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 welldefinedness 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.2010 Mathematics Subject Classification. 57M27, 57M25.

A new two-variable generalization of the Jones polynomial

“…Comparing (6.12) and (6.13) to Remarks 8 and we deduce that {{L}} coincides with the invariant V [V ] in the notation of [27] and, thus, we recover the invariant θ(q, E). Hence we have proved the following:…”

“…The cornerstone of this construction is the 3-variable generalization Θ of the invariants Θ d constructed in [4] and which will be presented here. The well-definedness of Θ has been proved both algebraically [4] and skein-theoretically [27,28]. We will present all approaches, as they are all very informative.…”

“…Consequently, new state sum models for the new invariants are defined in [28]. In the notation of [27] we have Θ(q, λ, E) = P [P ](ζ, a, E) for a = 1/ √ λ and ζ = q − q −1 .…”

“…An alternative, purely skein-theoretical approach, was also given in [4]. A self-contained proof of the well-definedness of Θ via this approach has been recently given by L. Kauffman and the second author [27], while in [28] they also construct new state sum models related to Θ. Furthermore, a closed combinatorial formula for the invariant Θ has been proved by W.B.R.…”

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

“…In fact, the same skein relation as the one satisfied by the Jones polynomial holds, but only on crossings between different components; we call these crossings mixed crossings. Using a recursive proof method developed originally by Lickorish-Millett [12] and adapted for θ by Kauffman-Lambropoulou [9], it can be shown that calculating the value of θ on a link L amounts to calculating the value of θ on links that are unions of unlinked knots and are obtained via the skein relation. More specifically, a series of switchings and smoothings of mixed crossings transforms the initial link to a family of unions of unlinked knots, called descending stacks.…”