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ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

  • MARK GRANT (a1) and AGATA SIENICKA (a2)

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.

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1.Alexander, J. W., A Lemma on Systems of Knotted Curves, Proc. Natl. Acad. Sci. USA. 9(3) (1923), 9395.
2.An, B. H., Automorphisms of braid groups on orientable surfaces, J. Knot Theor. Ramif. 25(5) (2016), 1650022, 32 pp.
3.Artin, E., der Zöpfe, Theorie, Abh. Math. Sem. Univ. Hamburg. 4(1) (1925), 4772.
4.Bellingeri, P., On automorphisms of surface braid groups, J. Knot Theor. Ramif. 17(1) (2008), 111.
5.Birman, J. S., Braids, links and mapping class groups (Princeton University Press, Princeton, New Jersey, 1974).
6.Burde, G. and Zieschang, H., Knots, De Gruyter Studies in Mathematics (De Gruyter, Berlin, 1985).
7.Farb, B. and Margalit, D., A primer on mapping class groups (Princeton University Press, Princeton, 2012).
8.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), 277296.
9.Gluck, H., The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308333.
10.Ivanov, N. V., Mapping class groups, Handbook of Geometric Topology (North-Holland, Amsterdam, 2002), 523633.
11.Kassel, C. and Turaev, V., Braid groups, Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008).
12.Kida, Y. and Yamagata, S., The co-Hopfian property of surface braid groups, J. Knot Theor. Ramif. 22(10) (2013), 1350055, 46 pp.
13.Markov, A., Über die freie Äquivalenz der geschlossenen Zöpfe, Recueil Mathématique Moscou, Mat. Sb. 1(43) (1936), 7378.
14.Morton, H. R., Infinitely many fibred knots having the same Alexander polynomial, Topology 17(1) (1978), 101104.
15.Morton, H. R., Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986), 247260.
16.Paris, L. and Rolfsen, D., Geometric subgroups of surface braid groups, Ann. Inst. Fourier (Grenoble). 49(2) (1999), 417472.
17.Skora, R. K., Closed braids in 3-manifolds, Math. Z. 211(2) (1992), 173187.
18.Sundheim, P. A., The Alexander and Markov Theorems via diagrams for links in 3-manifolds, Trans. Amer. Math. Soc. 337(2) (1993), 591607.
19.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), 4753.
20.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).

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ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

  • MARK GRANT (a1) and AGATA SIENICKA (a2)

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