Research ArticleNo Access

Big data approaches to knot theory: Understanding the structure of the Jones polynomial

Jesse S. F. Levitt

Creyon Bio, Carlsbad, CA, USA

E-mail Address: jslevitt@usc.edu

Search for more papers by this author

Mustafa Hajij

Univeristy of San Francisco, San Francisco, CA, USA

E-mail Address: mhajij@scu.edu

Search for more papers by this author

, and

Radmila Sazdanovic

North Carolina State University, Raleigh, NC, USA

E-mail Address: rsazdanovic@math.ncsu.edu

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S021821652250095XCited by:2 (Source: Crossref)

Abstract

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.

Keywords:

AMSC: 57K10, 57K14, 62R07, 62R40

References

1. J. W. Alexander , Topological invariants of knots and links, Trans. Amer. Math. Soc. 30(2) (1928) 275–306. Crossref, Google Scholar
2. C. Armond and O. T. Dasbach, Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial, preprint (2011), arXiv:1106.3948. Google Scholar
3. D. Bar-Natan and S. Garoufalidis , On the Melvin–Morton–Rozansky conjecture, Invent. Math. 125(1) (1996) 103–133. Crossref, Web of Science, Google Scholar
4. D. Bar-Natan et al., The Knot Atlas (2011–2019), http://katlas.org. Google Scholar
5. D. Bar-Natan and R. van der Veen, A polynomial time knot polynomial, preprint (2017), arXiv:1708.04853. Google Scholar
6. K. Bataineh, M. Elhamdadi and M. Hajij , The colored Jones polynomial of singular knots, New York J. Math 22 (2016) 1439–1456. Web of Science, Google Scholar
7. P. Beirne and R. Osburn , q-series and tails of colored Jones polynomials, Indag. Math. 28(1) (2017) 247–260. Crossref, Web of Science, Google Scholar
8. B. A. Burton et al., Regina: Software for low-dimensional topology (1999–2017), http://regina-normal.github.io/. Google Scholar
9. B. A. Burton, The next 350 million knots (2018), http://regina-normal.github.io/ data.html. Google Scholar
10. J. Cantarella, A. Henrich, E. Magness, O. O’Keefe, K. Perez, E. Rawdon and B. Zimmer , Knot fertility and lineage, J. Knot Theory Ramifications 26 (2017) 1750093. Link, Web of Science, Google Scholar
11. J. Cha and C. Livingston, Knotinfo: Table of knot invariants (2017), http://www. indiana.edu/ $\sim$ $\sim$ knotinfo. Google Scholar
12. O. T. Dasbach and X.-S. Lin , On the head and the tail of the colored Jones polynomial, Compos. Math. 142(5) (2006) 1332–1342. Crossref, Web of Science, Google Scholar
13. P. Dlotko, D. Gurnari and R. Sazdanovic, Knot invariants and their relations: a topological perspective (2021), arXiv:2109.00831. Google Scholar
14. Y. Diao, C. Ernst and A. Stasiak , A partial ordering of knots through diagrammatic unknotting, J. Knot Theory Ramifications 18 (2009) 505–522. Link, Web of Science, Google Scholar
15. C. H. Dowker and M. B. Thistlethwaite , Classification of knot projections, Topology Appl. 16(1) (1983) 19–31. Crossref, Web of Science, Google Scholar
16. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction (American Mathematical Society, 2010). Google Scholar
17. H. Edelsbrunner, D. Letscher and A. Zomorodian , Topological persistence and simplification, in Proc. 41st Annual Symp. Found. Comput. Sci. (IEEE, 2000), pp. 454–463. Crossref, Google Scholar
18. M. Elhamdadi and M. Hajij , Pretzel knots and $q$ -series, Osaka J. Math. 54(2) (2017) 363–381. Web of Science, Google Scholar
19. M. Elhamdadi and M. Hajij , Foundations of the colored Jones polynomial of singular knots, Bull. Korean Math. Soc. 55(3) (2018) 937–956. Web of Science, Google Scholar
20. M. Elhamdadi, M. Hajij and M. Saito , Twist regions and coefficients stability of the colored jones polynomial, Trans. Amer. Math. Soc. 370(7) (2018) 5155–5177. Crossref, Web of Science, Google Scholar
21. C. Ernst and D. W. Sumners , The growth of the number of prime knots, Math. Proc. Cambridge Philos. Soc. 102 (1987) 303–315. Crossref, Web of Science, Google Scholar
22. A. Flammini and A. Stasiak , Natural classification of knots, Proc. R. Soc. A 463 (2007) 569–582. Crossref, Google Scholar
23. S. Garoufalidis and T. T. Q. Lê , The colored Jones function is q-holonomic, Geom. Topol. 9(3) (2005) 1253–1293. Crossref, Web of Science, Google Scholar
24. M. Hajij , The tail of a quantum spin network, Ramanujan J. 40(1) 2016 135–176. Crossref, Web of Science, Google Scholar
25. M. Hajij , The colored Kauffman skein relation and the head and tail of the colored Jones polynomial, J. Knot Theory Ramifications 26(03) (2017) 1741002. Link, Web of Science, Google Scholar
