[Paper Review] Frobenius algebras and skein modules of surfaces in 3-manifolds

Uwe Kaiser|ArXiv.org|Feb 27, 2008

Geometric and Algebraic Topology10 references4 citations

TL;DR

This paper establishes a general framework linking Frobenius algebras to skein modules of surfaces in 3-manifolds by defining skein relations that arise from the kernel of an extended topological quantum field theory. The key contribution is a presentation theorem showing that the skein module is generated by incompressible surfaces colored with basis elements of the Frobenius algebra, with relations determined by tubing geometry and algebraic structure, and that the skein module of the 3-ball is isomorphic to the ground ring of the algebra.

ABSTRACT

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the manifold and relations of the algebra.

Motivation & Objective

• To develop a general skein theory for surfaces in 3-manifolds using Frobenius algebras, extending existing link-based skein modules to surface embeddings.
• To clarify the role of the kernel of an extended topological quantum field theory in generating local skein relations for surfaces.
• To prove a presentation theorem for skein modules of surfaces, identifying generators as incompressible surfaces colored by Frobenius algebra basis elements.
• To show that the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra, generalizing known results for link polynomials.
• To relate geometric invariants like tunnel number to algebraic relations in the skein module, demonstrating nontrivial global embedding phenomena.

Proposed method

• Define a skein module of surfaces in a 3-manifold M bounding a prescribed boundary curve system, using local relations derived from the kernel of an extended TQFT.
• Construct a map from colored graphs to surfaces via neck-cutting and sphere relations, encoding the skein relations geometrically.
• Introduce the module $Ø(M,\alpha,\mathfrak{g})$ generated by incompressible surfaces with colors from a generating set of the Frobenius algebra.
• Use tubing geometry to define relations between surfaces, capturing how compressions and handle attachments affect the skein module.
• Prove that the kernel of the TQFT extension is generated by neck-cutting and sphere relations, leading to a presentation of the skein module.
• Establish an isomorphism between the skein module of the 3-ball and the ground ring of the Frobenius algebra, using the presentation theorem.

Experimental results

Research questions

• RQ1 How can skein modules of surfaces in 3-manifolds be systematically constructed from Frobenius algebras?
• RQ2 What is the role of the kernel of an extended topological quantum field theory in generating local skein relations for surfaces?
• RQ3 How do geometric operations like tubing and compression of surfaces manifest algebraically in the skein module?
• RQ4 What is the structure of the skein module when the Frobenius algebra is free, and how can it be presented using incompressible surfaces and algebraic relations?
• RQ5 How do classical 3-manifold invariants like tunnel number appear in the relations of the skein module?

Key findings

• The skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra, generalizing the property of link skein modules.
• The skein module of surfaces in a 3-manifold admits a presentation with generators being incompressible surfaces colored by basis elements of the Frobenius algebra.
• Relations in the skein module are determined by tubing geometry and the algebraic relations of the Frobenius algebra, particularly neck-cutting and sphere relations.
• For a knot complement with tunnel number g, the relation $2^g t^{g/2}(S - 2) = 0$ holds when g is even, and $2^g t^{(g-1)/2}(S, X) = 0$ when g is odd, showing the tunnel number's role in the module structure.
• The construction shows that skein modules measure global embedding phenomena of surfaces, not just local topology, and are sensitive to nontrivial geometric invariants like tunnel number.