Exact computation of the n-loop invariants of knots

TL;DR

This paper presents a computational method implemented in SnapPy to precisely calculate the first six coefficients of the n-loop invariants of knots, which are conjecturally linked to the asymptotic behavior of the Kashaev invariant.

Contribution

The authors develop and demonstrate a practical computational approach to determine the initial coefficients of the n-loop invariants for knots, advancing the understanding of their structure.

Findings

Successfully computed the first six coefficients of the invariants.

Provided examples illustrating the computation method.

Confirmed the arithmetical properties of the coefficients.

Abstract

The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in $1/N$. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power series of a knot. We give examples that illustrate our method and its results.