Related papers: Knot state asymptotics I, AJ Conjecture and abelia…
It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral…
We prove the Geometric P=W conjecture in rank 3 on the three-punctured sphere. We describe the topology at infinity of the related character variety. We use asymptotic abelianization of harmonic bundles away from the ramification divisor…
In this paper, we study the asymptotics of the colored Jones polynomials of the Whitehead chains with one belt colored by $M_1$ and all the clasps colored by $M_2$ evaluated at the $(N+1/2)$-th root of unity $t=e^{\frac{2\pi i}{N+1/2}}$,…
In this article, we give a rough, and so not complete yet, proof of Kashaev's conjecture, that is, the volume conjecture for hyperbolic knots, where the hyperbolicity equations associated to knot diagrams appear as the stationary phase…
We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read--Rezayi state whose effective theory is the SU(2)_K Chern--Simons theory. As a generalization of…
In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.
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…
The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for…
A framework for studying knot and link invariants from any rational conformal field theory is developed. In particular, minimal models, superconformal models and $W_N$ models are studied. The invariants are related to the invariants…
This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…
We study the properties of classical vortex solutions in a non-Abelian gauge theory. A system of two adjoint Higgs fields breaks the SU(2) gauge symmetry to $Z_2$, producing 't Hooft-Polyakov monopoles trapped on cosmic strings, termed…
The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…
We study notions of complexity for link complement states in Chern Simons theory with compact gauge group $G$. Such states are obtained by the Euclidean path integral on the complement of $n$-component links inside a 3-manifold $M_3$. For…
It is shown that the four trefoil solitons that are described by the irreducible representations D^{3/2}_{mm'} of the quantum algebra SL_q(2) (and that may be identified with the four families of elementary fermions…
We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…
The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. D. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored…
A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds,…
In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove…
Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot…
The Slope Conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of incompressible surfaces. Our aim is to prove the Slope Conjecture for Montesinos knots, and to match parameters of a state-formula for…