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Detecting and visualizing $3$ $3$ -dimensional surgery

Stathis Antoniou

School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

E-mail Address: santoniou@math.ntua.gr

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Louis H. Kauffman

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA

Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia

E-mail Address: kauffman@uic.edu

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Sofia Lambropoulou

School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

E-mail Address: sofia@math.ntua.gr

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https://doi.org/10.1142/S0218216519400157Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue: Knots, Low-Dimensional Topology and Applications — Part II

Abstract

Topological surgery in dimension $3$ $3$ is intrinsically connected with the classification of $3$ $3$ -manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of $3$ $3$ -dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how $3$ $3$ -dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of $3$ $3$ -dimensional surgery as rotations of the decompactified $2$ $2$ -sphere. Each rotation produces a different decomposition of the $3$ $3$ -sphere which corresponds to a different visualization of the $4$ $4$ -dimensional process of $3$ $3$ -dimensional surgery.

Keywords:

AMSC: 57M05, 57M25, 57M27, 57R65

References

1. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, 2004). Google Scholar
2. S. Antoniou, Mathematical Modeling Through Topological Surgery and Applications, Springer Theses Book Series (Springer International Publishing, 2018), ISBN:978-3-319-97066-0. https://doi.org/10.1007/978-3-319-97067-7 Crossref, Google Scholar
3. S. Antoniou, L. H. Kauffman and S. Lambropoulou, Black holes and topological surgery (2018), https://arXiv.org/pdf/1808.00254. Google Scholar
4. S. Antoniou, L. H. Kauffman and S. Lambropoulou, Topological surgery in cosmic phenomena, Advances in Theoretical and Mathematical Physics 23(3) (2019) 701–765. https://doi.org/10.4310/ATMP.2019.v23.n3.a3 Crossref, Web of Science, Google Scholar
5. S. Antoniou and S. Lambropoulou, Extending topological surgery to natural processes and dynamical systems, PLoS One 12(9) (2017) e0183993. https://doi.org/10.1371/journal.pone.0183993 Crossref, Web of Science, Google Scholar
6. S. Antoniou and S. Lambropoulou, Topological surgery in nature, in Algebraic Modeling of Topological and Computational Structures and Applications, Springer Proceedings in Mathematics and Statistics, Vol. 219 (Springer International Publishing, 2017), pp. 313–336. https://doi.org/10.1007/978-3-319-68103-0 Crossref, Google Scholar
7. R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol. 20 (American Mathematical Society, 1999). Crossref, Google Scholar
8. L. H. Kauffman, S. Lambropoulou and D. Buck, Chapter on three dimensional topology for DNA, in DNA Topology (book in preparation). Google Scholar
9. R. Kirby, A Calculus for framed links in $S^{3}$ $S^{3}$ , Invent. Math. 45 (1978) 35–56. Crossref, Web of Science, Google Scholar
10. R. C. Kirby and M. G. Scharlemann, Eight faces of the Poincaré homology 3-sphere, Uspekhi Mat. Nauk 37(5) (1982) 139–159. https://doi.org/10.1016/B978-0-12-158860-1.50015-0 Google Scholar
11. C. Kosniowski, A First Course in Algebraic Topology (Cambridge University Press, Cambridge, 1980), ISBN:0521298644. https://doi.org/10.1017/CBO9780511569296 Crossref, Google Scholar
12. S. Lambropoulou and S. Antoniou, Topological surgery, dynamics and applications to natural processes, J. Knot Theory Ramifications 26(9) (2016) 1743002. https://doi.org/10.1142/S0218216517430027 Link, Web of Science, Google Scholar
13. S. Lambropoulou, N. Samardzija, I. Diamantis and S. Antoniou, Topological surgery and dynamics, Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2014, Workshop: Algebraic Structures in Low-Dimensional Topology (2014), doi:10.4171/OWR/2014/26. Google Scholar
14. J. Levin, Topology and the cosmic microwave background, Phys. Rep. 365 (2002) 251–333. Crossref, Web of Science, Google Scholar
15. W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. (2) 76 (1962) 531–540. Crossref, Web of Science, Google Scholar
16. J.-P. Luminet, J. R. Weeks, A. Riazuelo, R. Lehoucq and J.-P. Uzan, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature 425 (2003) 593–595. Crossref, Web of Science, Google Scholar
17. J. Milnor, A Procedure for Killing the Homotopy Groups of Differentiable Manifolds, Symposia in Pure Mathematics, Vol. 3 (American Mathematical Society, 1961), pp. 39–55. Crossref, Google Scholar
18. J. Milnor, On the 3-dimensional Brieskorn manifolds $M (p, q, r)$ $M (p, q, r)$ , in Knot Groups and $3$ $3$ -Manifolds: Papers Dedicated to the Memory of R.H. Fox, Annals of Mathematics Studies, Vol. 84 (Princeton University Press, Princeton, NJ, 1975). Google Scholar
19. A. H. Moore and M. Vazquez, A note on band surgery and the signature of a knot (2018), https://arXiv.org/pdf/1806.02440. Google Scholar
20. J. R. Munkres, Topology, 2nd edn. (Prentice Hall, 2000). Google Scholar
21. V. V. Prasolov and A. B. Sossinsky, Knots, Links, Braids and 3-Manifolds, Translations of Mathematical Monographs, Vol. 154 (American Mathematical Society, 1997). Crossref, Google Scholar
22. A. Ranicki, Algebraic and Geometric Surgery, Oxford Mathematical Monographs (Clarendon Press, 2002). Crossref, Google Scholar
23. D. Rolfsen, Knots and Links (Publish or Perish Inc/AMS Chelsea Publishing, 2003). Crossref, Google Scholar
24. N. Samardzija and L. Greller, Explosive route to chaos through a fractal torus in a generalized Lotka–Volterra model, Bull. Math. Biol. 50(5) (1988) 465–491. https://doi.org/10.1007/BF02458847 Crossref, Web of Science, Google Scholar
25. J. Stillwell, Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics (Springer-Verlag, New York, 1993). Crossref, Google Scholar
26. H. Threlfall and W. Seifert, A Textbook of Topology (Academic Press, 1980). Google Scholar
27. V. Volterra, Lecons sur la Theorie Mathematique de la lutte pour la vie (Gauthier-Villars, Paris, 1931), reissued, 1990. https://doi.org/10.1090/S0002-9904-1936-06292-0 Google Scholar
28. A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503–528. Crossref, Google Scholar
29. J. R. Weeks, The Shape of Space (CRC Press, 2001). Crossref, Google Scholar
30. M. Wilhelm, A. Karrass and D. Solitar, Combinatorial Group Theory (Dover Publications, New York, 2004). Google Scholar