Universal quantum knot invariants

Quantum topology is the branch of mathematics that studies objects from the realm of low-dimensional topology, such as knots, surfaces or 3-manifolds, using algebraic structures that arose in the 1980s inspired by ideas from theoretical physics. This fascinating new area of topology combines various fields of mathematics, including monoidal category theory, Hopf algebras, representation theory, Lie algebras, topological quantum field theories, algebraic topology, etc., all revolving around the so-called quantum knot invariants.

This book focuses on two such invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the Reshetikhin-Turaev invariants coming from the representation theory of the Hopf algebra— and the Kontsevich invariant —which is universal among finte type invariants—. More precisely, Gaussian calculus techniques developed by Bar-Natan and van der Veen are exploited to study a family of knot polynomial invariants that determines the universal invariant subject to a specific algebra, and to investigate how the most elementary polynomial of this collection is closely related to the so-called 2-loop polynomial of the knot, an invariant that encodes a certain part of its Kontsevich invariant.

Although this book covers several topics of current research, it has been written entirely in a self-contained way, so that it can be accessible to a broad audience.