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

Authors:	Lee Rudolph

Institution:	Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts 01610

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, 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 . 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 , in terms of quasipositive subsurfaces of .

Keywords:	Murasugi sum plumbing quasipositive slice genus Thom conjecture

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