quotient. Together with step 1.1 this proves $J$ and $Ψ$ are inverse isomorphisms, so the displayed classes are an $R$ -basis.

Depends on

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Sources

Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §3, Theorem 3.1 and proof (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, Proposition 6.2, and §6.2, Example 6.5, printed pp. 30–31 (standard reference, not scraped)