26. T. Hastie, R. Tibshirani, J. Friedman and J. Franklin , The elements of statistical learning: Data mining, inference and prediction, Math. Intelligencer 27(2) (2005) 83–85. Crossref, Google Scholar
27. J. Hoste , The enumeration and classification of knots and links, in Handbook of Knot Theory (Elsevier, 2005), pp. 209–232. Crossref, Google Scholar
28. J. Hoste, M. Thistlethwaite and J. Weeks , The first 1,701,936 knots, Math. Intelligencer 20(4) (1998) 33–48. Crossref, Web of Science, Google Scholar
29. M. C. Hughes , A neural network approach to predicting and computing knot invariants, J. Knot Theory Ramifications 29(03) (2020) 2050005. Link, Web of Science, Google Scholar
30. Inkscape Project, Inkscape (2011), https://inkscape.org. Google Scholar
31. S. V. Jablan and S. Radmila , Linknot: Knot Theory by Computer, Vol. 21 (World Scientific, 2007). Link, Google Scholar
32. V. Jejjala, A. Kar and O. Parrikar , Deep learning the hyperbolic volume of a knot, Phys. Lett. B 799 (2019) 135033. Crossref, Web of Science, Google Scholar
33. V. F. R. Jones , A polynomial invariant for knots via von neumann algebras, in Fields Medallists’ Lectures (World Scientific, 1997), pp. 448–458. Link, Google Scholar
34. R. M. Kashaev , The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39(3) (1997) 269–275. Crossref, Web of Science, Google Scholar
35. M. Khovanov , Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14(01) (2005) 111–130. Link, Web of Science, Google Scholar
36. T. T. Q. Lê , The colored Jones polynomial and the A-polynomial of knots, Adv. Math. 207(2) (2006) 782–804. Crossref, Web of Science, Google Scholar
37. T. T. Q. Lê et al., Integrality and symmetry of quantum link invariants, Duke Math. J. 102(2) (2000) 273–306. Crossref, Web of Science, Google Scholar
38. T. T. Q. Lê, A. T. Tran and V. Q. Huynh , On the AJ conjecture for knots, Indiana Univ. Math. J. 64(4) (1905) 1103–1151. Google Scholar
39. C. R. S. Lee , A trivial tail homology for non-A–adequate links, Algebr. Geom. Topol. 18(3) (2018) 1481–1513. Crossref, Web of Science, Google Scholar
40. W. B. R. Lickorish , An Introduction to Knot Theory, Vol. 175 (Springer Science & Business Media, 2012). Google Scholar
41. C. Livingston and A. H. Moore, Knotinfo: Table of knot invariants (2022), knotinfo.math.indiana.edu. Google Scholar
42. J. Lovejoy and R. Osburn , The Bailey chain and mock theta functions, Adv. Math. 238 (2013) 442–458. Crossref, Web of Science, Google Scholar
43. H. Murakami , An introduction to the volume conjecture, in Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 1–40. Crossref, Google Scholar
44. H. Murakami and J. Murakami , The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186(1) (2001) 85–104. Crossref, Web of Science, Google Scholar
45. D. Paweł, D. Gurnari and R. Sazdanovic, Knot invariants and their relations: A topological perspective, preprint (2021), arXiv:2109.00831. Google Scholar
46. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot and E. Duchesnay , Scikit-learn: Machine learning in Python, J. Mach. Learn. Res. 12 (2011) 2825–2830. Web of Science, Google Scholar
47. J. Rasmussen , Khovanov homology and the slice genus, Invent. Math. 182(2) (2010) 419–447. Crossref, Web of Science, Google Scholar
48. N. Yu. Reshetikhin and V. G. Turaev , Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26. Crossref, Web of Science, Google Scholar
49. D. Rolfsen , Knots and Links, Vol. 346 (American Mathematical Society, 1976). Google Scholar
50. J. Shlens, A tutorial on principal component analysis, preprint (2014), arXiv:1404.1100. Google Scholar
51. A. Shumakovitch , Private Communication (George Washington University, 2019). Google Scholar
52. W. Thomson , On vortex motion, Trans. R. Soc. Edinburgh 25 (1869) 217–260. Crossref, Google Scholar
53. V. G. Turaev , The Yang-Baxter equation and invariants of links, Invent. Math. 92(3) (1988) 527–553. Crossref, Web of Science, Google Scholar
54. R. E. Tuzun and A. S. Sikora , Verification of the jones unknot conjecture up to 22 crossings, J. Knot Theory Ramifications 27(03) (2018) 1840009. Link, Web of Science, Google Scholar
55. R. E. Tuzun and A. S. Sikora , Verification of the jones unknot conjecture up to 24 crossings, J. Knot Theory Ramifications 30(03) (2021) 2150020. Link, Web of Science, Google Scholar
56. M. Ward, Using neural networks to classify knots: Data mining and deep learning in knot theory (2018). Google Scholar
57. S. Wold, K. Esbensen and P. Geladi , Principal component analysis, Chemom. Intell. Lab. Syst. 2(1–3) (1987) 37–52. Crossref, Web of Science, Google Scholar
58. Inc., Wolfram Research, Mathematica, Version 12.0 (2019). Google Scholar