2012/09/02 by Blair, Ryan, Campisi, Marion, Johnson, Jesse +2
#57M25 #57M27 #57M50 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1209.0197
We show that the distance of a link K with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of Seifert genus, or indeed by the genus of (almost) any essential surface or Heegaard surface in the surgered manifold.