POLYNOMIAL INVARIANTS OF LINKS SATISFYING CUBIC SKEIN RELATIONS

POLYNOMIAL INVARIANTS OF LINKS SATISFYING CUBIC SKEIN RELATIONS PAOLO BELLINGERI AND LOUIS FUNAR Abstract. The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G2 quantum invariants. Our method consists in the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach. 1. Introduction 1.1. Preliminaries. J.Conway showed that the Alexander polynomial of a knot, when suitably normalized, satisfies the following skein relation: ? ( ) ?? ( ) = (t?1/2 ? t1/2)? ? ? ? ? Given a knot diagram one can always change some of the crossings such that the modified diagram represents the unknot. Therefore one can use the skein relation for a recursive computation of ?, although this algorithm is rather time consuming, since it is exponential. In the mid eighties V.Jones discovered another invariant verifying a different but quite similar skein relation, namely: t?1V ( ) ? t V ( ) = (t?1/2 ? t1/2)V ? ? ? ? , which was further generalized to a 2-variable invariant by replacing the factor (t1/2 ? t?1/2) with a new variable x.

• skein relation


• equation satisfied


• relation corresponding


• polynomial invariants


• ?2?2 ?


• group representations


• algebra associated


• satisfying cubic


• hecke algebra