Volume 6, issue 2 (2006)

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Ozsváth–Szabó and Rasmussen invariants of doubled knots

Charles Livingston and Swatee Naik

Algebraic & Geometric Topology 6 (2006) 651–657

arXiv: math.GT/0505361

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) = (p 1)(q 1)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.

Keywords
doubled knot, Ozsvath-Szabo invariant, Rasmussen invariant
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25
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Publication
Received: 26 February 2006
Accepted: 2 March 2006
Published: 18 May 2006
Authors
Charles Livingston
Department of Mathematics
Indiana University
Bloomington, IN 47405
USA
Swatee Naik
Department of Mathematics and Statistics
University of Nevada
Reno, NV 89557
USA