Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 25, pages 555-562.

Positive links are strongly quasipositive

Lee Rudolph

Abstract. Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the `local Thom Conjecture', has various interesting consequences; for instance, it yields an easily-computed estimate for the slice euler characteristic of the link L(D) (where D is arbitrary) that extends and often improves the `slice--Bennequin inequality' for closed-braid diagrams; and it leads to yet another proof of the chirality of positive and almost positive knots.

Keywords. Almost positive link, Murasugi sum, positive link, quasipositivity, Seifert's algorithm

AMS subject classification. Primary: 57M25. Secondary: 32S55, 14H99.

E-print: arXiv:math.GT/9804003

Submitted: 31 July 1998. (Revised: 18 March 1999.) Published: 21 November 1999.

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Lee Rudolph
Department of Mathematics, Clark University, Worcester MA 01610, USA

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