Instantons and Annular Khovanov Homology

Abstract

In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an application of this spectral sequence, we prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous). Another application is a new proof of Grigsby and Ni's result that tangle Khovanov homology distinguishes braids from other tangles.

Khovanov homology and the cinquefoil

“…We then use Stoffregen-Zhang's work [SZ18] relating the Khovanov spectrum of K to the annular Khovanov spectrum of its unknotted quotient A (with respect to the embedding of A in the solid torus complement of a neighborhood of B) 3 , together with the sl 2 (C)-action on annular Khovanov homology defined by Grigsby-Licata-Wehrli [GLW18], to prove that the annular Khovanov homology of A in its maximal nonzero annular grading is 1-dimensional. Combined with the spectral sequence from annular Khovanov homology to annular instanton homology due to Xie [Xie18] and studied further by Xie-Zhang [XZ19], this implies that A is braided with respect to B. In other words, A and B are mutually braided unknots.…”

Instantons and Khovanov skein homology on $I\times T^2$

“…The choice of u 0 is unique up to a sign. By definition I( Ŷ , L, ω) = AHI(∅), and the u 0 defined above agrees with the choice of the generator of AHI(∅) in [Xie18], which was also denoted by u 0 .…”

“…In order to find the E 2 -page of the spectral sequence given by Lemma 5.4, we need to calculate the differentials on the E 1 -page. Our strategy is to use the isomorphisms in Section 4 to reduce the computation to a known result on the annular instanton Floer homology from [Xie18].…”

Categorical lifting of the Jones polynomial: a survey

“…The proof uses Kronheimer-Mrowka's spectral sequence and its extension to annular links [179] (building on [8,150,58]); Batson-Seed's spectral sequence; and N. Dowlin's spectral sequence mentioned below. In other papers, the authors classify all links with Khovanov homology of dimension ≤ 8 [182] and show that Khovanov homology detects, for instance, L7n1 [183].…”

“…The proof of the first statement uses Kronheimer-Mrowka's spectral sequence, the second uses Dowlin's spectral sequence, and the third uses an annular version of the Kronheimer-Mrowka spectral sequence [179,181], Dowlin's spectral sequence, the spectral sequences for periodic knots [168,25] mentioned above, further hard results on Floer homology [179,181,97,104,36], and the sl 2 (C)-action on annular Khovanov homology [57]. Some of these, like the spectral sequences from equivariant Khovanov homology, lift, or at least recall, well-known properties of the Jones polynomial, such as K. Murasugi's formula [126].…”