[Paper Review] An invariant of link cobordisms from symplectic Khovanov homology

Jack W. Waldron|arXiv (Cornell University)|Dec 27, 2009

Geometric and Algebraic Topology21 references11 citations

TL;DR

This paper constructs a functorial invariant of smooth link cobordisms in R⁴ using symplectic Khovanov homology, defining maps between symplectic Khovanov homology groups up to sign ambiguity. It generalizes Seidel's relative invariant to exact Morse-Bott-Lefschetz fibrations with non-compact singular loci, proving that these maps are invariant under isotopy of cobordisms, thus establishing a topological quantum field theory structure for symplectic Khovanov homology.

ABSTRACT

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy invariant of smooth surfaces in 4D; a first step in proving the conjectured isomorphism of symplectic Khovanov homology and Khovanov homology. The maps themselves are defined using a generalisation of Seidel's relative invariant of exact Lefschetz fibrations to exact Morse-Bott-Lefschetz fibrations with non-compact singular loci.

Motivation & Objective

• To define a functorial invariant of smooth link cobordisms in R⁴ using symplectic Khovanov homology.
• To extend Seidel's relative invariant for exact Lefschetz fibrations to the case of non-compact singular loci and Morse-Bott degenerations.
• To show that the induced maps on symplectic Khovanov homology are invariant under isotopy of cobordisms, up to a global sign ambiguity.
• To provide a first step toward proving the conjectured isomorphism between symplectic Khovanov homology and Khovanov homology by endowing the former with a functorial structure.

Proposed method

• Generalizes Seidel's relative invariant for exact Lefschetz fibrations to exact Morse-Bott-Lefschetz fibrations with non-compact singular loci.
• Constructs a regular S¹-equivariant almost complex structure on a symplectic associated bundle to ensure regularity of holomorphic curves.
• Uses holomorphic sections of the fibration to define the relative invariant in the Morse-Bott setting.
• Applies the construction to symplectic Khovanov homology by realizing link cobordisms as isotopy classes in R³×[0,1] and defining maps via the generalized invariant.
• Establishes functoriality by proving invariance under movie moves and isotopies, including the switching move and stabilization/destabilization.
• Uses bridge diagrams and saddle cobordisms as building blocks to define maps for general smooth cobordisms via composition.

Experimental results

Research questions

• RQ1 Can symplectic Khovanov homology be endowed with a functorial structure under smooth link cobordisms in R⁴?
• RQ2 How can Seidel's relative invariant for exact Lefschetz fibrations be extended to handle non-compact singular loci and Morse-Bott degeneracies?
• RQ3 Are the induced maps on symplectic Khovanov homology invariant under isotopy of cobordisms, up to sign ambiguity?
• RQ4 Can the construction be used to prove the conjectured isomorphism between symplectic Khovanov homology and Khovanov homology?
• RQ5 Do the maps defined via cobordisms satisfy the axioms of a topological quantum field theory?

Key findings

• The paper constructs well-defined maps on symplectic Khovanov homology groups associated to isotopy classes of smooth link cobordisms in R³×[0,1], up to a global sign ambiguity.
• The maps are invariant under isotopy of cobordisms, including movie moves such as 10 and 12, and the switching move, ensuring consistency across different diagram representations.
• The construction proves that symplectic Khovanov homology admits a functorial structure from the category of links and smooth cobordisms to the category of abelian groups and homomorphisms up to sign.
• Stabilization and destabilization maps are shown to be inverses up to sign, confirming consistency in the cobordism calculus.
• The maps induced by creation, annihilation, and saddle cobordisms are well-defined on the symplectic Khovanov homology of crossing diagrams, independent of the choice of representative curves.
• Invariance under distant cobordism commutation is immediate due to the locality of the construction, confirming the TQFT-like behavior of the invariant.