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We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel…
$q,t$-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal $\mathfrak{gl}_1$ algebra. These representations are described in terms of finite difference equations…
We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization…
Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in…
The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call…
Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For…
The universal invariant with respect to a given ribbon Hopf algebra is a tangle invariant that dominates all the Reshetikhin-Turaev invariants built from the representation theory of the algebra. We construct a canonical strict monoidal…
Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that…
Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a…
We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed…
We consider a 2-complex in a particular form, called the Quinn model of a 2-complex. It can be sliced in graphs, where a change from one graph to another can be organized by a sequence of local transitions, which are described in a list of…
We classify exact indecomposable module categories over the representation category of all non-trivial Hopf algebras with coradical S_3 and S_4. As a byproduct, we compute all its Hopf-Galois extensions and we show that these Hopf algebras…
We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively,…
Hopf algebras appear in connection with various problems in Pure Mathematics and Theoretical Physics, mainly through their categoriesof representations, which are examples of tensor categories. In recent years, there have been major…
Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant…
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant…
Generalized Macdonald polynomials (GMP) are eigenfunctions of specifically-deformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar…
Differential expansion (DE) for a Wilson loop average in representation $R$ is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the…
We compute the invariants for a class of knots and links in arbitrary representations in $S^3/\mathbb{Z}_p$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=\lambda$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories.…
The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion…