Bulletin of the American Mathematical Society

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Book Information:

Authors: Christian Kassel and Vladimir Turaev
Title: Braid groups (With the graphical assistance of Olivier Dodane)
Additional book information: Graduate Texts in Mathematics, 247, Springer Science and Business Media, New York, 2008, xii+340 pp., ISBN 978-0-387-33841-5, hardcover, US$59.95

Author: Patrick Dehornoy
Title: Ordering braids
Additional book information: Mathematical Surveys and Monographs 148, American Mathematical Society, Providence, Rhode Island, 2008, x+323 pp., ISBN 978-0-8218-4431-1, hardcover, US$89.95

S. I. Adyan, Fragments of the word $\Delta$ in the braid group, Mat. Zametki 36 (1984), no. 1, 25–34 (Russian). MR 757642

J.W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. 9 (1923), 93-95.

E. Artin, Theorie der Zopfe, Abh. Math. Sem. Hamburg, 4 (1925), 47-72.

V. Bardakov, R. Mikhailov, V. Vershinin, J. Wu, Brunnian braids on surfaces, preprint arXiv:0909.3387v1 [math.GT] 18 Sept. 2009.

A. Berrick, F. Cohen, E. Hanbury, Y. Wong and J. Wu, Braids: Introductory lectures on braids, configurations and their applications, National Univ. of Singapore, Lecture Notes Series 19, World Scientific Publishing 2010.

A. J. Berrick, F. R. Cohen, Y. L. Wong, and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006), no. 2, 265–326. MR 2188127, DOI 10.1090/S0894-0347-05-00507-2

Joan S. Birman and William W. Menasco, Stabilization in the braid groups. I. MTWS, Geom. Topol. 10 (2006), 413–540. MR 2224463, DOI 10.2140/gt.2006.10.413

Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology 22 (1983), no. 1, 47–82. MR 682059, DOI 10.1016/0040-9383(83)90045-9

W. Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Hamburg II (1936), 171-178.

Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776

R. Courant, Differential and Integral Calculus, 2, Nordeman Publishing, 1948.

Patrick Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), no. 1, 115–150. MR 1214782, DOI 10.1090/S0002-9947-1994-1214782-4

Patrick Dehornoy, Groupes de Garside, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 267–306 (French, with English and French summaries). MR 1914933, DOI 10.1016/S0012-9593(02)01090-X

Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, Mathematical Surveys and Monographs, vol. 148, American Mathematical Society, Providence, RI, 2008. MR 2463428, DOI 10.1090/surv/148

I. A. Dynnikov, Arc-presentations of links: monotonic simplification, Fund. Math. 190 (2006), 29–76. MR 2232855, DOI 10.4064/fm190-0-3

David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982

David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694

Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517

R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest, Ordering the braid groups, Pacific J. Math. 191 (1999), no. 1, 49–74. MR 1725462, DOI 10.2140/pjm.1999.191.49

F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254. MR 248801, DOI 10.1093/qmath/20.1.235

Étienne Ghys, Knots and dynamics, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 247–277. MR 2334193, DOI 10.4171/022-1/11

E. Ghys and J. Leys, Lorenz and Modular Flows: A Visual Introduction, AMS Feature Column, Nov. 2006, http://www.ams.org/featurecolumn/archive/lorenz.html.

E. A. Gorin and V. Ja. Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sb. (N.S.) 78 (120) (1969), 579–610 (Russian). MR 0251712

A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, DOI 10.1007/BF01199469

T. Ito, Braid ordering and the geometry of a closed braid, arXiv:0805.1447 [math:GT].

V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403

Christian Kassel and Vladimir Turaev, Braid groups, Graduate Texts in Mathematics, vol. 247, Springer, New York, 2008. With the graphical assistance of Olivier Dodane. MR 2435235, DOI 10.1007/978-0-387-68548-9

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Science 20 (1963), 130-141.

D. Mackenzie, What’s Happening in the Mathematical Sciences 7 (2009), American Mathematical Society, 2-17.

A.A. Markov, Uber die freie Äquivalenz der geschlossenen Zopfe, Recueil Mathematique Moscou, Mat Sb 1 43 (1936), 73-78.

William Menasco and Morwen Thistlethwaite (eds.), Handbook of knot theory, Elsevier B. V., Amsterdam, 2005. MR 2179010

John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612

J. Nielsen, Jakob Nielsen: Collected Mathematical Papers, $\#$[N-18, 20, 21], Birkhäuser, 1986.

S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779–810. MR 1679799, DOI 10.1016/S0040-9383(98)00021-4

Stepan Yu. Orevkov, Quasipositivity problem for 3-braids, Turkish J. Math. 28 (2004), no. 1, 89–93. MR 2056762

T. Pinsky, private conversations at Technion, Haifa, Israel, in January 2008 and again in January 2010.

Józef H. Przytycki, Classical roots of knot theory, Chaos Solitons Fractals 9 (1998), no. 4-5, 531–545. Knot theory and its applications. MR 1628740, DOI 10.1016/S0960-0779(97)00107-0

Lee Rudolph, Algebraic functions and closed braids, Topology 22 (1983), no. 2, 191–202. MR 683760, DOI 10.1016/0040-9383(83)90031-9

Hamish Short and Bert Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), no. 3-4, 279–312. MR 1805402

Pierre Vogel, Representation of links by braids: a new algorithm, Comment. Math. Helv. 65 (1990), no. 1, 104–113. MR 1036132, DOI 10.1007/BF02566597

Shuji Yamada, The minimal number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), no. 2, 347–356. MR 894383, DOI 10.1007/BF01389082

