In this thesis a HOMFLY polynomial is found for knots and links in a lens space L(p, q). Further study of this polynomial invariant finds a relationship with the classical invariants of Legendrian and transverse links, when L(p, q) is endowed with a universally tight contact structure. In fact certain criteria are found which, if satisfied by any numerical invariant of links in L(p, q), guarantee that the invariant fits into a Bennequin type inequality. A linear function of the degree of the HOMFLY polynomial is then shown to satisfy these criteria. A corollary is that certain "simple" Legendrian and transverse realizations of knots admitting grid number one diagrams maximize the classical invariants in their knot type. In order to obtain the above results, formulae are found for computing the classical invariants of Legendrian and transverse links from a toroidal front projection. Having these formulae, and known results about fibered links that support a given contact structure, it is found whether the duals of some families of Berge knots support the universally tight contact structure.
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