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

Powell, Mark

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¯¯¯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.

Citation

Powell, M. (2017). The four-genus of a link, Levine–Tristram signatures and satellites. Journal of Knot Theory and Its Ramifications, 26(02), Article 1740008. https://doi.org/10.1142/s0218216517400089

• Journal Article Type: Article
• Acceptance Date: May 22, 2016
• Online Publication Date: Nov 21, 2016
• Publication Date: Feb 1, 2017
• Deposit Date: Oct 3, 2017
• Publicly Available Date: Nov 21, 2017
• Journal: Journal of Knot Theory and Its Ramifications
• Print ISSN: 0218-2165
• Electronic ISSN: 1793-6527
• Publisher: World Scientific Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 02
• Article Number: 1740008
• DOI: https://doi.org/10.1142/s0218216517400089
• Public URL: https://durham-repository.worktribe.com/output/1343827
• Related Public URLs: https://arxiv.org/abs/1605.06833