The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces and of group presentations.Beginning with a simple diagrammatic approach to the study of knots, reflecting the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent advances in our understanding to areas of current research. Topics included are straightforward introductions to topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. These topics combine into a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces.Both as an introduction to several areas of prime importance to the development of pure mathematics today, and as an account of pure mathematics in action in an unusual context, this book presents novel challenges to students and other interested readers.