Link homology theories from symplectic geometry

Abstract

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl ( n ) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n = 2 .