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Primeness of twisted knots


Author: Kimihiko Motegi
Journal: Proc. Amer. Math. Soc. 119 (1993), 979-983
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9939-1993-1181171-5
MathSciNet review: 1181171
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Abstract: Let $ V$ be a standardly embedded solid torus in $ {S^3}$ with a meridian-preferred longitude pair $ (\mu ,\lambda )$ and $ K$ a knot contained in $ V$. We assume that $ K$ is unknotted in $ {S^3}$. Let $ {f_n}$ be an orientation-preserving homeomorphism of $ V$ which sends $ \lambda $ to $ \lambda + n\mu $. Then we get a twisted knot $ {K_n} = {f_n}(K)$ in $ {S^3}$.

Primeness of twisted knots is discussed and we prove: A twisted knot $ {K_n}$ is prime if $ \vert n\vert > 5$. Moreover, $ {\{ {K_n}\} _{n \in Z}}$ contains at most five composite knots.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1181171-5
Keywords: Knot, twisting, primeness
Article copyright: © Copyright 1993 American Mathematical Society

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