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Ordering homotopy string links over surfaces

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador Sãocarlense, 400, Centro, 13566-590 São Carlos, SP, Brazil

E-mail Address: jthlima@icmc.usp.br

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and

Denise de Mattos

Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador Sãocarlense, 400, Centro, 13566-590 São Carlos, SP, Brazil

E-mail Address: deniseml@icmc.usp.br

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https://doi.org/10.1142/S0218216516500012Cited by:2 (Source: Crossref)

Abstract

Let $M$ be a closed, connected and oriented surface of genus $g \geq 1$ . In this work, we prove that the homotopy string links over $M$ , namely, ${\hat{P B}}_{n} (M)$ , are bi-orderable, i.e. there is an (explicit) left and right order for such group.

Keywords:

AMSC: 57M25, 20F60

References

1. E. Artin, Theory of braids, Ann. of Math. 48 (1946) 101–126. Crossref, Web of Science, Google Scholar
2. G. Baumslag, On generalized free products, Math Z. 78 (1962) 423–438. Crossref, Google Scholar
3. J. S. Birman, Braids, Links and Mapping class Groups, Annals of Mathematics Studies, Vol. 82 (Princeton University Press, 1973). Google Scholar
4. C. H. Goldberg, An exact sequence of braid groups, Math. Scand. 33 (1973) 69–82. Crossref, Web of Science, Google Scholar
5. D. L. Goldsmith, Homotopy of braids — In answer to a question of E. Artin. In Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973), pp. 91–96. Lecture Notes in Math., Vol. 375. Springer, Berlin, 1974. Google Scholar
6. D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 186 (2004) 187–218. Crossref, Web of Science, Google Scholar
7. J. González-Meneses, New presentations of surface braid groups, J. Knot Theory Ramifications 10(3) (2001) 431–451. Link, Web of Science, Google Scholar
8. J. González-Meneses, Ordering pure braid groups on compact, connected surfaces, Pacific J. Math. 3(2) (2002) 369–378. Crossref, Web of Science, Google Scholar
9. J. González-Meneses and L. Paris, Vassiliev invariants for braids on surfaces, Trans. Amer. Math. Soc. 356(1) (2004) 219–243. Crossref, Web of Science, Google Scholar
10. N. Habbeger and X. Lin, The classification of links up to Link-Homotopy, J. Amer. Math. Soc. 3(2) (1990) 389–419. Crossref, Google Scholar
11. J. Levine, An approach to homotopy classification of links, Trans. Amer. Math. Soc. 306 (1988) 361–387. Crossref, Web of Science, Google Scholar
12. W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations (Dover Publications, 1996). Google Scholar
13. J. Milnor, Link groups, Ann. Math. 59 (1954) 177–195. Crossref, Web of Science, Google Scholar
14. L. Paris and D. Rolfsen, Geometric subgroups of surface braid groups, Ann. Inst. Fourier (Grenoble) (2) 49 (1999) 101–156. Web of Science, Google Scholar
15. D. Rolfsen, P. Dehornoy, I. Dynnikov and B. Wiest, Ordering Braids (American Mathematical Society, Providence, RI, 2008). Google Scholar
16. E. Yurasovskaya, Homotopy string links over surfaces, Ph.D. thesis, The University of British Columbia (2008). Google Scholar