Quasipositivity as an obstruction to sliceness

Abstract

For an oriented link $L \subset S^3 = \Bd\!D^4$, let $\chi_s(L)$ 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 {\it slice} if $\chi_s(K)=1$. Realize $D^4$ in $\C^2$ as $\{(z,w):|z|^2+|w|^2\le1\}$. 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 $\Pscr(-3,5,7)$; 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''.