Alexander Polynomials of Knots Which Are Transformed into the Trefoil Knot by a Single Crossing Change

- Journal title : Kyungpook mathematical journal
- Volume 52, Issue 2, 2012, pp.201-208
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2012.52.2.201

Title & Authors

Alexander Polynomials of Knots Which Are Transformed into the Trefoil Knot by a Single Crossing Change

Nakanishi, Yasutaka;

Nakanishi, Yasutaka;

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.

Keywords

Alexander polynomials;Crossing change;Trefoil knot;Figure-eight knot;

Language

English

Cited by

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

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.