Detecting Incompressibility of Boundary in 3-Manifolds |
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Authors: | Colin Adams |
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Institution: | (1) Department of Mathematics, Williams College, Williamstown, MA, 01267, U.S.A. |
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Abstract: | A construction is presented which can be utilized to prove incompressibility of boundary in a 3-manifold W. One constructs a new 3-manifold DW by doubling W along a subsurface in its boundary. If DW is hyperbolic, and if W has compressible boundary, then DW must have a longitude of 'length' less than 4. This can be applied to show that an arc that is a candidate for an unknotting tunnel in a 3-manifold cannot be an unknotting tunnel. It can also be used to show that a 'tubed surface' is incompressible. For knot and link complements in S
3, and an unknotting tunnel, DW is almost always hyperbolic. Empirically, this construction appears to provide a surprisingly effective procedure for demonstrating that specific arcs are not unknotting tunnels. |
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Keywords: | hyperbolic 3-manifold unknotting tunnel boundary incompressibility |
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