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Algorithms for Computing Some Invariants for Discrete Knots

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DOI: 10.4236/am.2013.411206    2,626 Downloads   3,897 Views  

ABSTRACT

Given a cubic knot K, there exists a projection  of the Euclidean space R3 onto a suitable plane  such that p(K) is a knot diagram and it can be described in a discrete way as a cycle permutation. Using this fact, we develop an algorithm for computing some invariants for K: its fundamental group, the genus of its Seifert surface and its Jones polynomial.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Hinojosa, G. , Torres, D. and Valdez, R. (2013) Algorithms for Computing Some Invariants for Discrete Knots. Applied Mathematics, 4, 1526-1530. doi: 10.4236/am.2013.411206.

References

[1] M. Boege, G. Hinojosa and A. Verjovsky, “Any Smooth Knot Sn Rn+2 Is Isotopic to a Cubic Knot Contained in the Canonical Scaffolding of Rn+2,” Revista Matemática Complutense, Vol. 24, No. 1, 2011, pp. 1-13.
http://dx.doi.org/10.1007/s13163-010-0037-4
[2] G. Hinojosa, A. Verjovsky and C. V. Marcotte, “Cubulated Moves and Discrete Knots,” 2013, pp. 1-40.
http://arxiv.org/abs/1302.2133
[3] D. Rolfsen, “Knots and Links,” AMS Chelsea Publishing, American Mathematical Society, Providence Rhode Island, 2003.
[4] R. H. Fox, “A Quick Trip through Knot Theory. Topology of 3-Manifolds and Related Topics,” Prentice-Hall, Inc., Upper Saddle River, 1962.
[5] “The Knot Atlas,” 2013. http://katlas.math.toronto.edu

  
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