# The Alexander polynomial as a universal invariant

### Rinat Kashaev

Université de Genève, Switzerland

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## Abstract

Let $B_{1}$ be the polynomial ring $C[a_{±1},b]$ 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 $2×2$ matrices of the form $(a0 b1 )$. We prove that the universal invariant of a long knot $K$ associated with $B_{1}$ is the reciprocal of the canonically normalised Alexander polynomial $Δ_{K}(a)$. Given the fact that $B_{1}$ admits a $q$-deformation $B_{q}$ 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.