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2006 Ozsváth–Szabó and Rasmussen invariants of doubled knots
Charles Livingston, Swatee Naik
Algebr. Geom. Topol. 6(2): 651-657 (2006). DOI: 10.2140/agt.2006.6.651

Abstract

Let ν be any integer-valued additive knot invariant that bounds the smooth 4–genus of a knot K, |ν(K)|g4(K), and determines the 4–ball genus of positive torus knots, ν(Tp,q)=(p1)(q1)2. Either of the knot concordance invariants of Ozsváth-Szabó or Rasmussen, suitably normalized, have these properties. Let D±(K,t) denote the positive or negative t–twisted double of K. We prove that if ν(D+(K,t))=±1, then ν(D(K,t))=0. It is also shown that ν(D+(K,t))=1 for all t TB(K) and ν(D+(K,t))=0 for all t TB(K), where  TB(K) denotes the Thurston-Bennequin number.

A realization result is also presented: for any 2g×2g Seifert matrix A and integer a, |a|g, there is a knot with Seifert form A and ν(K)=a.

Citation

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Charles Livingston. Swatee Naik. "Ozsváth–Szabó and Rasmussen invariants of doubled knots." Algebr. Geom. Topol. 6 (2) 651 - 657, 2006. https://doi.org/10.2140/agt.2006.6.651

Information

Received: 26 February 2006; Accepted: 2 March 2006; Published: 2006
First available in Project Euclid: 20 December 2017

zbMATH: 1096.57010
MathSciNet: MR2240910
Digital Object Identifier: 10.2140/agt.2006.6.651

Subjects:
Primary: 57M27
Secondary: 57M25

Rights: Copyright © 2006 Mathematical Sciences Publishers

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