Knot polynomials for twist satellites

Abstract

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in those of the original knot, where contributing are adjoint and other representations from the “E8-sector”, what makes the story closely related to Vogel's universality. We also point out a problem with lifting the decomposition rule to the level of superpolynomials — it looks like such rule, if any, should be different for positive and negative twistings.