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The notion of Hempel distance has become a useful tool in measuring how complicated a Heegaard splitting of a particular 3-manifold is and how complex the incompressible surfaces it contains may be. If one is able to put a bound on this distance based on topological features of such a surface, then one can use this bound in reverse to say that "high distance implies high complexity" - typically, this amounts to saying that "high distance implies high genus."
In this talk we define a notion of distance as it applies to specific circular decompositions of certain types of knots in the 3-sphere - the so-called circular distance of the decompositions. For an incompressible surface in the knot complement, we show that there is an upper bound for the circular distance that is linear in the genus of this surface. In particular, we surmise that high distance implies that the complement is atoroidal.
This is joint work with Patrick Weed.