Isotopy and homeomorphism of closed surface braids

Mark Grant, Agata Sienicka

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The closure of a braid in a closed orientable surface Σ is a link in Σ × 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 center.
Original languageEnglish
Pages (from-to)297-306
Number of pages10
JournalGlasgow Mathematical Journal
Issue number2
Early online date15 May 2020
Publication statusPublished - May 2021


  • Surface braids
  • link isotopy
  • link homeomorphism
  • Birman exact sequence
  • Dehn–Nielsen–Baer Theorem
  • 57N05
  • 57M27
  • 2010 Mathematics Subject Classification
  • 20F36


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