This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its modifications and generalizations, including a mathematical treatment of Jones–Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of $3$-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones–Witten invariants are constructed via the rigorous skein algebra approach.

Unlike several other books, where the introduction of all of these invariants requires the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites in this book are minimal. Numerous figures and problems make it suitable as a course text and for self-study.

Readership

Undergraduates, graduate students, research mathematicians and physicists interested in differential geometry.