HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct

TL;DR

This paper develops a HOMFLY skein theory framework using parabolic restriction and quantum representations, leading to a universal formalism that includes the Turaev coproduct and its compatibility with surface operations.

Contribution

It introduces a HOMFLY category for quantum parabolic representations and constructs a universal parabolic restriction formalism with applications to skein theory and the Turaev coproduct.

Findings

Constructed a HOMFLY version of the quantum parabolic representation category.

Developed a universal formalism for parabolic restriction functors.

Reproduced the Turaev coproduct within the skein theory framework.

Abstract

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.