Topological strings and categorification of knot invariants, part 1.
Abstract for both talks
We start with a brief introduction into knot homology theories and categorification of polynomial knot invariants. In particular, we review the construction and the basic properties of a new homological knot invariant, recently introduced by Khovanov and Rozansky. To every knot diagram, it associates a bigraded chain complex whose graded Euler characteristic is the quantum-group sl(N) invariant. Motivated by the ideas from physics, we then present a reformulation of the sl(N) knot homology in terms of new triply-graded knot invariants. This leads to new conjectures on the structure of the homological knot invariants and suggests a larger theory which unifies 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 polynomial. We also describe the geometric meaning of the new knot invariants in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain Calabi-Yau three-fold.