arXiv research

Formulae for Rasmussen invariant of satellite knots with wrapping number 2 proved.

Abstract invariant cannot be expressed using various slice-torus invariants.

We produce chain-level generators of the virtual Lee complex $Kh ' (V )$ and use them to convert the computable bounds on the Rasmussen invariant of classical knots due to Kawamura and Lobb into bounds on the virtual Rasmussen invariant as defined by Dye, Kaestner, and Kauffman. We also exhibit a class of diagrams fo…

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Fo…

Defines a new Rasmussen invariant over integers and improves knot slice genus bounds.

Defines a Rasmussen invariant for links in RP^3.

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most…

Proves a conjecture about concordance invariant simplifying its relation to Rasmussen's invariant.

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.

We use simple properties of the Rasmussen invariant of knots to study its asymptotic behaviour on the orbits of a smooth volume preserving vector field on a compact domain in the 3-space. A comparison with the asymptotic signature allows us to prove that asymptotic knots are non-alternating, in general. Further we show…

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot…

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we p…

In this paper we provide a new Bennequin-type inequality for the Rasmussen- Beliakova-Wehrli invariant, featuring the numerical transverse braid invariants (the c-invariants) introduced by the author. From the Bennequin type-inequality, and a combinatorial bound on the value of the c-invariants, we deduce a new computa…

We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications…

We present a large family of knots for which the Rasmussen s-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer-Mrowka and Cochran-Harvey-Horn.

In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{á}th-Szab{ó} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.

Diagrammatic method calculates knot invariant from tangle decompositions.

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov defi…

A knot is not squeezable if it fails to meet certain invariant criteria.

We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a $(p,q,q)$-pretzel knot with integers $p,q \ge 2$ admits no Seifert fibered surgery.

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen's concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the …

New invariants from divisibility of Lee classes for slice-torus.

A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpar…

Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.

The Conway knot can't be smoothly tied to any other knot an infinite number of times.

Study shows concordance invariants bound Turaev genus.

We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…

We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.

Study symmetries in equivariant Khovanov homology.

Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refin…

New proof shows exotic 4-manifolds exist without complex calculations.

We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…

New link homology theories for $\mathbb{RP}^3$ yield distinct invariants.

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to de…

Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.

A virtual knot is an equivalence class of embeddings of $S^1$ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined,…

The paper calculates the slicing degree of knots using advanced homology theories.

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

New inequality for refined knot invariants in a specific space.

We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus kn…

We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect s…

New spanning tree model connects knot homology, s-invariant, and exotic discs.

New local equivalence groups refine Rasmussen's s-invariant.

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.

New concordance invariant from spectral sequence on Khovanov homology.

Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.

Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are…