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Derivatives of knots
and second-order signatures
Tim D Cochran, Shelly Harvey and Constance Leidy |
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Algebraic & Geometric Topology 10 (2010) 739–787
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Abstract |
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We define a set of “second-order” –signature invariants for any algebraically slice knot. These obstruct a knot’s being a slice knot and generalize Casson–Gordon invariants, which we consider to be “first-order signatures”. As one application we prove: If is a genus one slice knot then, on any genus one Seifert surface , there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new relation, generalizing homology cobordism, called null-bordism. |
Keywords
knot concordance, slice knot, $n$–solvable, signature
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Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M10
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References |
Publication
Received: 29 December 2008
Revised: 17 January 2010
Accepted: 17 January 2010
Published: 22 March 2010
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Authors |
| Tim D Cochran | |
| Department of Mathematics
MS-136 Rice University PO 1892 Houston, Texas 77251-1892 USA |
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| http://math.rice.edu/~cochran/ | |
| Shelly Harvey | |
| Department of Mathematics
MS-136 Rice University PO 1892, Houston, Texas 77251-1892 USA |
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| http://math.rice.edu/~shelly/ | |
| Constance Leidy | |
| Department of Mathematics Wesleyan University Wesleyan Station Middletown, CT 06459 USA |
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| http://cleidy.web.wesleyan.edu/ | |
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