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Detecting causality with symplectic quandles

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Abstract

We investigate the capability of symplectic quandles to detect causality for (2+1)-dimensional globally hyperbolic spacetimes (X). Allen and Swenberg showed that the Alexander–Conway polynomial is insufficient to distinguish connected sum of two Hopf links from the links in the family of Allen–Swenberg 2-sky like links, suggesting that it cannot always detect causality in X. We find that symplectic quandles, combined with Alexander–Conway polynomial, can distinguish these two types of links, thereby suggesting their ability to detect causality in X. The fact that symplectic quandles can capture causality in the Allen–Swenberg example is intriguing since the theorem of Chernov and Nemirovski, which states that Legendrian linking equals causality, is proved using Contact Geometry methods.

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Code Availability

The Wolfram Mathematica [19] code files to solve invariant for the connected sum of Hopf links and Allen–Swenberg links are published with open access at:

https://www.wolframcloud.com/obj/nilesh80/Published/SymplecticHopf2x2-Z5.nb

https://www.wolframcloud.com/obj/nilesh80/Published/SymplecticAllenSwT1-2x2-Z5.nb

https://www.wolframcloud.com/obj/nilesh80/Published/SymplecticAllenSwT2-2x2-Z5.nb

The code can also be requested from the author if the above links do not work for any reason.

Data availability statement

Data sharing does not apply to this article as no new data were created or analyzed.

Notes

  1. Allen and Swenberg [7] identified the infinite family of links as \(C(1*T_0(n))\); C(0) represents connected sum of two Hopf links H.

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Acknowledgements

This project was completed in the Summer of 2023 as a part of the Advanced Mathematics course by Horizon Academic Research Program. The program was supervised by Professor Vladimir Chernov of Dartmouth College and Professor Emanuele Zappala of Yale University. I would like to thank Professor Chernov and Professor Zappala for their constant support in this project.

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  1. The Shri Ram School Aravali, Gurugram, New Delhi, India

    Ayush Jain

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Jain, A. Detecting causality with symplectic quandles. Lett Math Phys 114, 63 (2024). https://doi.org/10.1007/s11005-024-01808-w

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