English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto\mathbf{Phi}_\a(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in$\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundaryparabolic $SL_2(\mathbb{C})$ representations of the fundamental group of theknot. These matrix invariants have a rich structure: (a) their$(\sigma_0,\sigma_1)$ entry, where $\sigma_0$ is the trivial and $\sigma_1$ thegeometric representation, is the power series expansion of the Kashaevinvariant of the knot around the root of unity $e^{2 \pi i \alpha}$ as anelement of the Habiro ring, and the remaining entries belong to generalizedHabiro rings of number fields; (b) the first column is given by theperturbative power series of Dimofte--Garoufalidis; (c)~the columns of$\mathbf{Phi}$ are fundamental solutions of a linear $q$-difference equation;(d)~the matrix defines an $SL_2(\mathbb{Z})$-cocycle $W_{\gamma}$ inmatrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smoothfunction on $\mathbb{R}$ and even to holomorphic functions on suitable complexcut planes, lifting the factorially divergent series $\mathbf{Phi}(h)$ toactual functions. The two invariants $\mathbf{Phi}$ and $W_{\gamma}$ arerelated by a refined quantum modularity conjecture which we illustrate indetail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$pretzel knots. This paper has two sequels, one giving a different realizationof our invariant as a matrix of convergent $q$-series with integer coefficientsand the other studying its Habiro-like arithmetic properties in more depth.

Knots, perturbative series and quantum modularity