High distance knots in closed 3-manifolds

Details High distance knots in closed 3-manifolds High distance knots in closed 3-manifolds

2009-11-18

arxiv.org/abs/0911.3675v2

Mathematics

Geometric Topology

Certain misstatements about the pair of pants decompositions constructed for reducible Heegaard splittings have been corrected

Scientific paper

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).

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