Khovanov homotopy type, Burnside category, and products

TL;DR

This paper introduces a new construction of the Khovanov homotopy type, proving its equivalence to previous models and exploring its behavior under various operations, leading to new insights into knot slice genera.

Contribution

A novel construction of the Khovanov homotopy type that is shown to be equivalent to existing models and respects key topological operations.

Findings

Construction is stably homotopy equivalent to previous models.

The construction behaves well with disjoint unions, connected sums, and mirrors.

New results on the slice genera of certain knots.

Abstract

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a]. Finally, combining these results with computations from [LS14c] and the refined s-invariant from [LS14b] we obtain new results about the slice genera of certain knots.