Quasipositivity as an obstruction to sliceness

Abstract:

For an oriented link $L \subset {S^3} = \partial {D^4}$, let ${\chi _s}{\text {(}}L{\text {)}}$ be the greatest Euler characteristic $\chi (F)$ of an oriented 2-manifold F (without closed components) smoothly embedded in ${D^4}$ with boundary L. A knot K is slice if ${\chi _s}(K) = 1$. Realize ${D^4}$ in ${\mathbb {C}^2}$ as $\{ (z,w):|z{|^2} + |w{|^2} \leq 1\}$. It has been conjectured that, if V is a nonsingular complex plane curve transverse to ${S^3}$, then ${\chi _s}(V \cap {S^3}) = \chi (V \cap {D^4})$. Kronheimer and Mrowka have proved this conjecture in the case that $V \cap {D^4}$ is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the "slice-Bennequin inequality" for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like $\mathcal {P}( - 3,5,7)$; all knots obtained from a positive trefoil $O\{ 2,3\}$ by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the "topologically locally-flat Thom conjecture".