Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Polynomial Unknotting and Singularity Index

  • Mishra, Rama (Indian Institute of Science Education and Research)
  • Accepted : 2013.03.19
  • Published : 2014.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce a new method to transform a knot diagram into a diagram of an unknot using a polynomial representation of the knot. Here the unknotting sequence of a knot diagram with least number of crossing changes can be realized by a family of polynomial maps. The number of singular knots in this family is defined to be the singularity index of the diagram. We show that the singularity index of a diagram is always less than or equal to its unknotting number.

Keywords

References

  1. N. A'Campo, Le Groupe de Monodromie du Deploiement des Singularites Isolees de Courbes Planes I, Math. Ann., 213(1975), 1-32. https://doi.org/10.1007/BF01883883
  2. N. A'Campo, Singularities and Related Knots, University of Tokyo, Notes by William Gibson and Masaharu Ishikawa.
  3. J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Studies, 61, Princeton Univ. Press, 1968.
  4. K. Murasugi, Knot Theory and Its Applications, Modern Birkhauser Classics, 1996.
  5. P. B. Kronheimer, T. S. Mrowka, "Gauge theory for embedded surfaces, I", Topology, 32(1993), 773-826. https://doi.org/10.1016/0040-9383(93)90051-V
  6. J. Rasmussen, Khovanov homology and the slice genus. arXiv:math.GT/0402131v1, 2004, 20 pages, 8 figures.
  7. R. Mishra, Polynomial representation of torus knots of type (p; q), J. Knot Theory Ramifications, 8(1999), 667-700. https://doi.org/10.1142/S0218216599000432
  8. R. Mishra, Minimal degree sequence for torus knots, J. Knot Theory Ramifications, 9(2000), 759-769. https://doi.org/10.1142/S0218216500000438
  9. R. Mishra, Polynomial representation of strongly invertible and strongly negative am-phicheiral knots, Osaka Journal of Mathematics, 43(2006), 625-639.
  10. P. Madeti, R. Mishra, Minimal Degree Sequence for Torus Knots of Type (p; 2p - 1), J. Knot Theory Ramifications, 15(2006), 1141-1151. https://doi.org/10.1142/S021821650600497X
  11. P. Madeti, R. Mishra, Minimal Degree Sequence for 2-bridge knots, Fund. Math., 190(2006), 191-210. https://doi.org/10.4064/fm190-0-7
  12. P. Madeti and R. Mishra, Minimal Degree Sequence for Torus Knots of Type (p; q), J. Knot Theory Ramifications, 18(2009), 485-491. https://doi.org/10.1142/S021821650900704X
  13. D. Pecker, Sur le theorµeme local de Harnack, C. R. Acad. Sci. Paris 326, Serie 1(1998), 573-576.
  14. R. Shukla, A. Ranjan, On Polynomial Representation of Torus knots, J. of Knot Theory Ramifications, 5(1996), 279-294. https://doi.org/10.1142/S0218216596000199
  15. A. R. Shastri, Polynomial representations of knots, Tohoku Math. J., (2) 44(1992), 11-17. https://doi.org/10.2748/tmj/1178227371
  16. R. Shukla, On polynomial isotopy of knot-types, Proc. Indian Acad. Sci. Math. Sci., 104(1994), 543-548. https://doi.org/10.1007/BF02867119