Estimates for the Bennequin number of Legendrian links from state models for knot polynomials
Consider a 3-dimensional manifold \(M\) equipped with a contact structure, that is, a completely nonintegrable 2-dimensional distribution. Then a Legendrian knot (resp. link) in \(M\) is a knot (resp. link) which is always tangent to the contact distribution. One fundamental paper for the study of Legendrian knots is the paper by \textit{D. Bennequin} [Astérisque 107/108, 87-161 (1983; Zbl 0573.58022)], where a famous inequality is proved stating that in the case of the 3-space \(\mathbb{R}^3\) equipped with its standard contact structure, the ``Bennequin number'' of a Legendrian knot is less than twice its genus. Other Bennequin number estimates have been derived later on, in particular by D. Fuchs and the author using previous work of J. Franks and R. Williams and of H. Morton on the Homfly polynomial. The estimates were then extended by S. Chmutov and V. Goryunov (titles of all these papers are provided in the paper that is reviewed). These new inequalities state that the Bennequin number of a Legendrian link in the standard contact structure is bounded above by the least degree of the Homfly polynomial and by the least degree of the Kauffman polynomial. In this paper, the author deduces these Bennequin number inequalities from the Yang-Baxter state models for knot polynomials.
scientific article; zbMATH DE number 1019793
scientific article