arXiv research

New symplectic annular Khovanov homology connects knot theory to Floer homology.

Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.

In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an…

New equivariant version of Khovanov homology for annuli.

New link detection results using closures of 3-braids.

New stable homotopy refinement of quantum annular Khovanov homology.

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Ho…

Study detects a specific type of link using annular Khovanov homology.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.

Spectral sequence connects knot homologies to quotient knots.

Extends quantum annular homology to infinite sets.

We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1|1)$ which is preserved under annular Reidemeister moves.

Let $\widetilde{K}$ be a two-periodic braid and let $K$ be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $\widetilde{K}$ to the next-to-top winding number grading of the sutured annular Khovanov hom…

We construct a braid conjugacy class invariant $κ$ by refining Plamenevskaya's transverse element $ψ$ in Khovanov homology via the annular grading. While $κ$ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using …

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

This paper upgrades Khovanov homology to an L-infinity module structure.

We define a third grading on Khovanov homology, which is an invariant of annular links but changes by $\pm 1$ under stabilization. We illustrate the use of our computer implementation, and give some example calculations.

Short note observes quantum Hochschild homology as a composition of known operations.

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n…

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use…

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

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as …

We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalge…

We construct equivariant Khovanov spectra for periodic links, using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in $S^3$.

New Khovanov homology for links with multiple punctures.

Unified theories for colored sl(2) knot homology.

Kirby color defined in Khovanov homology for 4D handlebodies.

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…

This paper defines a functor for $L_\infty$-modules and applies it to Khovanov homology.

Khovanov homology helps create quantum error-correcting codes.

New homology for links in annulus discovered.

Paper extends link Floer homology detection to almost braided links.

New link detection results using knot and link Floer homology.

We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…

Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B, in S^3, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previou…

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such th…

New categories help understand knot algebra.

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots i…

For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}].$ This spectral sequence splits along qu…

Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.

Proves a conjecture for annular links using homology classes.

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived …

Homologies of Jones and partition algebras match cyclic and symmetric groups.

The paper evaluates homology for links in a solid torus with special boundary conditions.

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelf…

Detects figure-eight knot using Khovanov homology.