Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Alexander Polynomials of Knots Which Are Transformed into the Trefoil Knot by a Single Crossing Change

  • Received : 2009.08.18
  • Accepted : 2012.05.25
  • Published : 2012.06.23
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Abstract

By the works of Kondo and Sakai, it is known that Alexander polynomials of knots which are transformed into the trivial knot by a single crossing change are characterized. In this note, we will characterize Alexander polynomials of knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change.

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References

  1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts Math., 84, Second Edition, Springer-Verlag, New York, 1990.
  2. H. Kondo, Knots of unknotting number 1 and their Alexander polynomials, Osaka J. Math., 16(1979), 551-559
  3. J. Levine, A characterization of knot polynomials, Topology, 4(1965), 135-141. https://doi.org/10.1016/0040-9383(65)90061-3
  4. Y. Nakanishi, Local moves and Gordian complexes, II, Kyungpook Math. J., 47(2007), 329-334.
  5. D. Rolfsen, A surgical view of Alexander's polynomial, in Geometric Topology (Proc. Park City, 1974), Lecture Notes in Math. 438, Springer-Verlag, Berlin and New York, 1974, pp. 415-423.
  6. D. Rolfsen, Knots and Links, Math. Lecture Series 7, Publish or Perish Inc., Berkeley, 1976.
  7. T. Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ., 5(1977), 451-456.
  8. H. Seifert, Uber das Geschlecht von Knoten, Math. Ann., 110(1934), 571-592.
  9. T. Takagi, Shotou Seisuuron Kougi (in Japanese) [Lectures on Elementary Number Theory], Second Edition, Kyoritsu Shuppan, Tokyo, 1971.

Cited by

  1. Differences of Alexander polynomials for knots caused by a single crossing change, II vol.26, pp.14, 2017, https://doi.org/10.1142/S0218216517500973
  2. Knot adjacency from a surgical view of Alexander invariants vol.26, pp.14, 2017, https://doi.org/10.1142/S0218216517500985