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Double branched covers of theta-curves

Jack S. Calcut

Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA

E-mail Address: jcalcut@oberlin.edu

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and

Jules R. Metcalf-Burton

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

E-mail Address: julesmb@umich.edu

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https://doi.org/10.1142/S0218216516500462Cited by:3 (Source: Crossref)

Abstract

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot $κ$ is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over $κ$ produces a prime knot. We apply this result to Kinoshita’s theta-curve.

Keywords:

AMSC: Primary: 57M12, Primary: 57M25, Secondary: 57M35, Secondary: 57Q91

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