Abstract
Let \(n \in \mathbb {Z}^+\). We provide two short proofs of the following classical fact, one using Khovanov homology and one using Heegaard–Floer homology: if the closure of an n-strand braid \(\sigma \) is the n-component unlink, then \(\sigma \) is the trivial braid.
J. Elisenda Grigsby—Partially supported by NSF grant DMS-0905848 and NSF CAREER award DMS-1151671.
Stephan M. Wehrli—Partially supported by NSF grant DMS-1111680.
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J.A. Baldwin, J.E. Grigsby, Categorified invariants and the braid group. Proc. Am. Math. Soc. 143(7), 2801–2814 (2015)
J.S. Birman, H.M. Hilden, On isotopies of homeomorphisms of Riemann surfaces. Annl. Math. 2(97), 424–439 (1973)
J.S. Birman, W.W. Menasco, Studying links via closed braids. V. The unlink. Trans. Am. Math. Soc. 329(2), 585–606 (1992)
T.D. Cochran, Non-trivial links and plats with trivial Gassner matrices. Math. Proc. Cambr. Philos. Soc. 119(1), 43–53 (1996)
P. Ghiggini, P. Lisca, Open book decompositions versus prime factorizations of closed, oriented 3–manifolds. math.GT/1407.2148, (2014)
E. Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, vol. 2 (Higher Education Press, Beijing, 2002), pp. 405–414
M. Hedden, Y. Ni, Khovanov module and the detection of unlinks. math.GT/1204.0960, (2012)
M. Hedden, L. Watson, On the geography and botany of knot Floer homology. math.GT/1404.6913, (2014)
K. Honda, W.H. Kazez, G. Matić, Right-veering diffeomorphisms of compact surfaces with boundary. Inven. Math. 169(2), 427–449 (2007)
K. Honda, W.H. Kazez, G. Matić, On the contact class in Heegaard Floer homology. J. Differ. Geom. 83(2), 289–311 (2009)
J. Johnson, Heegaard splittings and open books. math.GT/1110.2142, (2011)
M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)
M. Khovanov, Patterns in knot cohomology. I. Exp. Math. 12(3), 365–374 (2003)
Y. Ni, Homological actions on sutured Floer homology. math.GT/1010.2808, (2010)
P. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds. Annl. Math. 159(3), 1027–1158 (2004)
P. Ozsváth, Z. Szabó, Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39–61 (2005)
O. Plamenevskaya, Transverse knots and Khovanov homology. Math. Res. Lett. 13(4), 571–586 (2006)
Acknowledgments
We thank Ken Baker, John Baldwin, Rob Kirby, Tony Licata, and Danny Ruberman for interesting conversations and Joan Birman and Bill Menasco for a useful e-mail correspondence. We are especially grateful to Ian Biringer for telling us about Hopfian groups, to Matt Hedden for pointing out that Proposition 2 appears in [14], and to Tim Cochran for making us aware that historical references to Proposition 1 in the literature appear under the slogan, “Milnor’s invariants detect the trivial braid.” We would also like to thank the referee for making a number of useful suggestions that greatly improved the exposition.
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Grigsby, J.E., Wehrli, S.M. (2016). An Elementary Fact About Unlinked Braid Closures. In: Letzter, G., et al. Advances in the Mathematical Sciences. Association for Women in Mathematics Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-34139-2_2
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DOI: https://doi.org/10.1007/978-3-319-34139-2_2
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