[Paper Review] Contributions to Khovanov Homology

Stephan M. Wehrli|ArXiv.org|Oct 5, 2008

Geometric and Algebraic Topology25 references3 citations

TL;DR

This thesis advances Khovanov homology by proving it detects Conway mutation through explicit examples of mutant links with distinct homology, introducing a spanning tree model that explains the sparsity of Khovanov homology, extending Rasmussen's invariant to links, and generalizing colored Khovanov bracket functoriality for framed link cobordisms via a novel movie move theory for framed tangles.

ABSTRACT

Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are related to spanning trees of the Tait graph, and we exploit this result to investigate the structure of Khovanov homology for alternating knots. Further, we extend Rasmussen's invariant to links. Finally, we generalize Khovanov's categorifications of the colored Jones polynomial, and study conditions under which our categorifications are functorial with respect to colored framed link cobordisms. In this context, we develop a theory of Carter--Saito movie moves for framed link cobordisms.

Motivation & Objective

• To demonstrate that Khovanov homology is not invariant under Conway mutation, contrary to many classical invariants.
• To explain the observed sparsity of Khovanov homology ranks via a spanning tree model of the chain complex.
• To extend Rasmussen's knot invariant to links and show it provides stronger sliceness obstructions than the multivariable signature.
• To generalize Khovanov's colored Jones polynomial categorifications and establish conditions for functoriality under framed link cobordisms.
• To develop a theory of Carter–Saito movie moves for framed link cobordisms to support functoriality in the colored case.

Proposed method

• Proved that Khovanov's chain complex retracts to a subcomplex indexed by spanning trees of the Tait graph, reducing the chain complex size and explaining rank sparsity.
• Used this spanning tree model to reprove Lee’s result on alternating knots having homology concentrated on two lines in the bigrading plane.
• Constructed canonical chain maps for cup and cap cobordisms in Lee homology using Frobenius algebra relations and sign-free constructions.
• Introduced a modified colored Khovanov bracket with parameters (α,β) to control functoriality, proving chain maps for (α,β) = (1,1) and (0,1).
• Developed a movie move calculus for framed link cobordisms to verify functoriality up to sign and homotopy, enabling a 4-dimensional lift of the theory.
• Applied computer calculations via KhoHo to verify mutation non-invariance in specific mutant link pairs.

Experimental results

Research questions

• RQ1 Can Khovanov homology detect Conway mutation, and if so, how can such examples be constructed?
• RQ2 Why do Khovanov homology ranks tend to be significantly smaller than the number of generators in the chain complex?
• RQ3 Can Rasmussen’s s-invariant be extended to links, and does it provide stronger sliceness obstructions than existing invariants?
• RQ4 Under what conditions is the colored Khovanov bracket functorial with respect to framed link cobordisms?
• RQ5 What is the appropriate movie move calculus for framed link cobordisms to support functoriality in the categorified setting?

Key findings

• Provided the first explicit examples of mutant links with distinct Khovanov homology, proving it is strictly stronger than the Jones polynomial.
• Proved that Khovanov’s chain complex retracts to a subcomplex indexed by spanning trees of the Tait graph, explaining the rank sparsity phenomenon.
• Reproved Lee’s result on alternating knots having homology concentrated on two bigrading lines using the spanning tree model.
• Extended Rasmussen’s invariant to links and showed it detects sliceness in cases where the multivariable Levine–Tristram signature fails.
• Established that the Lee homology chain maps for cup and cap cobordisms are well-defined and induce chain transformations, enabling a 4-dimensional theory.
• Proposed a conjecture that the functorial colored Khovanov bracket descends to a well-defined functor on the quotient category of framed link cobordisms modulo movie moves.