No Access

The Jones polynomials of three-bridge knots via Chebyshev knots and billiard table diagrams

Moshe Cohen

Mathematics Department, State University of New York at New Paltz, New Paltz, New York, USA

https://doi.org/10.1142/S0218216521410066Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue — Dedicated to the 60th Birthday of J. H. Przvtycki
Guest Editors: M. K. Dabkowski, V. Harizanov, L. H. Kauffman, J. H. Przytycki, R. Sazdanovic and A. Sikora

Abstract

This work presents formulas for the Kauffman bracket and Jones polynomials of three-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of two-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.

Keywords:

AMSC: 57M25, 57M27, 05C31, 05B45

References

1. D. Bar-Natan et al., The Knot Atlas, http://katlas.org. Google Scholar
2. M. Beck, C. Haase and S. V. Sam , Grid graphs, Gorenstein polytopes, and domino stackings, Graphs Comb. 25(4) (2009) 409–426. Crossref, Web of Science, Google Scholar
3. A. T. Benjamin, L. Ericksen, P. Jayawant and M. Shattuck , Combinatorial trigonometry with Chebyshev polynomials, J. Stat. Plann. Inference 140(8) (2010) 2157–2160. Crossref, Web of Science, Google Scholar
4. A. T. Benjamin and D. Walton, Counting on Chebyshev polynomials, Math. Mag. 82(2) (2009) 117–126. Google Scholar
5. J. S. Birman , On the Jones polynomial of closed 3-braids, Invent. Math. 81 (1985) 287–294. Crossref, Web of Science, Google Scholar
6. M. G. V. Bogle, J. E. Hearst, V. F. R. Jones and L. Stoilov , Lissajous knots, J. Knot Theory Ramifications 3(2) (1994) 121–140. Link, Google Scholar
7. A. Boocher, J. Daigle, J. Hoste and W. Zheng , Sampling Lissajous and Fourier knots, Experiment. Math. 18(4) (2009) 481–497. Crossref, Web of Science, Google Scholar
8. M. Cohen , A determinant formula for the Jones polynomial of pretzel knots, J. Knot Theory Ramifications 21(6) (2012) 1250062, 23. Link, Web of Science, Google Scholar
9. M. Cohen, O. T. Dasbach and H. M. Russell , A twisted dimer model for knots, Fundam. Math. 225(1) (2014) 57–74. Crossref, Web of Science, Google Scholar
10. M. Cohen and M. Teicher , Kauffman’s clock lattice as a graph of perfect matchings: A formula for its height, Electron. J. Combin. 21(4) (2014) #P4.31. Crossref, Web of Science, Google Scholar
11. E. H. Comstock , The real singularities of harmonic curves of three frequencies, Trans. Wisconsin Acad. Sci. Arts Lett. 11 (1896–1897) 452–464. Google Scholar
12. P. R. Cromwell , Knots and Links (Cambridge University Press, Cambridge, 2004). Crossref, Google Scholar
13. O. T. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin and N. W. Stoltzfus , The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98(2) (2008) 384–399. Crossref, Web of Science, Google Scholar
14. B. M. M. de Weger , Padua and Pisa are exponentially far apart, Publ. Mat., Barc. 41(2) (1997) 631–651. Crossref, Web of Science, Google Scholar
15. S. Duzhin and M. Shkolnikov , A formula for the HOMFLY polynomial of rational links, Arnold Math. J. 1(4) (2015) 345–359. Crossref, Google Scholar
16. R. Euler , The Fibonacci number of a grid graph and a new class of integer sequences, J. Integer Seq. 8(2) (2005) art. 05.2.6, 16. Google Scholar
17. J. Gu and H. Jockers , A note on colored HOMFLY polynomials for hyperbolic knots from WZW models, Comm. Math. Phys. 338(1) (2015) 393–456. Crossref, Web of Science, Google Scholar
18. J. Hoste and L. Zirbel , Lissajous knots and knots with Lissajous projections, Kobe J. Math. 24(2) (2007) 87–106. Google Scholar
19. F. Jaeger, D. L. Vertigan and D. J. A. Welsh , On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Cambridge Philos. Soc. 108(1) (1990) no. 1, 35–53. Crossref, Web of Science, Google Scholar
20. V. F. R. Jones and J. H. Przytycki , Lissajous knots and billiard knots, in Knot Theory, Banach Center Publications, Vol. 42 (Polish Academy Science Institute of Mathematics, Warsaw, 1998), pp. 145–163. Google Scholar
21. T. Kanenobu , Examples on polynomial invariants of knots and links II, Osaka J. Math. 26(3) (1989) 465–482. Web of Science, Google Scholar
22. T. Kanenobu , Jones and Q polynomials for 2-bridge knots and links, Proc. Am. Math. Soc. 110(3) (1990) 835–841. Web of Science, Google Scholar
