# The Alexander polynomial as a universal invariant

Rinat Kashaev

Rinat Kashaev

**21/7/20**Published in : arXiv:2007.11036

Let 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 B_1 is the reciprocal of the canonically normalised Alexander polynomial

\Delta_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-Nathan and Garoufalidis, and Garoufalidis and Lê.