Khovanov Homotopy Type, Burnside Category, and Products

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.