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

Jorge Becerra

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.

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

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.
Paper Structure (4 sections, 10 theorems, 51 equations)

This paper contains 4 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Commutative XC-algebras
  3. The Sweedler algebra
  4. Computations

Key Result

Theorem 1.1

Any XC-algebra structure on the Sweedler algebra produces a framed knot invariant that only depends on the framing: In particular, this invariant is trivial for any 0-framed knot.

Theorems & Definitions (23)

  • Theorem 1.1: \ref{['thm:main']}
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • ...and 13 more