Advances in Knot Theory

This volume contains the proceedings of the Special Session on Knot Theory and Applications, which was held during the 2nd International AMS-UMI Joint Meeting, July 23-26, 2025, in Palermo, Italy. Knots are ubiquitous-from closed loops of DNA and tangled polymers to the vortex filaments of turbulent fluids. Knot theory provides a language to understand these intricate patterns, and in recent years it has become one of the most dynamic branches of modern mathematics. This volume brings together an international group of leading researchers. The contributions in the volume concern the following topics: geometric insights into knots and links and novel approaches to braid theory and 4-manifolds, new diagrammatic invariants, and connections to physics and materials science. Alongside classical theory, readers will find generalizations such as pseudo-knots, welded knots, virtual knots, knotoids, and other structures that expand the boundaries of knot theory. Whether you are a specialist in topology or simply curious about how mathematics untangles complexity, this volume offers a window into the living landscape of knot theory today.