Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the defining representation from GL_m+n to GL_m x GL_n. The restriction functor, however, is not braided, and therefore there is a priori no reason for the induced linear map between the corresponding skein algebras to be multiplicative. In this talk, I will address this problem using defect skein theory and the formalism of parabolic restriction.

In the first part of the talk, I will introduce skein theory for 3-manifolds with both surface and line defects. Local relations near the defects are produced from the algebraic data of a central algebra (codimension 1) and a centred bimodule (codimension 2). Examples of such structures are provided by the formalism of parabolic restriction. In the second part of the talk, I will explain how to construct a universal version of this formalism. Finally, we will see how Turaev’s coproduct extends to the entire skein category using the previous constructions.