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Tabulation of Prime Knots in Lens Spaces

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Abstract

Using computational techniques, we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces \(L(p,q)\). For these knots, we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for \(p\ge 13\).

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References

  1. Cattabriga, A., Manfredi, E., Mulazzani, M.: On knots and links in lens spaces. Topol. Appl. 160, 430–442 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Culler, M., Dunfield, N., Weeks, J.: Snappy, a program for studying the topology and geometry of 3-manifolds (2016). http://www.math.uic.edu/t3m/SnapPy/

  3. Gabrovšek, B.: Classification of knots in \(L(p,q)\), C++ source code (2016). https://github.com/bgabrovsek/lpq-classification

  4. Gabrovšek, B., Mroczkowski, M.: Knots in the solid torus up to 6 crossings. J. Knot Theory Ramif. 21, 11, 1250106 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Gabrovšek, B., Mroczkowski, M.: The HOMFLYPT skein module of the lens spaces \(L(p,1)\). Topol. Appl. 175, 72–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diamantis, I., Lambropoulou, S.: Braid equivalence in 3-manifolds with rational surgery description. Topol. Appl. 194, 269–295 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diamantis, I., Lambropoulou, S.: A new basis for the Homflypt skein module of the solid torus. J. Pure Appl. Algebra 220(2), 577–605 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diamantis, I., Lambropoulou, S., Przytycki, J.H.: Topological steps toward the Homflypt skein module of the lens spaces \(L(p,1)\) via braids. J. Knot Theory Ramif. 25(14), (2016). doi:10.1142/S021821651650084X

  9. Drobotukhina, YuV: Classification of link in \(\mathbb{R}\text{ P }^3\) with at most six crossings. Adv. Sov. Math. 18(1), 87–121 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Epstein, D.B.A., Penner, R.C.: Eucledian decompositions of noncompact hyperbolic manifolds. J. Diff. Geom. 27, 67–80 (1988)

    Article  MATH  Google Scholar 

  11. Gauss, C.F.: Werke (8), pp. 282–286. Teubner, Leipzig (1900)

    Google Scholar 

  12. Gordon, C., Luecke, J.: Knots are determined by their complements. J. Am. Math. Soc. 2, 371–415 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoste, J., Kidwell, M.: Dichromatic link invariants. Trans. Am. Math. Soc. 321(1), 197–229 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoste, J., Thistlethwaite, M., Weeks, J.: The first 1,701,936 knots. Math. Intell. 20, 33–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hoste, J., Przytycki, J.H.: The \((2,\infty )\)-skein module of lens spaces; a generalization of the Jones polynomial. J. Knot Theory Ramif. 2(3), 321–333 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hoste, J., Przytycki, J.H.: The Kauffman bracket skein module of \(S^1 \times S^2\). Math. Z. 220(1), 63–73 (1995)

    Article  MathSciNet  Google Scholar 

  17. Lambropoulou, S.: Knot theory related to generalized and cyclotomic Hecke algebras of type B. J. Knot Theory Ramif. 8(5), 621–658 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lambropoulou, S.: \(L\)-moves and Markov theorems. J. Knot Theory Ramif. 16(10), 1459–1468 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lambropoulou, S.: Solid torus links and Hecke algebras of \(B\)-type. In: Yetter, D.N. (ed.) Quantum Topology, pp. 225–245. World Scientific Press, Singapore (1994)

    Google Scholar 

  20. Lambropoulou, S., Rourke, C.P.: Markov’s theorem in 3-manifolds. Topol. Appl. 78, 95–122 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lambropoulou, S., Rourke, C.P.: Algebraic Markov equivalence for links in 3-manifolds. Compos. Math. 142, 1039–1062 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lovászand, L., Marx, M.L.: A forbidden substructure characterization of Gauss codes. Bull. Am. Math. Soc. 82(1), 121–122 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Marx, M.L.: The Gauss realizability problem. Proc. Am. Math. Soc. 22, 610–613 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mroczkowski, M.: Kauffman Bracket Skein module of the connected sum of two projective spaces. J. Knot Theory Ramif. 20(5), 651–675 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mroczkowski, M.: Kauffman bracket skein module of a family of prism manifolds. J. Knot Theory Ramif. 20(1), 159–170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mroczkowski, M.: Polynomial invariants of links in the projective space. Fund. Math. 184(1), 223–267 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mroczkowski, M., Dabkowski, M.K.: KBSM of the product of a disk with two holes and \(S^1\). Topol. Appl. 156(10), 1831–1849 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Prasad, G.: Strong rigidity of Q-rank 1 lattices. Invent. Math 21, 255–286 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  29. Przytycki, J.H.: Algebraic topology based on knots: an introduction. In: Proceedings of Knots ’96, pp. 279–297. World Sci, River Edge, NJ (1997)

  30. Przytycki, J.H.: Fundamentals of Kauffman bracket skein modules. Kobe J. Math. 16(1), 45–66 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Reidemeister, K.: Homotopieringe und Linsenräume. Abh. Math. Semin. Hamb. Univ. 11, 102–109 (1935)

    Article  MATH  Google Scholar 

  32. Schubert, H.: Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949(3), 57–104 (1949)

  33. Tait, P.G.: On knots II. Trans. R. Soc. Edinb. 32, 327–342 (1883-4)

  34. Turaev, V.G.: The Conway and Kauffman modules of the solid torus, Zap. Nauchn. Sem. Lomi 167:79–89 (1998, English translation: J. Soviet Math., 2799–2805 (1990))

  35. Weeks, J.: Convex hulls and isometries of cusped hyperbolic 3-manifolds. Topol. Appl. 52, 127–149 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Whitten, W.C.: A pair of non-invertible links. Duke Math. J. 36, 695–698 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  37. Whitten, W.C.: On noninvertible links with invertible proper sublinks. Proc. Am. Math. Soc. 26(2), 341–346 (1970)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, 1000, Ljubljana, Slovenia

    Boštjan Gabrovšek

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  1. Boštjan Gabrovšek
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Gabrovšek, B. Tabulation of Prime Knots in Lens Spaces. Mediterr. J. Math. 14, 88 (2017). https://doi.org/10.1007/s00009-016-0814-5

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  • DOI: https://doi.org/10.1007/s00009-016-0814-5

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