# Deep Learning the Hyperbolic Volume of a Knot

## TL;DR

This paper demonstrates that a deep neural network can accurately predict the hyperbolic volume of a knot directly from its Jones polynomial, suggesting a closer link than previously understood.

## Contribution

It introduces a neural network approach to estimate hyperbolic volume from the Jones polynomial, revealing a potential direct relationship.

## Key findings

- Neural network predicts volume with 97.6% accuracy.
- Training on 10% of data yields robust predictions.
- Supports a direct connection between Jones polynomial and hyperbolic volume.

## Abstract

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ can be recovered directly from the original Jones polynomial ($N = 2$). In this report we use a deep neural network to approximate $\text{Vol}(K)$ from the Jones polynomial. Our network is robust and correctly predicts the volume with $97.6\%$ accuracy when training on $10\%$ of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial.