Obstructions to slicing knots and splitting links

Abstract

In this thesis, we use invariants inspired by quantum field theory to study the smooth topology of links in space and surfaces in space-time. In the first half, we use Khovanov homology to the study the relationship between links in R3 and their components. We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the split union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer- Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink. In the second half, we consider knots as potential cross-sections of surfaces in R4. We use Heegaard Floer homology to show that certain knots never occur as cross-sections of surfaces with small first Betti number. (It was previously thought possible that every knot was a cross-section of a connect sum of three Klein bottles.) In particular, we show that any smooth surface in R 4 with cross-section the (2k, 2k - 1) torus knot has first Betti number at least 2k - 2.