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2020 The dihedral genus of a knot
Patricia Cahn, Alexandra Kjuchukova
Algebr. Geom. Topol. 20(4): 1939-1963 (2020). DOI: 10.2140/agt.2020.20.1939

Abstract

Let KS3 be a Fox p–colored knot and assume K bounds a locally flat surface SB4 over which the given p–coloring extends. This coloring of S induces a dihedral branched cover XS4. Its branching set is a closed surface embedded in S4 locally flatly away from one singularity whose link is K. When S is homotopy ribbon and X a definite four-manifold, a condition relating the signature of X and the Murasugi signature of K guarantees that S in fact realizes the four-genus of K. We exhibit an infinite family of knots Km with this property, each with a Fox 3–colored surface of minimal genus m. As a consequence, we classify the signatures of manifolds X which arise as dihedral covers of S4 in the above sense.

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Patricia Cahn. Alexandra Kjuchukova. "The dihedral genus of a knot." Algebr. Geom. Topol. 20 (4) 1939 - 1963, 2020. https://doi.org/10.2140/agt.2020.20.1939

Information

Received: 29 December 2018; Revised: 24 July 2019; Accepted: 25 October 2019; Published: 2020
First available in Project Euclid: 1 August 2020

zbMATH: 07226708
MathSciNet: MR4127087
Digital Object Identifier: 10.2140/agt.2020.20.1939

Subjects:
Primary: 57M12, 57M25, 57Q60

Rights: Copyright © 2020 Mathematical Sciences Publishers

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