Finding planar surfaces in knot- and link-manifolds
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2009-03-01
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Journal of Knot Theory and its Ramifications
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WORLD SCIENTIFIC PUBL CO PTE LTD
Melbourne Authors
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Mathematics and Statistics
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Journal Article
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Jaco, W., Rubinstein, J. H. & Sedgwick, E. (2009). Finding planar surfaces in knot- and link-manifolds. Journal of Knot Theory and Its Ramifications, 18 (3), pp.397-446. https://doi.org/10.1142/S0218216509006987.
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It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. Two similar results are obtained with added boundary conditions. Namely, given a link-manifold M, a component B of ∂M, and a slope γ on B, there is an algorithm to decide if there is an embedded punctured-disk in M with boundary γ and punctures in ∂M\B; and given a link-manifold M, a component B of ∂M, and a meridian slope μ on B, there is an algorithm to decide if there is an embedded punctured-disk with boundary a longitude on B and punctures in ∂M\B. In both cases, if there is one, the algorithm will construct one. The proofs introduce a number of new methods and differ from the classical proofs, using normal surfaces, as solutions may not be found among the fundamental solutions.
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Pure Mathematics