The asymptotic behaviors of the colored Jones polynomials of the figure-eight knot, and an affine representation

Murakami, Hitoshi

doi:10.2140/agt.2025.25.3523

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

The asymptotic behaviors of the colored Jones polynomials of the figure-eight knot, and an affine representation

Hitoshi Murakami

Algebraic & Geometric Topology 25 (2025) 3523–3583

DOI: 10.2140/agt.2025.25.3523

Abstract

We study the asymptotic behavior of the $N$ -dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp (κ + 2 p π \sqrt{- 1} ∕ N)$ , where $κ : = arccosh (\frac{3}{2})$ and $p$ is a positive integer. We can prove that it grows exponentially with growth rate determined by the Chern–Simons invariant of an affine representation from the fundamental group of the knot complement to the Lie group $SL (2; ℂ)$ .

Keywords

colored Jones polynomial, figure-eight knot, volume conjecture, Chern–Simons invariant, affine representation

Mathematical Subject Classification

Primary: 57K14

Secondary: 57K10

References

Publication

Received: 23 July 2023

Revised: 3 May 2024

Accepted: 10 July 2024

Published: 1 October 2025

Authors

Hitoshi Murakami
Graduate School of Information Sciences Tohoku University Sendai Japan

Open Access made possible by participating institutions via Subscribe to Open.

Volume 25, issue 6 (2025)