Knot Theory: The Quadratic Linking Degree

This lecture introduces the concept of the quadratic linking degree within the framework of knot theory. It begins with fundamental definitions, explaining knots as closed topological subspaces of the 3-sphere and introducing oriented knots and links. The instructor discusses the linking number, which quantifies how many times one component of a link winds around another. The formal definition of the linking number is presented, alongside the concept of Seifert surfaces and their intersections. The lecture progresses to algebraic geometry, defining oriented links with two components and their orientation classes. The instructor elaborates on the Hopf link and the Solomon link, illustrating their properties and linking numbers. The discussion extends to Chow groups and intersection theory, addressing challenges in defining boundary maps and orientations. The quadratic linking degree is defined as an analog of the linking number, with examples demonstrating its computation. The lecture concludes with insights into invariants and their significance in distinguishing between different links, emphasizing the complexity of the topic.