Theta Functions and Knots

This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Razvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. People with an interest in theta functions or knot theory can learn how the two are related. Those intere.

4.3.4 The algebra of linear operators on the space of theta functions and the quantum torus.

Preface; Contents; 1. Prologue; 1.1 The history of theta functions; 1.1.1 Elliptic integrals and theta functions; 1.1.2 The work of Riemann; 1.2 The linking number; 1.2.1 The definition of the linking number; 1.2.2 The Jones polynomial; 1.2.3 Computing the linking number from skein relations; 1.3 Witten's Chern-Simons theory; 2. A quantum mechanical prototype; 2.1 The quantization of a system of finitely many free one-dimensional particles; 2.1.1 The classical mechanics of finitely many free particles in a one-dimensional space; 2.1.2 The Schrodinger representation; 2.1.3 Weyl quantization.

2.2 The quantization of finitely many free one-dimensional particles via holomorphic functions2.2.1 The Segal-Bargmann quantization model; 2.2.2 The Schrodinger representation and the Weyl quantization in the holomorphic setting; 2.2.3 Holomorphic quantization in the momentum representation; 2.3 Geometric quantization; 2.3.1 Polarizations; 2.3.2 The construction of the Hilbert space using geometric quantization; 2.3.3 The Schrodinger representation from geometric considerations; 2.3.4 Passing from real to Kahler polarizations; 2.4 The Schrodinger representation as an induced representation.

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