The Alexander polynomial as a universal invariant

Abstract

Let ${\mathsf{B}}_1$ be the polynomial ring $\mathbb{C}[a^{\pm 1},b]$ 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 $2\times 2$ matrices of the form $\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)$. We prove that the universal invariant of a long knot $K$ associated with ${\mathsf{B}}_1$ 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-Natan and Garoufalidis, and Garoufalidis and Lê.