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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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References

REFERENCES

Alexander, J. W., A Lemma on Systems of Knotted Curves, Proc. Natl. Acad. Sci. USA. 9(3) (1923), 93–95.CrossRefGoogle ScholarPubMed
An, B. H., Automorphisms of braid groups on orientable surfaces, J. Knot Theor. Ramif. 25(5) (2016), 1650022, 32 pp.CrossRefGoogle Scholar
Artin, E., der Zöpfe, Theorie, Abh. Math. Sem. Univ. Hamburg. 4(1) (1925), 47–72.CrossRefGoogle Scholar
Bellingeri, P., On automorphisms of surface braid groups, J. Knot Theor. Ramif. 17(1) (2008), 1–11.CrossRefGoogle Scholar
Birman, J. S., Braids, links and mapping class groups (Princeton University Press, Princeton, New Jersey, 1974).Google Scholar
Burde, G. and Zieschang, H., Knots, De Gruyter Studies in Mathematics (De Gruyter, Berlin, 1985).Google Scholar
Farb, B. and Margalit, D., A primer on mapping class groups (Princeton University Press, Princeton, 2012).Google Scholar
Gillette, R. and Van Buskirk, J., The word problem and consequences for the braid groups and mapping class groups of the 2-sphere, Trans. Amer. Math. Soc. 131 (1968), 277–296.Google Scholar
Gluck, H., The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333.CrossRefGoogle Scholar
Ivanov, N. V., Mapping class groups, Handbook of Geometric Topology (North-Holland, Amsterdam, 2002), 523–633.CrossRefGoogle Scholar
Kassel, C. and Turaev, V., Braid groups, Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008).Google Scholar
Kida, Y. and Yamagata, S., The co-Hopfian property of surface braid groups, J. Knot Theor. Ramif. 22(10) (2013), 1350055, 46 pp.Google Scholar
Markov, A., Über die freie Äquivalenz der geschlossenen Zöpfe, Recueil Mathématique Moscou, Mat. Sb. 1(43) (1936), 73–78.Google Scholar
Morton, H. R., Infinitely many fibred knots having the same Alexander polynomial, Topology 17(1) (1978), 101–104.CrossRefGoogle Scholar
Morton, H. R., Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986), 247–260.CrossRefGoogle Scholar
Paris, L. and Rolfsen, D., Geometric subgroups of surface braid groups, Ann. Inst. Fourier (Grenoble). 49(2) (1999), 417–472.CrossRefGoogle Scholar
Skora, R. K., Closed braids in 3-manifolds, Math. Z. 211(2) (1992), 173–187.CrossRefGoogle Scholar
Sundheim, P. A., The Alexander and Markov Theorems via diagrams for links in 3-manifolds, Trans. Amer. Math. Soc. 337(2) (1993), 591–607.Google Scholar
Zhang, P., Automorphisms of braid groups on S 2, T 2, P 2 and the Klein bottle K, J. Knot Theor. Ramif. 17(1) (2008), 47–53.CrossRefGoogle Scholar
Zieschang, H., Vogt, E. and Coldewey, H.-D., Surfaces and planar discontinuous groups (Translated from the German by John Stillwell), Lecture Notes in Mathematics, vol. 835 (Springer, Berlin, 1980).CrossRefGoogle Scholar