A Chapter in Physical Mathematics: Theory of Knots in the Sciences

In the last twenty years a body of mathematics has evolved with strong direct input from theoretical physics, for example from classical and quantum field theories, statistical mechanics and string theory. In particular, in the geometry and topology of low dimensional manifolds (i.e. manifolds of dimensions 2, 3 and 4) we have seen new results, some of them quite surprising, as well as new ways of looking at known results. Donaldson’s work based on his study of the solution space of the Yang-Mills equations, Monopole equations of Seiberg-Witten, Floer homology, quantum groups and topological quantum field theoretical interpretation of the Jones polynomial and other knot invariants are some of the examples of this development. Donaldson, Jones and Witten have received Fields medals for their work [4]. We think the name “Physical Mathematics” is appropriate to describe this new, exciting and fast growing area of mathematics. Recent developments in knot theory make it an important chapter in “Physical Mathematics”. Untill the early 1980s it was an area in the backwaters of topology. Now it is a very active area of research with its own journal.