Khovanov homology and strong inversions

Abstract

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author (Adv. Math. 313 (2017), 915-946). In particular, we provide a counterexample to Conjecture 29 therein, as well as a refinement of and additional evidence for Conjecture 28.The Brieskorn spheres (2, q, 2nq ∓ 1) may be obtained by Dehn surgery on a torus knot in the three-sphere, namely, these are the integer homology spheres S 3 ±1/n (T 2,q ), where T 2,q is the positive (2, q) torus knot. These homology spheres admit Seifert fibrations, with base orbifold S 2 (2, q, 2nq ∓ 1). Denoting by (A, b) the two-fold branched cover of A with branch set b, each of these manifolds admits two descriptions as a two-fold branched cover:. This construction might be best termed as classical; for our purposes it is helpful to review the notation introduced in [9]. The first of these two-fold branched covers results from an involution on the Seifert fibred space that preserves an orientation on the fibres -we refer to this as the Seifert involution. The second of these arises from the Montesinos involution, which reverses an orientation on the fibres; the branch set in question arises from the Montesinos trick, that is, by first constructing a tangle (B 3 , τ ) over which the exterior of T 2,q is realized as a two-fold branched cover. We review the construction below as it is central to our enumeration of Kotelskiy is supported by an AMS-Simons travel grant. Watson is supported by an NSERC discovery/accelerator grant. Zibrowius is supported by the Emmy Noether Programme of the DFG, Project number 412851057. MSC2020: 57K10, 57K18.

A mnemonic for the Lipshitz–Ozsváth–Thurston correspondence

“…As stated, this is a special case of a theorem proved in [12,Section 5] appealing to techniques from [4] (see also [23]). The observation could alternatively be extracted from [4,Section 3.4] (see the aside starting on page 2527 below accompanying Figure 8), and also follows from work of Haiden, Katzarkov and Kontsevich [2]; see Section 1.8 of [12] for more discussion. We will review the algebraic objects in Section 1 and, without reproducing the proof in full, explain some key steps in this special case in Section 2.…”

“…Recent work interprets relative versions of homological invariants in terms of immersed curves, including Heegaard Floer homology for manifolds with torus boundary (see Hanselman, Rasmussen and Watson [4]) as well as link Floer homology (see Zibrowius [23]), singular instanton knot homology (see Hedden, Herald and Kirk [7]), and Khovanov homology (see Kotelskiy, Watson and Zibrowius [12]) for 4-ended tangles. In particular, Section 5 of [12] classifies type D structures over a quiver algebra associated with a surface with boundary in terms of immersed curves on this surface; compare Haiden, Katzarkov and Kontsevich [2] and Hanselman, Rasmussen and Watson [4]. Denoting a field by k, perhaps the simplest algebra to illustrate these classification results is R D kOEu; v=.uv/.…”

Thin links and Conway spheres

“…As a result, the bottom braid move relating the pretzel tangles and lifts to a linear transformation of the planar cover. For the class of tangles admitting an unknot closure, there is a sense in which the behaviour one sees is not more complicated than that observed in this single example; see [KWZ22] for more. This is an ungraded statement, however: the grading information is subtle and important.…”

“…Of note is the fact that the behaviour one encounters in practice is relatively tame, by comparison with the delicate casework seen in the proofs. In particular, if one chooses to focus on the invariants that are encountered in nature, for instance in the examples computed in [KWZ22], most of the forgoing material simplifies considerably. We attempt to highlight this here, and with the reader who has skipped directly to this section from the introduction in mind, our aim is to present this material in a vaguely self-contained way.…”

“…The principle exploited above can be thought of as the Montesinos trick: given a strongly invertible knot , there is an involution on the complement with quotient a tangle . (This is the idea behind the enumeration of tangles given in [KWZ22]. ) This tangle will always have the property of being cap trivial , that is, the -filling of is unknotted.…”