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On surgery curves for
genus-one slice knots
Patrick M. Gilmer and Charles Livingston |
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Vol. 265 (2013), No. 2, 405–425
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Abstract |
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If a knot K bounds a genus-one Seifert surface F ⊂ S3 and F contains an essential simple closed curve α that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice and the determinant of K is not 1, then there are strong constraints on the algebraic concordance class of such α, and it was thought that these constraints might imply that α is at least algebraically slice. We present a counterexample; in the process we answer negatively a question of Cooper and relate the result to a problem of Kauffman. Results of this paper depend on the interplay between the Casson–Gordon invariants of K and algebraic invariants of α. |
Keywords
knot, slice knot, genus 1
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Mathematical Subject Classification 2010
Primary: 57M25
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Milestones
Received: 7 September 2012
Revised: 27 December 2012
Accepted: 17 January 2013
Published: 28 August 2013
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Authors |
| Patrick M. Gilmer | |
| Department of Mathematics Louisiana State University 376 Lockett Hall Baton Rouge, LA 70803 United States |
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| Charles Livingston | |
| Department of Mathematics Indiana University Rawles Hall Bloomington, IN 47405-5701 United States |
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