Abstract
This chapter describes how Magma [3] was used to investigate and understand a phenomenon observed when implementing a conjugacy test for elements of a braid group. These investigations ultimately lead to the discovery of a new invariant of conjugacy classes in braid groups, to an efficient way of computing this invariant, and in particular to a much more powerful conjugacy test than the one which was originally to be implemented [11].
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References
1. E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126.
2. Joan S. Birman, Ki Hyoung Ko, Sang Jin Lee, The in.mum, supremum, and geodesic length of a braid conjugacy class, Adv. Math. 164-1 (2001), 41–56.
3. Wieb Bosma, John Cannon, Catherine Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.
4. Patrick Dehornoy, A fast method for comparing braids, Adv. Math. 125-2 (1997), 200–235.
5. Patrick Dehornoy, Groupes de Garside, Ann. Sci. école Norm. Sup. (4), 35-2 (2002), 267–306.
6. Patrick Dehornoy, Luis Paris, Gaussian groups and Garside groups, two generalizations of Artin groups, Proc. London Math. Soc. (3) 79-3 (1999), 569–604.
7. Elsayed A. El-Rifai, H. R. Morton, Algorithms for positive braids, Quart. J. Math. Oxford Ser. (2) 45-180 (1994), 479–497.
8. David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, William P. Thurston, Word processing in groups, chapter 9, Boston: Jones and Bartlett Publishers, 1992.
9. Nuno Franco, Juan González-Meneses, Conjugacy problem for braid groups and Garside groups, J. Algebra, 266-1 (2003), 112–132.
10. F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2), 20 (1969), 235–254.
11. Volker Gebhardt, A new approach to the conjugacy problem in Garside groups, Journal of Algebra 292-1 (2005), 282–302.
12. Volker Gebhardt, Braid groups, Chapter 31, pp. 963–1014 in: John Cannon, Wieb Bosma (eds.), Handbook of Magma Functions, Version 2.11, Volume 3, Sydney, 2004.
13. Ki Hyoung Ko, Sang Jin Lee, Jung Hee Cheon, Jae Woo Han, Ju-sung Kang, Choonsik Park, New public-key cryptosystem using braid groups, pp. 166–183 in: Advances in cryptology-CRYPTO 2000 (Santa Barbara, CA), Lecture Notes in Comput. Sci. 1880, Berlin: Springer, 2000.
14. Matthieu Picantin, The conjugacy problem in small Gaussian groups, Comm. Algebra, 29-3 (2001), 1021–1039.
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Gebhardt, V. (2006). Computer aided discovery of a fast algorithm for testing conjugacy in braid groups. In: Bosma, W., Cannon, J. (eds) Discovering Mathematics with Magma. Algorithms and Computation in Mathematics, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37634-7_12
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DOI: https://doi.org/10.1007/978-3-540-37634-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37632-3
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