The Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond

We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.