# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Quasipositive plumbing (constructions of quasipositive knots and links, V)HTML articles powered by AMS MathViewer

by Lee Rudolph
Proc. Amer. Math. Soc. 126 (1998), 257-267 Request permission

## 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
• 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
• © Copyright 1998 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 126 (1998), 257-267
• MSC (1991): Primary 57M25; Secondary 32S55, 14H99
• DOI: https://doi.org/10.1090/S0002-9939-98-04407-4
• MathSciNet review: 1452826