Georgia Institute of Technology College of Sciences

This minicourse provides a friendly, step-by-step introduction to the Kontsevich integral. We begin by demystifying the formula and its construction, showing how it serves as a far-reaching generalization of the classical Gauss linking integral. To establish the invariance of the Kontsevich integral, we explore the holonomy of the Knizhnik–Zamolodchikov (KZ) connection on configuration spaces, utilizing the framework of Chen’s iterated integrals. We will then discuss the universality of the Kontsevich integral for both finite-type (Vassiliev) and quantum invariants, culminating in a concrete combinatorial formula expressed through Drinfeld’s associators. Time permitting, we will conclude by constructing the LMO invariant, demonstrating how it functions as a 3-manifold analog of the Kontsevich integral.