A lift of the colored Jones polynomial of a knot
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866914363140472832 |
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| author | Garoufalidis, Stavros Wheeler, Campbell |
| author_facet | Garoufalidis, Stavros Wheeler, Campbell |
| contents | Habiro lifted the Witten-Reshetikhin-Turaev invariant of an integer homology 3-sphere (a complex-valued function on the set of complex roots of unity) to an element of the Habiro ring. We lift the colored Jones polynomial of a knot, with Alexander polynomial $Δ(t)$, to the recently introduced Habiro ring of the étale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},Δ(t)^{-1}]$ (with Frobenius lifts $t\mapsto t^p$ for all primes $p$). This implies the existence of a loop expansion at roots of unity (confirming a conjecture of Habiro), and a lift of power series invariants of Ohtsuki for 3-manifolds with Betti number 1 to a Habiro ring. Our results have natural extensions to the skein module of a knot complement, and they suggest a natural lift of the colored Jones polynomial colored by representations of a simple Lie (super) algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01619 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A lift of the colored Jones polynomial of a knot Garoufalidis, Stavros Wheeler, Campbell Geometric Topology High Energy Physics - Theory Number Theory Habiro lifted the Witten-Reshetikhin-Turaev invariant of an integer homology 3-sphere (a complex-valued function on the set of complex roots of unity) to an element of the Habiro ring. We lift the colored Jones polynomial of a knot, with Alexander polynomial $Δ(t)$, to the recently introduced Habiro ring of the étale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},Δ(t)^{-1}]$ (with Frobenius lifts $t\mapsto t^p$ for all primes $p$). This implies the existence of a loop expansion at roots of unity (confirming a conjecture of Habiro), and a lift of power series invariants of Ohtsuki for 3-manifolds with Betti number 1 to a Habiro ring. Our results have natural extensions to the skein module of a knot complement, and they suggest a natural lift of the colored Jones polynomial colored by representations of a simple Lie (super) algebra. |
| title | A lift of the colored Jones polynomial of a knot |
| topic | Geometric Topology High Energy Physics - Theory Number Theory |
| url | https://arxiv.org/abs/2603.01619 |