[Paper Review] Mutation invariance of Khovanov homology over $\mathbb{F}_2$

Stephan M. Wehrli|ArXiv.org|Apr 22, 2009

Geometric and Algebraic Topology10 references7 citations

TL;DR

This paper proves that Khovanov homology and Lee homology over the field $ Ftwo$ are invariant under component-preserving link mutations. Using a formal Khovanov bracket with $ Ftwo$-coefficients and algebraic manipulations of dotted cobordisms, the author establishes invariance via an explicit chain map that commutes with differentials, resolving gaps in earlier arguments and confirming mutation invariance in this setting.

ABSTRACT

We prove that Khovanov homology and Lee homology with coefficients in $\mathbb{F}_2$ are invariant under component-preserving link mutations.

Motivation & Objective

• To resolve the open question of whether Khovanov homology is invariant under component-preserving link mutations when coefficients are in $ Ftwo$.
• To fill gaps in Bar-Natan's 2005 argument on mutation invariance by working over $ Ftwo$, where the necessary algebraic structures are well-behaved.
• To show that both Khovanov homology and Lee homology with $ Ftwo$ coefficients are invariant under component-preserving mutations.
• To provide a self-contained proof using formal Khovanov brackets with $ Ftwo$-linear combinations of decorated cobordisms and dot manipulation rules.

Proposed method

• Constructs a variant of the formal Khovanov bracket taking values in a category of $ Ftwo$-linear combinations of decorated 2-cobordisms with finitely many dots.
• Introduces algebraic operations on dots—specifically, dot sliding and dot removal via the $ Ftwo$-linear structure—to manipulate cobordism diagrams.
• Defines a chain map $ phi$ as a composition of endomorphisms $ phi_k$, each corresponding to a crossing in the tangle, and shows $ phi^2 = Id$ over $ Ftwo$.
• Uses the fact that dot operations commute with the differential and satisfy $ partial_{ullet} delta$-relations to prove that $ phi$ intertwines the differentials on the source and target complexes.
• Applies the decomposition of $y$-mutations into $z$-mutations and Reidemeister moves to reduce the general mutation invariance to the $z$-mutation case.
• Employs a telescope sum argument on dot operators to show that the total map $ phi$ satisfies $ phi circ d_A circ phi^{-1} = d_B$, proving chain homotopy equivalence.

Experimental results

Research questions

• RQ1 Is Khovanov homology with $ Ftwo$ coefficients invariant under component-preserving link mutations?
• RQ2 Can the gaps in Bar-Natan's 2005 argument on mutation invariance be filled using $ Ftwo$ coefficients?
• RQ3 Does Lee homology with $ Ftwo$ coefficients also exhibit mutation invariance?
• RQ4 Can the formal Khovanov bracket with $ Ftwo$-coefficients be used to prove invariance under $z$-mutations via algebraic dot operations?
• RQ5 Is the graded homotopy type of $Kh(L)$ preserved under component-preserving mutations over $ Ftwo$?

Key findings

• Khovanov homology and Lee homology with $ Ftwo$ coefficients are invariant under component-preserving link mutations.
• The proof establishes that the formal Khovanov bracket with $ Ftwo$-coefficients is invariant under $z$-mutations via a well-defined chain map $ phi$.
• The chain map $ phi$ satisfies $ phi circ d_A circ phi^{-1} = d_B$, proving that the differential is preserved under mutation.
• The key technical step is showing that the composition of dot operations leads to a telescoping sum that cancels intermediate terms, leaving only contributions from endpoints $a$ and $c$.
• The use of $ Ftwo$ coefficients ensures that all self-crossing operations square to identity, simplifying the algebraic structure and enabling the proof.
• The result confirms that mutation invariance holds in this setting, complementing Bloom's later result for odd Khovanov homology over integers.