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Skein modules for links in 3-manifolds — Examples and Counterexamples
1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
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- Construction of the Natural Numbers
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- Cosets, Index and Lagrange's Theorem
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2 · Logical flowchart
3 · Definitions, theorems and proofs
None yet.
4 · Examples, counterexamples and false statements
Forgetting framing identifies distinct skein classes
Statement refuted
Assume the Axiom of Choice and let . The assertion that the Kauffman bracket skein module can be formed from unframed link-isotopy classes without changing its classes is false. In , 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
Facts & Assumptions
Given: The Axiom of Choice, the framed Kauffman bracket skein modules of and a solid torus, and the two pairs of framed links in the statement.
The Axiom of Choice, written (The Axiom of Choice).
Countable choice, written , means every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ()).
In ZF, implies (AC implies DC implies countable choice).
The positive full twist multiplies any link-component class by ; on the two basis classes in the examples the resulting changes are nonzero (Framing transport and compatible collar actions under oriented diffeomorphisms).
The S³ empty class is a free basis and a zero-framed trivial circle represents (The Kauffman bracket skein module of S³ is rank one).
The solid-torus core classes form a free -basis, including (The solid-torus skein module has a core-multicurve basis).
Proof
In the zero-framed trivial circle has class by [F2]. Its positive framing twist has class by [F1]. These are unequal by the nonzero calculation in the framing-and-boundary lemma and the basis assertion [F2].
In the solid torus, the zero-framed core has class and its positive full framing twist has class by [F1]. Since is an element of a free basis by [F3] and 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.
The empty link, trivial circle, and annular core multicurves
Example
Assume the Axiom of Choice and let and . In , a zero-framed trivial circle equals times the empty-link generator. In the solid-torus module, one and two parallel blackboard-framed core curves have classes and , 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
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.
The Axiom of Choice, written (The Axiom of Choice).
Countable choice, written , means every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ()).
In ZF, implies (AC implies DC implies countable choice).
The module definition imposes the zero-framed trivial-circle relation (Kauffman bracket skein module of a three-manifold).
The empty link is a free basis of the S³ module and (The Kauffman bracket skein module of S³ is rank one).
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).
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
Substitute the empty link for in the trivial-circle relation [F1] to get ; [F2] identifies as the generator of the free rank-one module.
By [F3], the core multicurves and are distinct basis vectors and are linearly independent over . 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.
Sources
- Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §2, framing relation and Theorem 2.3(a)
- Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §3.1, pp. 11–12
- Józef H. Przytycki, Fundamentals of Kauffman Bracket Skein Modules, arXiv:math/9809113 (1998), §2, Theorem 2.3(a), and §3, Theorem 3.1
- Thang T. Q. Lê, The Colored Jones Polynomial and the AJ Conjecture, §6.1–§6.2, printed pp. 30–31