arXiv research

Combinatorial proof shows knot invariant in Lipshitz's grid homology.

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …

Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.

Link Floer homology detects split links.

Proves a formula in Heegaard Floer homology using combinatorial methods.

New homotopy types defined for links in thickened surfaces with higher genus.

Defines homotopy type for links in thickened surfaces.

Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khova…

New invariant for 4-manifolds with framed links, stronger than existing invariants.

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshit…

New knots found with non-trivial Steenrod operations on Khovanov homology.

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

Extends Heegaard Floer theory to surfaces of dimension one.

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.

In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invar…

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, the…

New operations match Steenrod squares on Khovanov homology.

Incompatible operations affect Khovanov homology and spectral sequences.

The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the autho…

New method connects knot Floer homology with bordered Floer homology.

A new method converts knot Floer homology to immersed curves.

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.

In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group define…

Constructs Khovanov spectra for periodic links, proving rank inequalities.

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Kh…

We compute Steenrod squares on Khovanov homology.

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric…

Functor decomposes Khovanov spectra for non-alternating diagrams.

Notes on Khovanov and knot Floer theories' stable homotopy types.

Prime homology detects split links in prime characteristic.

Study shows $(p,q)$-cables of non-trivial knots are not thin.

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…

Study on knots proves inequality in Floer homology.

Improved algorithm for calculating knot invariants.

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

The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as $m\rightarrow\infty$. We show that the Khovanov homotopy types of $(n,m)$ torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become s…

The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.

Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to e…

Defines a new symplectic Khovanov homology for links in fibered 3-manifolds.

For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore, the odd Kh…

Extends quantum annular homology to infinite sets.

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

We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the …

Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.

New algebras model Ozsváth-Szabó's Kauffman-states.

Given an $m$-periodic link $L\subset S^3$, we show that the Khovanov spectrum $\mathcal{X}_L$ constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of $\mathcal{X}_L$ to the equivariant Khovanov homology of $L$ constructed by the second author. The action of Steenrod algebra …

New homological results for bordered Floer algebras derived from hypertoric categories.