Arborescent links and modular tails

TL;DR

This work proves an explicit product formula for the tail $\Phi_L(q)$ of the colored Jones polynomial for alternating arborescent links $L$, expressing it as $\Phi_L(q)=\prod_{v\in\mathcal{V}_+} h_{w(v)+e(v)}$ when $0\notin w(\mathcal{V}_-)$. The proof leverages reduced Tait graphs and a key structural proposition showing the $+$-Tait graph decomposes into edge-connected polygons, enabling the product expression. The paper then applies the result to 2-bridge knots, Montesinos knots, and a non-Montesinos example (11a$_{250}$), providing explicit tail identities and recovering known identities in the literature. It also studies asymptotics in cases where the main theorem does not apply, providing numerical evidence that some tails are not products of $h_b$ and may not be modular, with detailed analyses for pretzel knots and the knot $8_{18}$. The work raises questions about a potential classification of alternating arborescent knots by modularity of tails and highlights connections to Volume Conjecture-type phenomena and the topology of the underlying Tait graphs.

Abstract

We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of the modularity of tails of the colored Jones polynomial for alternating knots.

Figures (40)

• Figure 1: $+$ and $-$ regions.
• Figure 2: Signed-colored diagram, $\mathcal{T}_{\pm}$ and $\mathcal{T}_{\pm}^{\prime}$ for $K=5_2$.
• Figure 4: A weighted tree for $5_2$.
• Figure 5: The two ribbons for Figure \ref{['fig:w52']}.
• Figure 6: The surface for the two plumbed ribbons and $5_2$.

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Definition 2.1
• Remark 2.2
• Example 2.3
• Remark 2.4
• Example 2.5
• Proposition 3.1
• proof