Special Issue for Tim Cochran; Guest Editors: J. E. Grigsby, S. Harvey, K. Orr and D. RubermanNo Access

The four-genus of a link, Levine–Tristram signatures and satellites

Mark Powell

Département de Mathématiques, Université du Québec à Montréal, Canada

https://doi.org/10.1142/S0218216517400089Cited by:12 (Source: Crossref)

Abstract

We give a new proof that the Levine–Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalized satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's $¯ ¯ ¯ μ$ $\bar{μ}$ -invariants of the closure of the infection string link vanish. We construct knots for which the combination of the two results determines the 4-genus.

Keywords:

AMSC: 57M25, 57M27, 57N70

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