No Access Submitted: 17 December 2020 Accepted: 09 February 2021 Published Online: 10 March 2021
J. Math. Phys. 62, 032503 (2021); https://doi.org/10.1063/5.0040956
more...View Affiliations
View Contributors
  • Samantha Allen
  • Jacob H. Swenberg
Let X be a (2 + 1)-dimensional globally hyperbolic spacetime with a Cauchy surface Σ whose universal cover is homeomorphic to R2. We provide empirical evidence suggesting that the Jones polynomial detects causality in X. We introduce a new invariant of certain tangles related to the Conway polynomial and prove that the Conway polynomial does not detect the connected sum of two Hopf links among relevant three-component links, which suggests that the Conway polynomial does not detect causality in the scenario described.
The authors would like to thank Vladimir Chernov and Ina Petkova for their guidance throughout the project. The authors would also like to thank Gage Martin for helpful comments. S.A. would like to thank Charles Livingston for a helpful conversation. J.H.S. would like to thank Vanessa Pinney for her insight on enumeration of links. J.H.S. received support from NSF Grant No. DMS-1711100.
  1. 1. V. Chernov, G. Martin, and I. Petkova, “Khovanov homology and causality in spacetimes,” J. Math. Phys. 61, 022503 (2020). https://doi.org/10.1063/5.0002297, Google ScholarScitation, ISI
  2. 2. J. Natário and P. Tod, “Linking, Legendrian linking and causality,” Proc. London Math. Soc. 88(3), 251–272 (2004). https://doi.org/10.1112/s0024611503014424, Google ScholarCrossref
  3. 3. V. Chernov and S. Nemirovski, “Legendrian links, causality, and the Low conjecture,” Geom. Funct. Anal. 19, 1320–1333 (2010). https://doi.org/10.1007/s00039-009-0039-x, Google ScholarCrossref
  4. 4. D. Bar-Natan, “On Khovanov’s categorification of the Jones polynomial,” Algebraic Geom. Topol. 2, 337–370 (2002). https://doi.org/10.2140/agt.2002.2.337, Google ScholarCrossref
  5. 5. L. H. Kauffman and M. Silvero, “Alexander-Conway polynomial state model and link homology,” J. Knot Theory Ramification 25, 1640005 (2016). https://doi.org/10.1142/s0218216516400058, Google ScholarCrossref
  6. 6. G. Martin, “Khovanov homology detects T(2,6),” arXiv:2005.02893 (2020). Google Scholar
  7. 7. J. A. Baldwin and J. E. Grigsby, “Categorified invariants and the braid group,” Proc. Am. Math. Soc. 143, 2801–2814 (2015). https://doi.org/10.1090/s0002-9939-2015-12482-3, Google ScholarCrossref
  8. 8. S. Eliahou, L. H. Kauffman, and M. B. Thistlethwaite, “Infinite families of links with trivial Jones polynomial,” Topology 42, 155–169 (2003). https://doi.org/10.1016/s0040-9383(02)00012-5, Google ScholarCrossref
  9. 9. C. Livingston and A. H. Moore, “Linkinfo: Table of link invariants,” https://linkinfo.math.indiana.edu, 2020. Google Scholar
  10. 10. L. H. Kauffman, “The Conway polynomial,” Topology 20, 101–108 (1981). https://doi.org/10.1016/0040-9383(81)90017-3, Google ScholarCrossref
  11. 11. L. H. Kauffman, “State models and the Jones polynomial,” Topology 26, 395–407 (1987). https://doi.org/10.1016/0040-9383(87)90009-7, Google ScholarCrossref
  12. 12. A. S. Sikora, “Tangle equations, the Jones conjecture, and quantum continued fractions,” arXiv:2005.08162 [math.GT] (2020). Google Scholar
  13. 13. O. T. Dasbach and S. Hougardy, “Does the Jones polynomial detect unknottedness?,” Exp. Math. 6, 51–56 (1997). https://doi.org/10.1080/10586458.1997.10504350, Google ScholarCrossref
  14. 14. Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL, 2020. Google Scholar
  1. © 2021 Author(s). Published under license by AIP Publishing.