Skip to main content
SpringerLink
Log in

Classification of Knots of Small Complexity in Thickened Tori

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We present a table of knots in a thickened torus T × I the diagrams of which have less than five crossing points. The knots are constructed by a three-step process: enumeration of regular graphs of degree 4, enumeration of all corresponding knot projections for each graph, and construction of minimal diagrams. The completeness of the table is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. G. Tait, “On knots,” Edinb. Trans. 28, 145–190 (1877); 32, 327–339, 493–506 (1885).

  2. J. W. Alexander and G. B. Briggs, “On types of knotted curves,” Ann. Math. 28, 562–586 (1927).

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Rolfsen, Knots and Links, Berkeley (1976).

  4. J. Hoste, M. Thistlethwaite, and J. Weeks, “The first 1,701,935 knots,” Math. Intell. 20, No. 4, 33–48 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  5. Yu. V. Drobotukhina, “An analogue of the Jones polynomial for links in ℝP 3 and a generalization of the Kauffman-Murasugi theorem” [in Russian], Algebra Anal. 2, No. 3, 171–191 (1990); English transl.: Leningr. Math. J. 2, No. 3, 613–630 (1991).

    MATH  MathSciNet  Google Scholar 

  6. Yu. V. Drobotukhina, “Classification of links in ℝP 3 with at most six crossings,” In: Topology of Manifolds and Varieties, Am. Math. Soc., Providence RI (1994).

  7. A. Bogdanov, V. Meshkov, A. Omelchenko, and M. Petrov, “Enumerating the k-tangle projections,” J. Knot Theory Ramifications 21, No. 7 (2012). DOI:10.1142/S0218216512500691.

  8. S. A. Grishanov, V. R. Meshkov, and A. V. Omelchenko, “Kauffman-type polynomial invariants for doubly periodic structures,” J. Knot Theory Ramifications 16, No. 6, 779–788 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  9. L. H. Kauffman, “State models and the Jones polynomial,” Topology 26, No. 3, 395–407 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. V. Prasolov and A. B. Sossinsky, Knots, Links, Braids, and 3-Manifolds, Am. Math. Soc., Providence RI ( 1997).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. South Ural State University, 76, Lenin prospekt, Chelyabinsk, 454080, Russia

    A. A. Akimova

  2. Chelyabinsk State University, 129, Brat’ev Kashirinyh St., Chelyabinsk, 454001, Russia

    S. V. Matveev

  3. Institute of Matematics and Mechanics, Ural Branch of RAS, 16, S. Kovalevskaya St., Ekaterinburg, 620219, Russia

    S. V. Matveev

Authors
  1. A. A. Akimova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Matveev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Akimova.

Additional information

Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 12, No. 3, 2012, pp. 10–21.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akimova, A.A., Matveev, S.V. Classification of Knots of Small Complexity in Thickened Tori. J Math Sci 202, 1–12 (2014). https://doi.org/10.1007/s10958-014-2029-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-014-2029-2

Keywords

Search

Navigation