The maximal Thurston-Bennequin number of a doubled knot
A Legendrian link in the 3-space \(\mathbb{R}^3\) equipped with the standard contact structure is a smooth embedding of disjoint circles into \(\mathbb{R}^3\) that is tangent to the contact structure at each point. The Thurston-Bennequin number of a Legendrian link is a Legendrian isotopy invariant that can be combinatorially computed via its front diagram. The largest value of this invariant among all Legendrian representatives of a given link \(L\) is known to be bounded above due to \textit{D. Bennequin} [Astérisque 107/108, 87--161 (1983; Zbl 0573.58022)], and this value is called the maximal Thurston-Bennequin number of the link. In [\textit{D. Fuchs} and \textit{S. Tabachnikov}, Topology 36, No.~5, 1025--1053 (1997; Zbl 0904.57006)] and [\textit{S. Tabachnikov}, Math. Res. Lett. 4, No.~1, 143--156 (1997; Zbl 0877.57001)], an upper bound of the maximal Thurston-Bennequin number of a link \(L\) in terms of the Kauffman polynomial was provided. Note that this bound is not always sharp. In the paper under review, the author proves that if the above upper bound is sharp for a given knot \(K\), then that for any twisted double of \(K\) is also sharp. For the proof of the sharpness, he uses the result of \textit{D. Rutherford} [Int. Math. Res. Not. 2006, No.~9, 15 p (2009; Zbl 1106.57012)] that provides the equivalence of the sharpness of the upper bound and the existence of a ruling of a front diagram of a Legendrian link. As a corollary, he provides formulas of positive doubles of torus knots and two bridge knots.
scientific article; zbMATH DE number 5692889
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