On Denominators of the Kontsevich Integral and the Universal Perturbative Invariant of 3-Manifolds

TL;DR

This paper investigates the denominators of the Kontsevich integral and universal perturbative invariants of 3-manifolds, providing bounds and divisibility properties that clarify their integrality and prime factorization characteristics.

Contribution

It establishes explicit divisibility bounds for the denominators of the Kontsevich integral and universal perturbative invariants, advancing understanding of their integrality properties.

Findings

Denominator of degree n Kontsevich integral divides (2!3!... n!)^4(n+1)!

Denominator of degree n universal invariant not divisible by primes > 2n+1

Provides new bounds on denominators related to 3-manifold invariants

Abstract

The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree $n$ part of the Kontsevich integral of any knot or link is a divisor of $(2!3!... n!)^4(n+1)!$. We also show that the denominator of of the degree $n$ part of the universal perturbative invariant of homology 3-spheres is not divisible by any prime greater than $2n+1$.