23. L. H. Kauffman , State models and the Jones polynomial, Topology 26(3) (1987) 395–407. Crossref, Web of Science, Google Scholar
24. R. Kenyon , Lectures on dimers, in Statistical Mechanics, IAS/Park City Mathematics Series, Vol. 16 (American Mathematical Society, Providence, RI, 2009), pp. 191–230. Crossref, Google Scholar
25. D. Klarner and J. Pollack , Domino tilings of rectangles with fixed width, Discrete Math. 32 (1980) 45–52. Crossref, Web of Science, Google Scholar
26. P.-V. Koseleff and D. Pecker , On Fibonacci knots, Fibonacci Q. 48(2) (2010) 137–143. Google Scholar
27. P.-V. Koseleff and D. Pecker , Chebyshev diagrams for two-bridge knots, Geom. Dedicata 150 (2011) 405–425. Crossref, Web of Science, Google Scholar
28. P.-V. Koseleff and D. Pecker , Chebyshev knots, J. Knot Theory Ramifications 20(4) (2011) 575–593. Link, Web of Science, Google Scholar
29. P.-V. Koseleff and D. Pecker , Harmonic knots, J. Knot Theory Ramifications 25(13) (2016) 1650074, 18. Link, Web of Science, Google Scholar
30. P.-V. Koseleff and D. Pecker , On Alexander–Conway polynomials of two-bridge links, J. Symb. Comput. 68 (2015) 215–229. Crossref, Web of Science, Google Scholar
31. P. B. Kronheimer and T. S. Mrowka , Khovanov homology is an unknot-detector, Publ. Math., Inst. Hautes Étud. Sci. 113 (2011) 97–208. Crossref, Web of Science, Google Scholar
32. C. Lamm , There are infinitely many Lissajous knots, Manuscripta Math. 93(1) (1997) 29–37. Crossref, Web of Science, Google Scholar
33. C. Lamm, Zylinder-Knoten und symmetrische Vereinigungen, Bonner Mathematische Schrif-ten [Bonn Mathematical Publications], 321, Universität Bonn Mathematisches Institut, Bonn, 1999, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (1999). Google Scholar
34. T. T. Q. Le and A. T. Tran , The Kauffman bracket skein module of two-bridge links, Proc. Amer. Math. Soc. 142 (2014) 1045–1056. Crossref, Web of Science, Google Scholar
35. E. Lee, S. Y. Lee and M. Seo , A recursive formula for the Jones polynomial of 2-bridge links and applications, J. Korean Math. Soc. 46(5) (2009) 919–947. Crossref, Web of Science, Google Scholar
36. S. Lewallen, Floergåsbord, arXiv:407.0769v1 (2014). Google Scholar
37. W. B. R. Lickorish and K. C. Millett , A polynomial invariant of oriented links, Topology 26 (1987) 107–141. Crossref, Web of Science, Google Scholar
38. L. Lovász and M. D. Plummer , Matching theory, in North-Holland Mathematics Studies, Vol. 121 (North-Holland Publishing Co., Amsterdam, 1986), Annals of Discrete Mathematics, 29, xxvii+544 pp. Google Scholar
39. B. Lu and J. K. Zhong , An algorithm to compute the Kauffman polynomial of 2-bridge knots, Rocky Mt. J. Math. 40(3) (2010) 977–993. Crossref, Web of Science, Google Scholar
40. H. Masur , Ergodic theory of translation surfaces, in Handbook of Dynamical Systems, Vol. 1B (Elsevier, Amsterdam, 2006), pp. 527–547. Google Scholar
41. H. Masur and S. Tabachnikov , Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A (North-Holland, Amsterdam, 2002), pp. 1015–1089. Google Scholar
42. H. Murakami , An introduction to the volume conjecture, in Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 1–40. Crossref, Google Scholar
43. S. Nakabo , Formulas on the HOMFLY and Jones polynomials of 2-bridge knots and links, Kobe J. Math. 17(2) (2000) 131–144. Google Scholar
44. S. Nakabo , Explicit description of the HOMFLY polynomials for 2-bridge knots and links, J. Knot Theory Ramifications 11(4) (2002) 565–574. Link, Web of Science, Google Scholar
45. The On-Iine Encyclopedia of Integer Sequences, (2014), Sequence A000931, http://oeis.org. Google Scholar
46. K. Qazaqzeh, M. Yasein and M. Abu-Qamar , The Jones polynomial of rational links, Kodai Math. J. 39(1) (2016) 59–71. Crossref, Web of Science, Google Scholar
47. R. C. Read , A note on tiling rectangles with dominoes, Fibonacci Q. 18 (1980) 24–27. Web of Science, Google Scholar
48. R. P. Stanley , On dimer coverings of rectangles of fixed width, Discrete Appl. Math. 12 (1985) 81–87. Crossref, Web of Science, Google Scholar
49. A. Stoimenow , Rational knots and a theorem of Kanenobu, Exp. Math. 9(3) (2000) 473–478. Crossref, Web of Science, Google Scholar
50. M. B. Thistlethwaite , A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987) 297–309. Crossref, Web of Science, Google Scholar