High distance knots in closed 3-manifolds

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is isomorphic to the mapping class group of (S, V,W).Comment: Certain misstatements about the pair of pants decompositions constructed for reducible Heegaard splittings have been corrected. The paper has also been restructured some to aid the exposition of the proof. Details have been provided for the end of the proof of the main theorem. And the first author's name has been change

Neighbors of knots in the Gordian graph

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.Comment: Accepted by American Mathematical Monthly. New version incorporates referee comment