Related papers: Fivebranes and Knots
In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and…
In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial…
Extending previous work that involved D3-branes ending on a fivebrane with $\theta_{\mathrm{YM}}\not=0$, we consider a similar two-sided problem. This construction, in case the fivebrane is of NS type, is associated to the three-dimensional…
It has been argued based on electric-magnetic duality that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four-dimension. And the Euler characteristic of…
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of…
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we…
Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the…
We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation)…
Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting…
We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized…
Chern-Simons field theory based on a compact non-abelian gauge group is studied as a theory of knots and links in three dimensions. A method to obtain the invariants for links made from braids of upto four strands is developed. This…
In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge…
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include…
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…
Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at C^3/Gamma orbifold singularities and vise versa, in a setup which gives rise to N=1 supersymmetric gauge theories in four dimensions. The Brane Box Models are…
We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle…
We consider a supersymmetric system of D-5-branes compactified on a 5-torus with a self-dual background field strength on a 4-torus and carrying left-moving momentum along a circle. The corresponding supergravity solution describes a…
We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot…
This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…