Quasipositive plumbing (constructions of quasipositive knots and links, V)
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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 $S$.References
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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
- 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
Dedicated: Dedicated to Professor Kunio Murasugi