New skein invariants of links

Abstract

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, [Formula: see text], [Formula: see text] and [Formula: see text], based on the invariants of knots, [Formula: see text], [Formula: see text] and [Formula: see text], denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants ([Formula: see text], [Formula: see text], [Formula: see text]) on sublinks of a given link [Formula: see text], obtained by partitioning [Formula: see text] into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.