Khovanov homology and the symmetry group of a knot

Abstract

Abstract. We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology -and hence the Jones polynomial -we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homology but, owing to the presence of a strong inversion, may be distinguished using our invariant. This work suggests a strengthened relationship between Khovanov homology and Heegaard Floer homology by way of two-fold branched covers that we formulate in a series of conjectures. There are two features common to each of these applications. First, the quantity extracted from Khovanov homology is an integer (a particular grading [28,35,37], a count of a collection of gradings [47], or a dimension count [21]); and second, the quantity measured is not one that can be extracted from the Jones polynomial -additional structure in Khovanov homology is essential in each case. The latter points to a clear advantage of

On the geography and botany of knot Floer homology

“…Update: This has been disproved by Baker and Luecke [6]. See also [97,Conjecture 30] and the accompanying discussion. Conjecture 6.9 ([49, Conjecture 4]).…”

A refinement of Khovanov homology

“…The knots 8 8 and 10 129 are shown in Figure 5. These knots have identical Khovanov cohomology but are known not to be related by mutation; see [26]. Note that, in this example, we appeal to the fact that 8 8 is a (non-torus) 2-bridge knot, hence it admits a pair of strong inversions and two involutive symmetric diagrams.…”

“…[9]). See [26] for this pair in context, and for a proof that they cannot be related by mutation. Appealing to symmetries, we have: To see that these two strong inversions are distinct, observe that the latter diagram has non-trivial invariant in k min = −3 by appealing to the support lemma (the reader can find a non-involutive alternating diagram with n − = 5 and supporting ordinary Khovanov cohomology in i = −5).…”

Double branched covers of knotoids

“…This section is devoted to the proof of the main result, Theorem 1.1. We should point out that the correspondence between knotoids and strongly invertible knots is partially inspired by the construction in [42], Section 2.2. We begin by giving a precise definition of what a strongly invertible knot is.…”