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-manifolds with planar presentations and the width of satellite knots

Authors:	Martin Scharlemann Jennifer Schultens

Institution:	Department of Mathematics, University of California, Santa Barbara, California 93106 ; Department of Mathematics, University of California, Davis, California 95616

Abstract:	We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of $S^{3}$ is a motivating example. To we associate a connectivity graph $\Gamma$ . For $M \subset S^{3}$ , $\Gamma$ is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of $S^{3} - M$ is a tree, then there is a level-preserving reimbedding of so that $S^{3} - M$ is a connected sum of handlebodies. Corollary. $\bullet$ The width of a satellite knot is no less than the width of its pattern knot and so $\bullet$ $w(K_{1} \char93 K_{2}) \geq max(w(K_{1}), w(K_{2}))$ .

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