Maximal Thurston-Bennequin numbers of alternating links (Q2502970)

A Legendrian link is a disjoint union of \(C^{\infty}\) imbedded circles in \(\mathbb R^3\) with the standard contact structure, so that the circles are everywhere tangent to the contact distribution (the standard contact structure is the one given by the \(1\)-form \(dz-y\,dx\)). The Thurston-Bennequin number of a Legendrian link \(L\) is denoted \(\mathbf {tb}(L)\). The maximum of \(\mathbf {tb}(L)\) over all Legendrian links ambient isotopic through Legendrian links to \(L\) is denoted TB\((L)\). Letting \(F_{(x,y)}(L)\) denote the Kauffmann polynomial, the author shows that TB\((L)=-\max-\deg_{x} F{(x,y)}-1\) when \(L\) is alternating. This makes sharp for these links the previously known upper bound.