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Abstract

These notes for lectures delivered for the Simons Semester at the Banach Center in Warsaw open with an accessible introduction to knot theory and quantum knot invariants. We then develop the theory of Kauffman bracket skein modules, skein algebras, and stated skein algebras. We explore their connections to character varieties, operator algebras, Floer homology, cluster algebras, quantum Teichmüller theory, and mathematical physics. Finally, we discuss generalizations to $G$-skein modules for simple Lie groups $G$, developed via Reshetikhin–Turaev theory and factorization homology. Throughout, we strive to present the material as clearly and elementarily as possible, deliberately avoiding technical details where they are not essential.