Geodesic
ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS
Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 297-306

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DOI

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. 
GRANT, MARK; SIENICKA, AGATA. ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS. Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 297-306. doi: 10.1017/S0017089520000208
@article{10_1017_S0017089520000208,
     author = {GRANT, MARK and SIENICKA, AGATA},
     title = {ISOTOPY {AND} {HOMEOMORPHISM} {OF} {CLOSED} {SURFACE} {BRAIDS}},
     journal = {Glasgow mathematical journal},
     pages = {297--306},
     year = {2021},
     volume = {63},
     number = {2},
     doi = {10.1017/S0017089520000208},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000208/}
}
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