• S.I. Adyan, Fragments of the word $\Delta$ in the braid group, Mat. Zametki 36, No. 1 (1984), 25-34. English translation Math. Notes 36 (1984), No. 1-2, 505-510. MR 757642 (86b:20048)
• J.W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. 9 (1923), 93-95.
• E. Artin, Theorie der Zopfe, Abh. Math. Sem. Hamburg, 4 (1925), 47-72.
• V. Bardakov, R. Mikhailov, V. Vershinin, J. Wu, Brunnian braids on surfaces, preprint arXiv:0909.3387v1 [math.GT] 18 Sept. 2009.
• A. Berrick, F. Cohen, E. Hanbury, Y. Wong and J. Wu, Braids: Introductory lectures on braids, configurations and their applications, National Univ. of Singapore, Lecture Notes Series 19, World Scientific Publishing 2010.
• A. Berrick, F. Cohen, Y. Wang and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc. 19, no. 2 (2005), 265-326. MR 2188127 (2007e:20073)
• J. Birman and W. Menasco, Stabilization in the braid groups I: MTWS, Geometry and Topology 10 (2006) 413-540. MR 2224463 (2007e:57005)
• J. Birman and R. Williams, Knotted periodic orbits in dynamical systems-I: Lorenz’s equations, Topology 22, No. 1 (1983), 47-82. MR 682059 (84k:58138)
• W. Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Hamburg II (1936), 171-178.
• G. Burde and H. Zieschang, Knots, de Gruyter, Berlin and New York, 1985. MR 808776 (87b:57004)
• R. Courant, Differential and Integral Calculus, 2, Nordeman Publishing, 1948.
• P. Dehornoy, Braid groups and non-distributive operations, Trans. Amer. Math. Soc. 345 (1994), No. 1, 115-151. MR 1214782 (95a:08003)
• P. Dehornoy, Groupes de Garside, Ann. Sci. Ecole Norm. Sup. 35 (2002), 267-306. MR 1914933 (2003f:20068)
• P. Dehornoy (with I. Dynnikov, D. Rolfsen and B. Wiest), Ordering Braids, Math. Surveys and Monographs 148, American Mathematical Society, Providence, RI, 2008. MR 2463428 (2009j:20046)
• I. A. Dynnikov, Arc-presentations of links: Monotonic simplification, Fund. Math. 190 (2006), 29–76. MR 2232855 (2007e:57006)
• D. Eisenbud and W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Ann. of Math. Stud. 110, Princeton Univ. Press, 1985. MR 817982 (87g:57007)
• D. Epstein, J. Cannon, F. Holt, S. Levy, M. Patterson and W. Thurston, Word processing in groups, Jones and Bartlett, Boston, MA, 1992, Chapter 9. MR 1161694 (93i:20036)
• E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. MR 0141126 (25:4537)
• R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke and B. Wiest, Ordering the braid groups, Pacific J. Math. 191 (1999), 49-74. MR 1725462 (2000j:20064)
• F. Garside, The braid group and other groups, Quart. J. Math Oxford 20 (1969), 235-254. MR 0248801 (40:2051)
• E. Ghys, Knots and dynamics, Proc. Int. Congress of Mathematicians 2007, Vol. 1, Zürich, European Math. Society, 247-277. MR 2334193 (2008k:37001)
• E. Ghys and J. Leys, Lorenz and Modular Flows: A Visual Introduction, AMS Feature Column, Nov. 2006, http://www.ams.org/featurecolumn/archive/lorenz.html.
• E. Gorin and V. Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sbornik, Volume 78 (1969), 579–610. MR 0251712 (40:4939)
• A. Hurwitz, Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-61. MR 1510692
• T. Ito, Braid ordering and the geometry of a closed braid, arXiv:0805.1447 [math:GT].
• V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. MR 908150 (89c:46092)
• C. Kassel and V. Turaev, Braid Groups, Graduate Texts in Mathematics 247, Springer Science and Business Media, New York, 2008. MR 2435235 (2009e:20082)
• E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Science 20 (1963), 130-141.
• D. Mackenzie, What’s Happening in the Mathematical Sciences 7 (2009), American Mathematical Society, 2-17.
• A.A. Markov, Uber die freie Äquivalenz der geschlossenen Zopfe, Recueil Mathematique Moscou, Mat Sb 1 43 (1936), 73-78.
• W. Menasco and M. Thistlethwaite, editors, Handbook of Knot Theory, Elsevier, 2005. MR 2179010 (2006h:57001)
• J. Milnor, Singular points of complex hypersurfaces. Ann. of Math. 61, Princeton University Press, 1968. MR 0239612 (39:969)
• J. Nielsen, Jakob Nielsen: Collected Mathematical Papers, $\#$[N-18, 20, 21], Birkhäuser, 1986.
• S. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38, no. 4 (1999), 779-810. MR 1679799 (2000b:14066)
• S. Orevkov, Quasipositivity problem for $3$-braids, Turkish J. Math. 28, no. 1 (2004), 89–93. MR 2056762 (2005d:20063)
• T. Pinsky, private conversations at Technion, Haifa, Israel, in January 2008 and again in January 2010.
• J. Przytycki, Classical roots of knot theory, Chaos, Solitons and Fractals 9 (1998), no. 4-5, 531-545. MR 1628740 (99c:57030)
• L. Rudolph, Algebraic functions and closed braids, Topology 22, no. 2 (1983), 191–202. MR 683760 (84e:57012)
• H. Short and B. Wiest, Orderings of mapping class groups after Thurston, Enseignement Math. (2), 46 (2000), No. 3-4, 279-312. MR 1805402 (2003b:57003)
• P. Vogel, Representations of links by braids: A new algorithm, Comment. Math. Helv. 65 (1990), 104-113. MR 1036132 (90k:57013)
• S. Yamada, The minimum number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), 347-356. MR 894383 (88f:57015)