An Algebra of Arcs and Knots on a Surface (Helen Wong, CMC)

The end of the previous century saw radical changes to three-dimensional topology, which arose from two completely different approaches. One breakthrough came from Bill Thurston’s introduction of hyperbolic geometry into the field. The second one came from the Vaughn Jones’s discovery of a new “quantum” invariant for knots that brought in insight and techniques from mathematical physics and non-commutative algebra. It is widely believed that the two approaches are related.

In this talk, we will focus on a point of connection introduced over five years ago by J. Roger and T. Yang.  They defined an algebra of arcs and knots on a surface that is both  obviously related to the quantum invariants and motivated by hyperbolic geometry.  We will give a overview of what’s known about this algebra of arcs and knots on a surface.  Warning: there will be lots of pictures of arcs and knots on surfaces.