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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The framed braid group and $ 3$-manifolds

Authors: Ki Hyoung Ko and Lawrence Smolinsky
Journal: Proc. Amer. Math. Soc. 115 (1992), 541-551
MSC: Primary 57N10; Secondary 20F36, 57M07
MathSciNet review: 1126197
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Abstract: The framed braid group on $ n$ strands is defined to be a semidirect product of the braid group $ {B_n}$ and $ {{\mathbf{Z}}^n}$. Framed braids represent $ 3$-manifolds in a manner analogous to the representation of links by braids. Consider two framed braids equivalent if they represent homeomorphic $ 3$-manifolds. The main result of this paper is a Markov type theorem giving moves that generate this equivalence relation.

References [Enhancements On Off] (What's this?)

  • [B] J. Birman, Braids, links and mapping class groups, Ann. of Math Stud., vol. 82, Princeton Univ. Press, Princeton, NJ, 1975. MR 0375281 (51:11477)
  • [C] F. R. Cohen, Artin's braid groups, classical homotopy theory and sundry other curiosities, Braids, Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI (1988), 167-206. MR 975079 (90a:20077)
  • [FR] R. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979), 1-15. MR 528232 (80c:57005)
  • [J] V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335-388. MR 908150 (89c:46092)
  • [K] R. C. Kirby, A calculus for framed links in $ {S^3}$, Invent. Math. 45 (1978), 35-56. MR 0467753 (57:7605)
  • [L] W. B. R. Lickorish, A representation of orientable, combinatorial $ 3$-manifolds, Ann. of Math. (2) 76 (1962), 531-540. MR 0151948 (27:1929)
  • [L1] -, $ 3$-Manifold invariants from the combinatorics of the Jones polynomial, Pacific J. Math., vol. 149, no. 2, (1991), 337-347. MR 1105702 (92d:57007)

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Keywords: $ 3$-manifolds, Braid group
Article copyright: © Copyright 1992 American Mathematical Society

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