Homological width and Turaev genus

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $\delta$-gradings $k$ and $-k$.

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

A well-known conjecture of Rasmussen states that for any knot K K in S 3 S^{3} , the rank of the reduced Khovanov homology of K K is greater than or equal to the rank of the reduced knot Floer homology of K K . This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating.

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, it allows one to answer questions like: Is there a direct relation between Khovanov homology and the A A -polynomial of a knot? Which would not have been asked otherwise. We will explain that the answer to this question is “yes” and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A A -polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the “color behavior” of the colored s l ( 2 ) \mathfrak {sl}(2) knot homology and, eventually, to a similar conjecture for the colored HOMFLY homology. The deformed invariant is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology.

We study a module structure on Khovanov homology, which we show is natural under the Ozsváth–Szabó spectral sequence to the Floer homology of the branched double cover. As an application, we show that this module structure detects trivial links. A key ingredient of our proof is that the Λ∗H_1–module structure on Heegaard Floer homology detects S^1 × S^2 connected summands.

Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology.

In this paper, we study the Khovanov homology of an alternating virtual link L and show that it is supported on g+2 diagonal lines, where g equals the virtual genus of L. Specifically, we show that Khi,j(L) is supported on the lines j=2i−σξ+2k−1 for 0≤k≤g+1 where σξ∗(L)+2g=σξ(L) are the signatures of L for a checkerboard coloring ξ and its dual ξ∗. Of course, for classical links, the two signatures are equal, and this recovers Lee’s H-thinness result for Kh∗,∗(L). Our result applies more generally to give an upper bound for the homological width of the Khovanov homology of any checkerboard virtual link L. The bound is given in terms of the alternating genus of L, which can be viewed as the virtual analogue of the Turaev genus. The proof rests on associating, with any checkerboard colorable link L, an alternating virtual link diagram with the same Khovanov homology as L. In the process, we study the behavior of the signature invariants under vertical and horizontal mirror symmetry. We also compute the Khovanov homology and Rasmussen invariants in numerous cases and apply them to show nonsliceness and determine the slice genus for several virtual knots. ??Table]ras-table at the end of the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant for alternating virtual knots up to six crossings.

Inspired by bordered Floer homology, we describe a type [math] structure in Khovanov homology, which complements the type [math] structure previously defined by the author. The type [math] structure is a differential module over a certain algebra. This can be paired with the type [math] structure to recover the Khovanov chain complex. The homotopy type of the type [math] structure is a tangle invariant, and homotopy equivalences of the type [math] structure result in chain homotopy equivalences on the Khovanov chain complex found from a pairing. We use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. Several examples are included, showing in particular how this approach computes the correct torsion summands for the Khovanov homology of connect sums. A lengthy appendix is devoted to establishing the theory of these structures over a characteristic-zero ring.

ABSTRACTIn the Khovanov homology of links, presence of -torsion is a very common phenomenon. Finite number of examples of knots with -torsion for n > 2 were also known, none for n > 8. In this article, we present several infinite families of links whose Khovanov homology contains -torsion for 2 < n < 9 and -torsion for s < 24. We introduce 4-braid links with -torsion which are counterexamples to parts of the PS braid conjecture. We also provide an infinite family of knots with -torsion in reduced Khovanov homology and -torsion in odd Khovanov homology.

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in S^{3} that do not become isotopic when their interiors are pushed into B^{4} . In particular, we identify examples where the surfaces are not even topologically isotopic in B^{4} , examples that are topologically but not smoothly isotopic, and examples of infinite families of surfaces that are distinct only up to isotopy rel. boundary. Our main proofs distinguish surfaces using the cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.

The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.

In order to find a knot (or link) invariant from a Seifert matrix, we need to look for something that will not change under the operations Λ1 and \(\Lambda^{\pm 1}_2\), defined in Theorem 5.4.1. We will see in this chapter that the Alexander polynomial is such an invariant. The Alexander polynomial is not the only important invariant that we can extricate from the Seifert matrix, the signature of a link can also be defined from it. In addition to defining these two invariants we shall, in this chapter, prove some of their basic characteristics. Nota bene, throughout this chapter we shall assume all the knots and links are oriented.

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology.We show that if a knot K has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all knots obtainable from K by a proper rational tangle replacement.It follows, for example, that unknotting number 1 knots have 2-torsion in their Khovanov homology. 57K18Shumakovitch conjectured that every nontrivial knot has an element of order 2 in its Khovanov homology [10, Conjecture 1].The conjecture has been verified for some infinite families of knots (see, for example, [2; 9; 10]) and has withstood large computational searches.In this note, we provide topological evidence for the conjecture, and we verify the conjecture for a large class of knots that include all unknotting number 1 knots.