In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still unclear.

Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó’s Upsilon. In this talk, I will describe the construction and properties of such an invariant.

There are already a few Upsilon-like invariants from Khovanov homology, beginning with Grigsby-Licata-Wehrli’s annular Rasmussen invariants d_t from annular Khovanov-Lee homology and Lewark-Lobb’s gimel_n from sl(n) link homology. More recently, Ballinger constructed concordance invariants Phi by combining Rasmussen’s E(-1) spectral sequence differentials with the Lee’s differentials on Khovanov homology (in the form of Khovanov-Rozansky sl(2) homology), and showed that this invariant satisfies many of the same properties as Upsilon. In contrast to Ballinger’s invariant, our construction does not add any more differentials, and is derived from a framework quite similar to that of Upsilon. I will discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.

This is based on joint work with Ross Akhmechet.

Zoom Link: https://ncsu.zoom.us/j/93394263183?pwd=TEZ0N2VaVFdRSmpjbXJUNlhpWDhnQT09
Meeting ID: 933 9426 3183
Passcode: 146889