Park City Mathematics Institute Research Course-Low-Dimensional Topology Project Abstract

Research Course-Low-Dimensional Topology

Authors: Brian Hopkins, Robin Dobashi, Kevin Ji, Westley Knight, Joshua Lopez, Katherine Maschmeyer, Patrick Mawn, Lauren Nowak

 

This teacher group participated in the Undergraduate Summer School course "Low-dimensional Topology" led by Dan Mathews (week 1) and Jessica Purcell (weeks 2 and 3), both of Monash University in Melbourne, Australia. The group attended the 1pm lecture and then spent an hour debriefing and working on problem sets. The course proceeded roughly with an additional dimension each week: knots (1-manifolds), surfaces (2-manifolds), then 3-manifolds. An important concept throughout the course was an invariant; here, a mathematical object that helps distinguish manifolds. We discussed various polynomial invariants for knots, computing bracket and Jones polynomials for several examples. For surfaces, we learned about Dehn twists and worked through Lickorish's 1962 proof that Dehn twists generate the mapping class group, an invariant for surfaces. Finally, we showed that any 3-manifold can be described by each of the techniques Heegaard splittings and Dehn fillings.

 

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