Jacob Rasmussen

The Superpolynomial for knot homologies

We propose a framework for unifying the sl(N) Khovanov– Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory that categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large-N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee. While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture that gives new predictions even for the original sl(2) Khovanov homology.

Odd Khovanov homology

We describe an invariant of links in the three-sphere which is closely related to Khovanovs Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanovs definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.

Torus knots and the rational DAHA

Author(s): Gorsky, E; Oblomkov, A; Rasmussen, J; Shende, V | Abstract:

Lens space surgeries and a conjecture of Goda and Teragaito

Using work of Ozsvath and Szabo, we show that if a nontrivial knot in S 3 admits a lens space surgery with slope p, then p � 4g+3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito.

Floer homology of surgeries on two-bridge knots

We compute the Ozsv ath-Szab o Floer homologies HF and d HF for three-manifolds obtained by integer surgery on a two-bridge knot. AMS Classication 57R58; 57M27

On stable Khovanov homology of torus knots.

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity.

Khovanov-Rozansky homology of two-bridge knots and links

We compute the reduced version of Khovanov and Rozanskys sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.

The decategorification of sutured Floer homology

We define a torsion invariant T for every balanced sutured manifold (M, γ), and show that it agrees with the Euler characteristic of sutured Floer homology (SFH). The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M, γ) is complementary to a Seifert surface of an alternating knot, then SFH(M, γ) is either 0 or ℤ in every Spin c structure. The torsion invariant T can also be used to show that a sutured manifold is not disc decomposable, and to distinguish between Seifert surfaces.The support of SFH gives rise to a norm z on H 2 (M, ∂ M; ℝ). The invariant T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x s . For closed 3-manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z<x s can happen in general. Finally, we compute T for several wide classes of sutured manifolds.

Floer Simple Manifolds and L-Space Intervals

Abstract An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call “Floer simple,” we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope from the intervals interior. As applications, we give a new proof of the classification of Seifert fibered L-spaces over S 2 , and prove a special case of a conjecture of Boyer and Clay [6] about L-spaces formed by gluing three-manifolds along a torus.

The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Author(s): Oblomkov, A; Rasmussen, J; Shende, V; Gorsky, E | Abstract: