The Alexander polynomial as a universal invariant

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.