Algebraic & Geometric Topology

Ozsváth–Szabó and Rasmussen invariants of doubled knots

Charles Livingston and Swatee Naik

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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.

Article information

Algebr. Geom. Topol., Volume 6, Number 2 (2006), 651-657.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

doubled knot Ozsvath-Szabo invariant Rasmussen invariant


Livingston, Charles; Naik, Swatee. Ozsváth–Szabó and Rasmussen invariants of doubled knots. Algebr. Geom. Topol. 6 (2006), no. 2, 651--657. doi:10.2140/agt.2006.6.651.

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