Basepoints in Khovanov homology and nonorientable surfaces

TL;DR

This paper introduces an enhanced Khovanov TQFT with basepoint actions over , revealing invariance properties related to nonorientable surfaces and establishing functoriality for a specific pointed Khovanov homology.

Contribution

It develops a new enhanced Khovanov TQFT with basepoint actions over , demonstrating invariance under certain connected sums and establishing functoriality for Baldwin-Levine-Sarkar's pointed Khovanov homology.

Findings

Enhanced Khovanov TQFT behaves like gauge/Floer invariants of double branched covers.

Invariance under connected sum with  with Euler number .

Vanishing after connected sum with  with Euler number 2.

Abstract

We enhance the Khovanov TQFT using basepoint actions, over the field with two elements. Our enhanced Khovanov TQFT behaves similarly to gauge/Floer theoretic invariants of the double branched cover with opposite orientation: they both are invariant, in a certain sense, under taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number -2, and they both vanish after taking the connected sum with the standard $\mathbb{RP}^{2}$ with Euler number 2. This invariance property answers a version of a question posed by Lipshitz and Sarkar. Furthermore, our construction establishes, as a special case, functoriality for the pointed Khovanov homology defined by Baldwin, Levine, and Sarkar.