Knot concordance and matrix factorizations

Abstract

In this thesis, I prove that for the $E(-1)$ spectral sequence constructed by Rasmussen, beginning at the Khovanov-Rozansky $\mathfrak{sl}(n)$ homology of a knot and converging to the homology of the unknot, all higher pages are knot invariants. This is then used to construct a number of numerical knot invariants, each of which is a concordance homomorphism, and these new invariants are applied to obstruct nonorientable surfaces or surfaces in connected sums of $\mathbb{C} P^2$ from bounding a knot, as well as to bounds on the smooth slice genus.