Skein Theory and the Quantum A-Polynomial - Part 1

Explore the foundational concepts of skein theory and its connection to the quantum A-polynomial in this lecture from the 2025 SCGCS School on Topological Symmetries and Defects in Quantum Field Theory. Delve into the mathematical framework that bridges knot theory and quantum invariants, examining how skein relations provide a systematic approach to studying polynomial invariants of knots and links. Learn about the construction and properties of the quantum A-polynomial, a powerful tool that encodes quantum geometric information about knot complements and their representations. Discover the interplay between classical and quantum aspects of knot theory, including how skein algebras emerge from quantum deformations of coordinate rings and their role in understanding the geometric structure of moduli spaces. Gain insights into the theoretical foundations that connect topological quantum field theory with the algebraic geometry of character varieties, providing essential background for understanding modern developments in quantum topology and its applications to quantum field theory.