The Alexander polynomial as a universal invariant

  • Rinat Kashaev

    Université de Genève, Switzerland
The Alexander polynomial as a universal invariant cover

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Let be the polynomial ring 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 matrices of the form . We prove that the universal invariant of a long knot associated with is the reciprocal of the canonically normalised Alexander polynomial . Given the fact that admits a -deformation 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.