New skein invariants of links

Abstract

We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a 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, $H[R]$, $K[Q]$ and $D[T]$, based on the invariants of knots, $R$, $Q$ and $T$, 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 ($R$, $Q$, $T$) on sublinks of a given link $L$, obtained by partitioning $L$ into collections of sublinks.

42 pages, 14 figures. This latest version of the paper contains new and revised work on the skein theory of the new invariants and does not have the material on state summations. See version 4 or [`Skein invariants of links and their state sum models', Symmetry 9 (2017), 226, doi:10.3390/sym9100226] for a complete treatment of the state summation