Khovanov-Rozansky homology over $\mathbb{F}[U,V]$

Abstract

We define a 'full' version of Khovanov-Rozansky homology: a chain complex CH(K) over F[U,V] whose chain homotopy type is a knot invariant and which has the property that setting U = V = 0 recovers the reduced Khovanov-Rozansky homology. We explore the algebraic structure of CH(K) and show that a variation of the structure theorem for knot Floer homology applies. This allows us to define counterparts to knot Floer concordance invariants $\tau$, $\epsilon$ and ${\varphi }_j$ in the Khovanov-Rozansky setting.