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Explore advanced concepts in skein theory and the quantum A-polynomial in this 59-minute lecture delivered by J. Brown from the University of Edinburgh as part of the 2025 SCGCS School on Topological Symmetries and Defects in Quantum Field Theory. Delve into the mathematical foundations that connect knot theory with quantum field theory, examining how skein relations provide algebraic tools for studying knots and links. Investigate the quantum A-polynomial as a powerful invariant that encodes geometric and topological information about knot complements. Build upon previous parts of this lecture series to understand the deeper connections between classical topology and quantum invariants, with applications to low-dimensional topology and mathematical physics. Gain insights into how these theoretical frameworks contribute to our understanding of topological symmetries and defects in quantum field theory contexts.
