Geodesic
Ozsváth–Szabó and Rasmussen invariants of doubled knots
Livingston, Charles1 ; Naik, Swatee2
1Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
2Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 2, pp. 651-657
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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) = (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.

DOI : 10.2140/agt.2006.6.651
Keywords: doubled knot, Ozsvath-Szabo invariant, Rasmussen invariant

Livingston, Charles  1   ; Naik, Swatee  2

1 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
2 Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA 
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Livingston, Charles; Naik, Swatee. Ozsváth–Szabó and Rasmussen invariants of doubled knots. Algebraic and Geometric Topology, Tome 6 (2006) no. 2, pp. 651-657. doi: 10.2140/agt.2006.6.651

[1] D Bar-Natan, The Knot Atlas

[2] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)", Astérisque 107, Soc. Math. France (1983) 87

[3] E Eftekhary, Longitude Floer homology and the Whitehead double,

[4] M Hedden, P Ording, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal,

[5] C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735

[6] L L Ng, Maximal Thurston–Bennequin number of two-bridge links, Algebr. Geom. Topol. 1 (2001) 427

[7] B Owens, S Strle, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal,

[8] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615

[9] O Plamenevskaya, Bounds for the Thurston–Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004) 399

[10] J A Rasmussen, Khovanov homology and the slice genus,

[11] L Rudolph, Constructions of quasipositive knots and links I, from: "Knots, braids and singularities (Plans-sur-Bex, 1982)", Monogr. Enseign. Math. 31, Enseignement Math. (1983) 233

[12] A Shumakovitch, Rasmussen invariant, Slice-Bennequin inequality, and sliceness of knots,

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