Knots and Quantum Physics

Knot theory was born, arguably, after an unsuccessful attempt by Kelvin to explain the variety of chemical elements in nature. After that, for about a century, the theory of knots was a subject of pure mathematics dealing with the problem of their classification. The return of knots into physics happened with the work of Jones, Witten and others, when it was shown that topological invariants of knots appear as observables (correlation functions) in certain gauge theories, dubbed topological. At approximately the same period importance of topological theories was appreciated in the novel ``topological'' phenomena in condensed matter physics. Since then, the relevance of knot theory has become evident in different models of condensed matter, quantum information theory, quantum gravity and even biology. In this talk I will introduce the subject of knot theory and place it in the context of problems of mathematical physics. I will discuss a few known applications in physics and review some more recent developments in the context of quantum information theory. I will argue that knots and topological quantum field theories provide an interesting way to understand fundamental features of quantum physics.