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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Lee Rudolph
Journal: Proc. Amer. Math. Soc. 126 (1998), 257-267
MSC (1991): Primary 57M25; Secondary 32S55, 14H99
MathSciNet review: 1452826
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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 $S$.


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Additional Information

Lee Rudolph
Affiliation: Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts 01610
Email: lrudolph@black.clarku.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04407-4
PII: S 0002-9939(98)04407-4
Keywords: Murasugi sum, plumbing, quasipositive, slice genus, Thom conjecture
Received by editor(s): October 1, 1995
Additional Notes: Partially supported by grants from CAICYT, NSF (DMS-8801915, DMS-9504832), and CNRS
Dedicated: Dedicated to Professor Kunio Murasugi
Communicated by: Ronald Stern
Article copyright: © Copyright 1998 American Mathematical Society