Algebraic Geometry and Topological Strings Seminar 

26/10/2005, 15:00 — 16:00 — Amphitheatre Pa2, Mathematics Building
Sergei Gukov, Harvard University, Cambridge

Knot Homology and Topological Strings - I

We start with a brief introduction into knot homology theories and categorification of polynomial knot invariants. Of particular interest are homology theories of Ozsvath-Szabo-Rasmussen and Khovanov-Rozansky which provide a homological lift of the Alexander polynomial and the quantum sl(N) invariant, respectively. Motivated by the ideas from physics, we then present 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 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.

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REFERENCES:

S. Gukov, A. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, hep-th/0412243.
N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for Knot Homologies, math.gt/0505662.