  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 $G_2$ quantum invariants. Our method consists in the
study of Markov traces on a suitable tower of quotients of cubic Hecke algebras
extending Jones approach.
