Knot Theory is a lively exposition of the mathematics of knotting, written to illustrate its foundation as well as the remarkable breadth of techniques it employs—combinatorial, algebraic, and geometric. Over the last century, knot theory has progressed from a study based largely on intuition and conjecture into one of the most active areas of today’s mathematical investigation. This book is an accessible presentation of the results of that progress.
The development of knot theory has taken place within the context of the growth of topology: Knot Theory illuminates that context. The classification of surfaces, one of the major achievements in topology, is described and then applied to prove the existence of prime decompositions of knots. The interplay between topology, and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and basic group theory are introduced to study the properties of knots, including the unknotting number, the braid index, and the bridge number. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.
No training beyond linear algebra is required. Livingston guides the reader through a general survey of knot theory and then demonstrates the application of the basic results to sophisticated problems in knotting. Symmetry, one of the most beautiful facets of knot theory, is described in detail. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject, the Conway, Jones, and Kauffman polynomials.
1. A Century of Knot Theory
2. What is a Knot?
3. Combinatorial Techniques
4. Geometric Techniques
5. Algebraic Techniques
6. Geometry, Algebra, and the Alexander Polynomial
7. Numerical Invariants
8. Symmetries of Knots
9. High-Dimensional Knot Theory
10. New Combinatorial Techniques
Appendix 1. Knot Table
Appendix 2. Alexander Polynomials
Charles Livingston did his undergraduate work at MIT, and received his Ph.D. from the University of California, Berkeley in 1980. He was a G. C. Evans Instructor at Rice University for one year before joining the faculty of Indiana University, where he is now a Professor of Mathematics. He served as the Director of Undergraduate Studies at Indiana University from 1987-1989 and continues to be actively involved with the undergraduate program, both in the department and the university. He is a popular lecturer and advisor to graduate students, and served for many years as the advisor to the Graduate Mathematics Club. In 1993, he won the Rothrock Teaching Award which is given annually to the outstanding teacher in the Mathematics Department.
His research has focused on fundamental questions in low-dimensional topology, including work on 3-manifolds, surfaces, and knot theory. Some of his recent work has focused on intriguing connections between knot theory and number theory.