arXiv research

New polynomial invariants for knots and links from quandle coloring quiver decategorification.

Decategorifies higher actions in Heegaard Floer homology.

In two previous papers, the author showed how to decompose the Khovanov homology of a link $\mathcal{L}$ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for $\mathcal{L}$ is decomposed into the union of two tangles. Since Khovanov homol…

We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsváth- Szabó's algebra $\mathcal{B}(n,\mathcal{S})$, an…

Witt algebra acts on categorified quantum groups in type A.

We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinator…

New algebraic structures help categorify link invariants.

New invariants for virtual knots and links defined via quiver representations.

New polynomial invariants from quandle action quivers.

New polynomial invariants derived from birack and switch structures.

Define quiver representation-valued invariants for classical and virtual knots

Theory for algebraic data on categories via concentration structures.

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…

Abstract TQFT for sutured manifolds using Floer homology.

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\o…

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism φfrom F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invari…

Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reid…

We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopo…

We define a torsion invariant T for every balanced sutured manifold (M,g), 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,g) is complementary to a Seifert surface of an alternating knot, …