MT 855: Combinatorial Methods in Knot Theory

Spring 2013

Time/Location: MW 12:30-1:45, O'Neill 256 (modulo cancelations and Friday make-up classes)

Professor: Josh Greene

Office: Carney 264

Course description: We will survey some of the important progress and open problems in knot theory. Our point of view will be based mainly on the intersection graph technique due to Litherland and championed by Gordon and Luecke. Starting from the ground up, we will discuss basic constructions involving knots, links, and 3-manifolds: Heegaard splittings, prime decomposition, Dehn surgery, thin position, the special role of surfaces of low complexity, etc. We will then turn our attention to the intersections of surfaces in 3-manifolds that arise in cases of special interest, such as when Dehn surgery along a knot results in the 3-sphere, a lens space, or a reducible manifold. By studying the combinatorics of these intersections, we will establish many significant foundational results: the knot complement problem (Gordon-Luecke), property R (Gabai), and properties of knots with a lens space, reducible, or toroidal surgery (Gordon-Luecke, Goda-Teragaito, Baker, etc.). Along the way we will encounter many interesting constructions and relationships between questions in low-dimensional topology. We will also hint at the perspective that Heegaard Floer homology offers. The course should be accessible to anyone with a background in algebraic topology.

References: For background on the differential topology underpinning the course, I suggest the standard references M. Hirsch, Differential Topology and V. Guilleman and A. Pollack, Differential Topology. For the basics about knots and 3-manifolds that we will develop, I suggest C. McA. Gordon, Some aspects of classical knot theory, Springer Lecture Notes 685 (1978) 1-60; J. Hempel, 3-manifolds; W.B.R. Lickorish, An Introduction to Knot Theory; and especially D. Rolfsen, Knots and Links. For the core material about Dehn surgery, we will closely follow C. McA. Gordon, Dehn surgery and 3-manifolds. Low dimensional topology, IAS / PCMI Series 15 (2009) 21-71. For the proof of the knot complement theorem, I suggest the original source C. McA. Gordon and J. Luecke, Knots are determined by their complements, JAMS 2 no. 2 (1989) 371-415 and the companion C. McA. Gordon, Combinatorial methods in Dehn surgery. Lectures at KNOTS '96 (1997) 263-290. More specific /additional references will appear in the notes as they become relevant. Registered students should check the Blackboard site for ``assistance'' with references that are more difficult to track down.