Abstract
Quantum topology is the branch of mathematics that studies objects from the realm of lowdimensional topology, such as knots, surfaces or 3manifolds, 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 socalled quantum knot invariants.
This book focuses on two such invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the ReshetikhinTuraev 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 BarNatan 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 socalled 2loop 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 selfcontained way, so that it can be accessible to a broad audience.
This book focuses on two such invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the ReshetikhinTuraev 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 BarNatan 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 socalled 2loop 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 selfcontained way, so that it can be accessible to a broad audience.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  31May2024 
Place of Publication  [Groningen] 
Publisher  
Print ISBNs  9789493330665 
DOIs  
Publication status  Published  2024 