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ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

Part of: Special aspects of infinite or finite groups Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  15 May 2020

MARK GRANT  and
AGATA SIENICKA
MARK GRANT
Affiliation:
Institute of Mathematics, University of Aberdeen, Aberdeen, UK, e-mail: mark.grant@abdn.ac.uk
AGATA SIENICKA
Affiliation:
Mathematical Institute, University of Bonn, Bonn, Germany, e-mail: s6agsien@uni-bonn.de
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Abstract

The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

MSC classification

Primary: 20F36: Braid groups; Artin groups
Secondary: 57N05: Topology of $E^2$, $2$-manifolds 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 297 - 306
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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