Character Varieties of Tangles and Singular Instanton Homology

Explore character varieties of tangles and singular instanton homology in this 56-minute lecture from the Western Hemisphere Virtual Symplectic Seminar. Delve into Kai Smith's research on using Lagrangian Floer Theory to study knots, following the program initiated by Hedden, Herald, and Kirk. Learn about decomposing knots, applying perturbations, and utilizing character varieties to obtain Lagrangian pairs in the pillowcase space. Discover how these Lagrangians' intersections can bound singular instanton homology of knots. Examine Smith's work on finding Lagrangians for tangle sums and understand why Lagrangian Floer homology alone is not a knot invariant. Investigate topics such as SU(2) character varieties, the 4-punctured sphere, perturbations, and the correct differential. Gain insights into the challenges of deriving knot invariants from Lagrangians and the need for more sophisticated methods in this field of study.