Quasipositive plumbing (constructions of quasipositive knots and links, V)

Abstract:

A Seifert surface $S\subset S^{3}=\partial D^{4}$ is a fiber surface if a push-off $S\to S^{3}\setminus S$ induces a homotopy equivalence; roughly, $S$ is quasipositive if pushing $\operatorname {Int} S$ into $\operatorname {Int} D^{4}\subset \mathbb {C}^{2}$ produces a piece of complex plane curve. A Murasugi sum (or plumbing) is a way to fit together two Seifert surfaces to build a new one. Gabai proved that a Murasugi sum is a fiber surface iff both its summands are; we prove the analogue for quasipositive Seifert surfaces. The slice (or Murasugi) genus $g_{s}(L)$ of a link $L\subset S^{3}$ is the least genus of a smooth surface $S\subset D^{4}$ bounded by $L$. By the local Thom Conjecture, $g_{s}(\partial S)=g(S)$ if $S\subset S^{3}$ is quasipositive; we derive a lower bound for $g_{s}(\partial S)$ for any Seifert surface $S$, in terms of quasipositive subsurfaces of $S$.