[Paper Review] An sl(2) tangle homology and seamed cobordisms

Carmen Caprau|ArXiv.org|Jul 20, 2007

Geometric and Algebraic Topology17 references16 citations

TL;DR

This paper constructs a bigraded sl(2) tangle homology theory over ℤ[i][a] using seamed cobordisms and foams, resolving the sign ambiguity in Khovanov's original sl(2) homology. By functorially assigning complexes of foams to tangle diagrams and applying a degree-preserving functor to ℤ[i][a]-modules, the theory yields a homology whose graded Euler characteristic matches the quantum sl(2) link invariant, with explicit computation showing 6-dimensional homology for the figure-eight knot at a=0.

ABSTRACT

We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee's modification of it) corresponds to a=0 (or a=1). In particular, our construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl(2) homology theory.

Motivation & Objective

• To construct a bigraded cohomology theory for oriented tangles categorifying the quantum sl(2) link invariant.
• To resolve the sign ambiguity in functoriality of Khovanov's original sl(2) homology by introducing a parameterized theory over ℤ[i][a].
• To extend Bar-Natan’s cobordism-based approach to tangles and Khovanov’s foam-based sl(3) theory to a unified framework for sl(2) using seamed cobordisms.
• To ensure the theory is properly functorial under tangle cobordisms via a functor from foams modulo local relations to ℤ[i][a]-modules.

Proposed method

• Constructs a cube of resolutions from tangle diagrams, assigning to each resolution a complex of foams between webs.
• Uses a Frobenius system defined by ℤ[i][X,a]/(X²−a) to model the quantum sl(2) invariant algebraically.
• Applies a degree-preserving functor ℱ from the category of foams modulo local relations to ℤ[i][a]-modules, mapping closed webs to a rank-2 module 𝒜 in degrees ±1.
• Imposes local relations on foams to ensure invariance under Reidemeister moves and movie moves.
• Computes the homology of the resulting complex via totalization and applies a base change to ℂ when setting a=0 to analyze specific examples.
• Uses matrix representations of cobordisms relative to the basis (1,X) of 𝒜 to compute differentials and homotopy equivalences.

Experimental results

Research questions

• RQ1 Can a bigraded sl(2) tangle homology be constructed that is properly functorial under tangle cobordisms, resolving the sign ambiguity in Khovanov’s original theory?
• RQ2 How can Khovanov’s sl(3) foam-based approach be adapted to construct a functorial sl(2) homology using seamed cobordisms and foams?
• RQ3 What is the structure of the homology for specific knots, such as the figure-eight knot, when the parameter a is set to 0 or 1?
• RQ4 How does the graded Euler characteristic of the constructed homology relate to the quantum sl(2) link invariant?

Key findings

• The theory is a bigraded cohomology theory over ℤ[i][a] whose graded Euler characteristic recovers the quantum sl(2) link invariant.
• At a=0, the homology of the figure-eight knot is 6-dimensional over ℂ, with generators in bidegrees (−2,5), (−1,1), (0,−1), (0,1), (1,−1), and (2,−5).
• The construction resolves the sign ambiguity in functoriality by introducing a parameterized framework over ℤ[i][a], ensuring well-defined maps under tangle cobordisms.
• The functor ℱ applied to the complex of foams yields a homotopy-invariant complex of ℤ[i][a]-modules, whose homology is a bigraded invariant of the tangle.
• The theory is invariant under Reidemeister moves and movie moves due to the imposition of local relations on foams.
• For a=1, the theory corresponds to Lee’s variant of Khovanov’s sl(2) homology, confirming consistency with known special cases.