An sl_n stable homotopy type for matched diagrams

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n >= 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n >= 3 the cohomology of the stable homotopy type agrees with the sl_n Khovanov-Rozansky cohomology of the underlying knot. We make some consistency checks of this sl_n stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP^2 in the sl_3 type for some diagram, and show that the sl_4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.Comment: 62 pages, color figure

A note on Gornik's perturbation of Khovanov-Rozansky homology

We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.Comment: 11 pages, 1 figure, proofs expanded, corollary adde

A counterexample to Batson's conjecture

We give a counter-example to Batson’s conjecture on the non-orientable smooth slice genera of torus knots