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Companionship of knots and the Smith conjecture


Author: Robert Myers
Journal: Trans. Amer. Math. Soc. 259 (1980), 1-32
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1980-0561820-8
MathSciNet review: 561820
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Abstract: This paper studies the Smith Conjecture in terms of H. Schubert's theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of $ {{\textbf{Z}}_p}$ on $ {S^3}$. Theorem. Every essential torus in an invariant knot space $ C(J)$ of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in $ C(J)$, this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0561820-8
Keywords: Smith Conjecture, knot, companion, periodic transformation, fibered knot, simple knot, cabled knot, doubled knot, braid, 3-manifold
Article copyright: © Copyright 1980 American Mathematical Society

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