Khovanov homology and ribbon concordances

  title={Khovanov homology and ribbon concordances},
  author={Adam Simon Levine and Ian Zemke},
  journal={Bulletin of the London Mathematical Society},
  • A. LevineIan Zemke
  • Published 4 March 2019
  • Mathematics
  • Bulletin of the London Mathematical Society
We show that a ribbon concordance between two links induces an injective map on Khovanov homology. 

Homotopy ribbon concordance and Alexander polynomials

We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L, then the Alexander polynomial of L divides the Alexander polynomial of J.

Ribbon distance and Khovanov homology

We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the X-action on

Link homology theories and ribbon concordances

It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We

Homotopy functoriality for Khovanov spectra

We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.

Ribbon homology cobordisms

Heegaard Floer homology and ribbon homology cobordisms

It has recently been shown by several authors that ribbon concordances, or certain variants thereof, induce an injection on knot homology theories. We prove a variant of this for Heegaard Floer

A mixed invariant of non-orientable surfaces in equivariant Khovanov homology

. We construct a mixed invariant of nonorientable surfaces from the Lee and Bar-Natan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including

Ribbon concordance and the minimality of tight fibered knots

. Agol proved that ribbon concordance forms a partial ordering on the set of knots in the 3-sphere. In this paper, we prove that all tight fibered knots are minimal in this partially ordered set. We

Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

Abstract We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine–Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that

Ribbon cobordisms as a partial order

. We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical 3-manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela-Vick and



Knot Floer homology obstructs ribbon concordance

  • Ian Zemke
  • Mathematics
    Annals of Mathematics
  • 2019
We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. We also generalize

A categorification of the Jones polynomial

Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

Band-sums are ribbon concordant to the connected sum

We show that an arbitrary band-connected sum of two or more knots are ribbon concordant to the connected sum of these knots. As an application we consider which knot can be a nontrivial

Khovanov Homology: Torsion and Thickness

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating

An invariant of tangle cobordisms

We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle

Khovanov's homology for tangles and cobordisms

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological

On the Khovanov and knot Floer homologies of quasi-alternating links

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer

An invariant of link cobordisms from Khovanov homology.

In (10), Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links

On Khovanov’s categorification of the Jones polynomial

The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of "the categorification of

sl(2) tangle homology with a parameter and singular cobordisms

We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan’s approach to tangles on one