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Résumé |
Coloured Jones and Alexander polynomials are quantum invariants originating in
representation theory and their geometric information is an important open problem
in quantum topology. We present a new topological perspective that unifies these
invariants through the topology of configuration spaces. First, for a fixed N, we
define new link invariants: “N th Unified Jones invariant” and “N th Unified
Alexander invariant” globalising all coloured Jones and ADO link polynomials of
(multi)-colours bounded by N. Asymptotically, Habiro defined his universal knot
invariant globalising coloured Jones polynomials by introducing the Habiro ring.
For the link case, such globalisation remained open for both sequences of
invariants.
We answer this problem coming from representation theory using topological tools.
On the representation theory side we define extensions of Habiro type rings. On
the topological side, we construct a universal Jones link invariant and a
universal Alexander link invariant. Putting these together, our universal
invariants of geometrical nature take values in the extended Habiro rings that we
construct |
