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A unification for the ADO and colored Jones polynomials of a knot

Abstract : The goal of this thesis is the study and unification of the ADO and colored Jones knot invariants. The latter is a family of polynomials deeply studied and already unified by Habiro. They are at the core of several constructions (3 manifolds invariants, semi simple TQFT, ...). The ADO polynomials are more recent and are of interest in the study of non semi simple 3 manifold invariants and TQFT. This work will allow us to construct an invariant unifying both families, to show that those two are in fact equivalent, and transfer some known properties of the colored Jones polynomials to the ADO. In particular, we will present some holonomic properties and finite type invariant expansion. We first show how to build an element unifying the ADO polynomials by looking up their differences. We need an Habiro type completion ring in order to allow evaluation at roots of unity. Then, we must show that it is a knot invariant. To do so, it suffice to show that we can obtain it from the universal quantum knot invariant and a Verma module. We can then link this work to Habiro's and use this connection in order to show that this new invariant also unify the colored Jones polynomials. Moreover, we can show that the unified invariant is solely determined by the colored Jones polynomials. This uniqueness will allow us to deduce a bunch of properties on the unified and ADO invariants. First, we can show an holonomic property of the ADO, meaning that there is an operator polynomial canceling every ADO polynomial. This polynomial is the quantum A polynomial. Another nice property is the expansion into a series with topological finite type invariant as coefficients. Finally, the braid representation approach will open the path to homological interpretation of the invariants. The link case will also be discussed, shedding some light on the obstructions to such unification, and presenting some partial results.
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Submitted on : Wednesday, May 11, 2022 - 11:47:31 AM
Last modification on : Monday, July 4, 2022 - 9:25:55 AM


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  • HAL Id : tel-03664751, version 1


Sonny Willetts. A unification for the ADO and colored Jones polynomials of a knot. Geometric Topology [math.GT]. Université Paul Sabatier - Toulouse III, 2021. English. ⟨NNT : 2021TOU30198⟩. ⟨tel-03664751⟩



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