The Alexander polynomial as a universal invariant

TL;DR

This paper demonstrates that the Alexander polynomial can be viewed as a universal invariant derived from a specific Hopf algebra, linking it to quantum invariants and providing a new conceptual perspective.

Contribution

It establishes that the Alexander polynomial is the universal invariant associated to a particular Hopf algebra, connecting classical knot invariants with quantum group deformations.

Findings

Universal invariant of a long knot equals the reciprocal of the Alexander polynomial.

Provides a conceptual interpretation of the Melvin–Morton–Rozansky conjecture.

Links the Alexander polynomial to quantum group deformations and colored Jones polynomials.

Abstract

Let $\mathsf{B}_1$ be the polynomial ring $\mathbb{C}[a^{\pm1},b]$ with the structure of a complex Hopf algebra induced from its interpretation as the algebra of regular functions on the affine linear algebraic group of complex invertible upper triangular 2-by-2 matrices of the form $\left( \begin{smallmatrix} a&b\\0&1 \end{smallmatrix}\right)$. We prove that the universal invariant of a long knot $K$ associated to $\mathsf{B}_1$ is the reciprocal of the canonically normalised Alexander polynomial $\Delta_K(a)$. Given the fact that $\mathsf{B}_1$ admits a $q$-deformation $\mathsf{B}_q$ which underlies the (coloured) Jones polynomials, our result provides another conceptual interpretation for the Melvin--Morton--Rozansky conjecture proven by Bar-Nathan and Garoufalidis, and Garoufalidis and L\^e.