Abstract
In this lecture, we show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 3-manifold is when the manifold has a properly embedded non-separating sphere. This change of framing is given by the Dirac trick, also known as the light bulb trick. We begin with background information on non-separating spheres, Dehn homeomorphisms, and map** class groups of 3-manifolds. The main tools used in the proof are based on Darryl McCullough’s work on the map** class groups of 3-manifolds and spin structures. We then relate these results to the theory of skein modules.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R. P. Bakshi, D. Ibarra, G. Montoya-Vega, J. H. Przytycki, D. Weeks, On framings of links in 3-manifolds. Canad. Math. Bull. 64 (2021), no. 4, 752–764. ar**v:2001.07782 [math.GT].
J. W. Barrett, Skein spaces and spin structures, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, 267–275. ar**v:gr-qc/9512041.
P. Cahn, V. Chernov, R. Sadykov, The number of framings of a knot in a 3-manifold, J. Knot Theory Ramifications 23 (2014), no. 13, 1450072, 9 pp. ar**v:1404.5851 [math.GT].
Q. Chen, The 3-move conjecture for 5-braids, Knots in Hellas’ 98; The Proceedings of the International Conference on Knot Theory and its Ramifications; Volume 1. In the Series on Knots and Everything, Vol. 24, September 2000, pp. 36–47.
S. Das, G. Francis, J. Hart, C. Hartman, J. Heyn-Cubacub, L. H. Kauffman, D. Plepys, D. J. Sandin, Air on the Dirac Strings. https://www.youtube.com/watch.
D. Gabai, G. R. Meyerhoff, N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic. Ann. of Math. (2) 157 (2003), no. 2, 335–431. ar** class groups of 3-manifolds, then and now. Geometry and topology down under, 53–63, Contemp. Math., 597, Amer. Math. Soc., Providence, RI, 2013.
J. Hoste, J. H. Przytycki, Homotopy skein modules of orientable 3-manifolds. Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 475–488.
J. F. P. Hudson, Piecewise linear topology. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees W. A. Benjamin, Inc., New York-Amsterdam 1969 ix+282 pp.
D. McCullough, Map**s of reducible 3-manifolds. Geometric and algebraic topology, 61–76, Banach Center Publ., 18, PWN, Warsaw, 1986.
D. McCullough, Homeomorphisms which are Dehn twists on the boundary. Algebr. Geom. Topol. 6 (2006), 1331–1340. ar**v:1604.02075 [math.GT].
D. McCullough, A. Miller, Homeomorphisms of 3-manifolds with compressible boundary. Mem. Amer. Math. Soc. 61 (1986), no. 344, xii+100 pp.
C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2) 66 (1957), 1–26.
V. Pasquier, H. Saleur, Common structures between finite systems and conformal field theories through quantum groups Nucl. Phys. B330, 1990 523–556.
J. H. Przytycki, A q-analogue of the first homology group of a 3-manifold, Contemporary Mathematics 214, Perspectives on Quantization (Proceedings of the joint AMS-IMS-SIAM conference on Quantization, Mount Holyoke College, 1996); Ed. L.A.Coburn, M.A.Rieffel, AMS 1998, 135–144.
J. H. Przytycki, A. S. Sikora, On skein algebras and \(SL(2,\mathbb {C})\)-character varieties. Topology 39 (2000), no. 1, 115–148. ar**v:q-alg/9705011[math.QA].
E. Stiefel, Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten. (German) Comment. Math. Helv. 8 (1935), no. 1, 305–353.
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. (German) Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Przytycki, J.H., Bakshi, R.P., Ibarra, D., Montoya-Vega, G., Weeks, D. (2024). Spin Structure and the Framing Skein Module of Links in 3-Manifolds. In: Lectures in Knot Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-031-40044-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-031-40044-5_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-40043-8
Online ISBN: 978-3-031-40044-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)