A refined functorial universal tangle invariant

TL;DR

This paper introduces a refined, canonical functorial universal tangle invariant that enhances existing invariants by preserving additional structures and is constructed using flexible XC-algebras, unifying various algebraic frameworks.

Contribution

It constructs a strict monoidal functor refining the Kerler-Kauffman-Radford invariant, preserving braiding, twist, and open trace, using XC-algebras.

Findings

Defines a canonical functor encoding the universal invariant.

Refines the Kerler-Kauffman-Radford invariant with additional structure preservation.

Uses XC-algebras to unify ribbon Hopf algebras and endomorphism algebras.

Abstract

The universal invariant with respect to a given ribbon Hopf algebra is a tangle invariant that dominates all the Reshetikhin-Turaev invariants built from the representation theory of the algebra. We construct a canonical strict monoidal functor that encodes the universal invariant of upwards tangles and refines the Kerler-Kauffman-Radford functorial invariant. Moreover, this functor preserves the braiding, twist and the open trace, the latter being a mild modification of Joyal-Street-Verity's notion of trace in a balanced category. We construct this functor using the more flexible XC-algebras, a class which contains both ribbon Hopf algebras and endomorphism algebras of representation of these.