Related papers: Colored sl(N) homology and SU(N) representations

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror…

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a…

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a non-commutative ring from H-triangulations of 3-manifolds and observed that for trefoil and…

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact…

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb].…

We generalize the works of Lee [arXiv:math/0210213v3] and Gornik [arXiv:math/0402266v2] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv:1002.2662v1]. As applications, we construct non-degenerate…

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 which categorifies the HOMFLY…

In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled…

We prove that Khovanov homology with Z/2-coefficients detects the link L7n1, and the union of a trefoil and its meridian.

The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for…

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of…

Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of…

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…

We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural…

In this paper we define, for each aspherical orientable 3-manifold $M$ endowed with a \emph{torus splitting} $\c{T}$, a 2-dimensional fundamental $l_1$-class $[M]^{\c{T}}$ whose $l_1$-norm has similar properties as the Gromov simplicial…

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…

Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to $SL(n) $ and tensor products of the standard…

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed…

For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescu series denoted $F^{\mathfrak{g}}_K$ which serves as an analytic continuation of the quantum knot invariants associated to finite dimensional irreducible…

We generalize the Morton-Franks-Williams inequality to the colored $\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our…