Integrality of Kauffman brackets of trivalent graphs

TL;DR

This paper demonstrates that Kauffman brackets of colored trivalent graphs can be renormalized into Laurent polynomials with integer coefficients, and explores their relations, identities, and integrality properties within quantum topology.

Contribution

It introduces a method to renormalize Kauffman brackets into integer Laurent polynomials and compares different state-sum models, providing new proofs of quantum $6j$-symbol identities.

Findings

Kauffman brackets can be renormalized to integer Laurent polynomials

Shadow-state sums and R-matrix state-sums are equivalent

Standard identities of quantum $6j$-symbols are proven with short proofs

Abstract

We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials depending only on the abstract combinatorial structure of the graph. Then we compare the shadow-state sums and the state-sums based on $R$-matrices and Clebsch-Gordan symbols, reprove their equivalence and comment on the integrality of the weight of the states. We also provide short proofs of most of the standard identities satisfied by quantum $6j$-symbols of $U_q(sl_2)$.