Introduction to Knot Theory

Explore the fundamental concepts of knot theory through this comprehensive lecture series delivered at the Park City Mathematics Institute. Delve into the mathematical study of knots and links, learning how to classify and distinguish different types of knots using various invariants and techniques. Master the basic definitions and terminology of knot theory, including knot diagrams, Reidemeister moves, and knot equivalence. Discover how to compute and apply knot invariants such as the Alexander polynomial, Jones polynomial, and linking numbers to analyze knot properties. Examine the relationship between knots and topology, understanding how knot theory connects to broader mathematical concepts in algebraic topology and geometry. Investigate advanced topics including knot homology theories and their applications in modern mathematics. Gain hands-on experience working with knot diagrams and performing calculations that demonstrate key theoretical principles. Build a solid foundation in this fascinating area of mathematics that has applications ranging from DNA research to quantum physics, presented through clear explanations and systematic development of the subject matter across multiple detailed sessions.