Knot invariants from XC-structures on the Sweedler algebra are trivial

TL;DR

This paper demonstrates that XC-structures on the Sweedler algebra yield trivial knot invariants dependent only on framing, and provides examples of such structures not arising from ribbon Hopf algebras.

Contribution

It shows all XC-structures on the Sweedler algebra produce trivial invariants and introduces new XC-structures without ribbon Hopf-algebra origins.

Findings

Knot invariants from XC-structures on the Sweedler algebra are trivial.

Explicit non-ribbon XC-structures on the Sweedler algebra are constructed.

Abstract

An XC-algebra is the minimum algebraic structure needed to define a framed, oriented knot invariant and generalises Lawrence's invariant obtained from ribbon Hopf algebras. In this note, we show that the knot invariant produced by any XC-structure on the Sweedler algebra is completely determined by the framing of the knot. Furthermore, we also exhibit explicit families of XC-structures on the Sweedler algebra that do not have a ribbon Hopf-algebraic origin.