HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct

We define a HOMFLY version of the category ${\text{Rep}}_q\text{P}$ of quantum representations of a parabolic subgroup $\text{P}\subseteq {\text{GL}}_{m+n}$ of block triangular matrices. Alongside this category, we construct functors that interpolate the usual restriction functors between ${\text{GL}}_{m+n}$, $\text{P}$ and the subgroup $\text{L}\subseteq {\text{GL}}_{m+n}$ of block-diagonal matrices, yielding a universal version of the formalism of parabolic restriction. Based on this formalism, we construct central algebras and centred bimodules which serve as algebraic ingredients for defining a skein theory on $3$-manifolds with surface and line defects. We recover the Turaev coproduct on the HOMFLY skein algebra as a particular instance of this theory. In particular, this coproduct is compatible with the cutting and gluing of surfaces.