Khovanov homology and the symmetry group of a knot

Liam Watson (UCLA)

Abstract: This talk will describe an invariant of (appropriately decorated) tangles by considering an natural inverse limit in Khovanov homology. This leads to an invariant of strongly invertible knots, in the form of a graded vector space, that vanishes if and only if the knot is trivial. The invariant appears to contain different information from the Khovanov homology of the knot. For example, it can detect non-amphicheirality in cases where Khovanov homology fails to and separates certain pairs that Khovanov homology does not. More generally, equivalence of strongly invertible knots is best formulated in terms of conjugacy classes of strong inversions in the symmetry goup of a knot, that is, the mapping class group of the knot exterior. This new invariant seems to be quite good at separating these conjugacy classes.