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Explore the intricate connections between knot theory and symplectic topology in this conference talk from the IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. Delve into how restrictions on knot types of periodic Reeb orbits, imposed by dynamical convexity assumptions, influence 4-dimensional symplectic topology through collaborative research with Pedro Salomão, Richard Siefring, Michael Hutchings, and Vinicius Ramos. Examine the properties of dynamically convex star-shaped domains in 4-dimensional symplectic vector spaces, focusing on how the minimal action among periodic Reeb orbits in the boundary that are unknotted with self-linking number -1 (Hopf orbits) satisfies normalized symplectic capacity axioms. Discover why this minimal action equals the cylindrical capacity and learn how Edtmair's complementary results establish this equality. Understand the connection to the first ECH capacity and how this relationship provides a purely symplectic geometric explanation without requiring Seiberg-Witten theory. Investigate why certain transverse knot types in the standard contact 3-sphere cannot be realized as periodic Reeb orbits of dynamically convex contact forms, gaining insights into the fundamental constraints that govern the intersection of knot theory and symplectic geometry.
