Ozsváth–Szabó and Rasmussen invariants of doubled knots

Livingston, Charles; Naik, Swatee

doi:10.2140/agt.2006.6.651

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Ozsváth–Szabó and Rasmussen invariants of doubled knots

Charles Livingston and Swatee Naik

Algebraic & Geometric Topology 6 (2006) 651–657

DOI: 10.2140/agt.2006.6.651

arXiv: math.GT/0505361

Abstract

Let $ν$ be any integer-valued additive knot invariant that bounds the smooth 4–genus of a knot $K$ , $| ν (K) | \leq g_{4} (K)$ , and determines the 4–ball genus of positive torus knots, $ν (T_{p, q}) = (p - 1) (q - 1) ∕ 2$ . Either of the knot concordance invariants of Ozsváth-Szabó or Rasmussen, suitably normalized, have these properties. Let $D_{\pm} (K, t)$ denote the positive or negative $t$ –twisted double of $K$ . We prove that if $ν (D_{+} (K, t)) = \pm 1$ , then $ν (D_{-} (K, t)) = 0$ . It is also shown that $ν (D_{+} (K, t)) = 1$ for all $t \leq TB (K)$ and $ν (D_{+} (K, t)) = 0$ for all $t \geq - TB (- K)$ , where $TB (K)$ denotes the Thurston-Bennequin number.

A realization result is also presented: for any $2 g \times 2 g$ Seifert matrix $A$ and integer $a$ , $| a | \leq g$ , 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

Volume 6, issue 2 (2006)