Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

TL;DR

This paper explores the relationship between positive knots and multiple zeta values (MZVs), showing that certain knots correspond to irreducible MZVs up to 15 crossings through Feynman diagram analysis.

Contribution

It establishes a novel connection between positive knots and MZVs using Feynman diagrams up to 9 loops, identifying all such knots up to 15 crossings.

Findings

Number of irreducible MZVs generated by a specific polynomial.

Positive knots with up to 9 crossings correspond to MZVs.

Positive knots with more than 9 crossings are not associated with MZVs.

Abstract

It is found that the number, $M_n$, of irreducible multiple zeta values (MZVs) of weight $n$, is generated by $1-x^2-x^3=\prod_n (1-x^n)^{M_n}$. For $9\ge n\ge3$, $M_n$ enumerates positive knots with $n$ crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops.