Parameterized Complexity of Quantum Invariants of Knots

Explore the parameterized complexity of quantum invariants of knots in this 50-minute lecture by Clément Maria. Delve into a general fixed parameter tractable algorithm for computing quantum invariants of knots presented by diagrams, with complexity singly exponential in the carving-width or tree-width of the knot diagram. Examine Reshetikhin-Turaev invariants derived from simple Lie algebras, including colored Jones polynomials and colored HOMFLYPT polynomial, and their connections to geometric topology conjectures. Learn about the algorithm's reliance on graphical calculus and tree embedding of low congestion. Cover topics such as defining quantum invariants, Penrose calculus, cutting wheel, algorithm structure, leaves, configurations, and merging techniques.