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A note on unknotting number

Published online by Cambridge University Press:  24 October 2008

Steven A. Bleiler
Steven A. Bleiler
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A.
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For a classical knot K in the 3-shere, the unknotting number u(K) is defined to be the smallest number of crossing changes required to obtain the unknot, the minimum taken over all the regular projections. This dependence on projection makes the unknotting number a difficult knot invariant. While some algebriac methods exist to give a lower bound for u(K) the unknotting number for approximately one-sixth of the 84 knots with nine or fewer crossings remains unddetermined, see [9] or [7]. For an upper bound, one is usually forced to intellgently experiment knotting various projections of the knot under study. Usual practice is to work with a minimal crossingprojection; indeed, it has long been a ‘folk’ conjecture that the unknotting number is realized in such a projection ([5], p. 21). This note shows by example that this conjecture is false. This remarkable knot is rational, i.e. a 2-bridge knot, and hence alternating. Thus there is also the surprising result that the unknotting number of an alternating kniot is not necessarily realized in a minimal alternating projecting.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 469 - 471
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Alexander, J. W. and Briggs, G. B.. On types of knotted curves. Ann. of Math. 28 (1927), 562–586.CrossRefGoogle Scholar
[2]Bleiler, S.. Unknotting rational knots (to appear).Google Scholar
[3]Caudron, A.. Classification des neuds et des enlacements. Publications Mathématiques d'Orsay 82–04, Bat. 425, 91405 Orsay, France.Google Scholar
[4]Conway, J. H.. An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Pergamon Press, 1969), 329–358.Google Scholar
[5]McA, C.. Gordon. Knot theory. A course at the University of California-Santa Barbara, Fall 1979. Lecture Notes by S. A. Bleiler, preprint.Google Scholar
[6]Hirzebruch, F., Neumann, W. D., Koh, S. S.. Differential Manifolds and Quadratic Forms, Lecture notes in pure and applied mathematics, vol. 4 (Marcel Dekker, 1971).Google Scholar
[7]Lickorish, W. R.. The unknotting number of a classical knot, preprint.Google Scholar
[8]Murasugi, K.. On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1965), 387–422.CrossRefGoogle Scholar
[9]Nakanishi, Y.. A Note on Unknotting Number, Math. Sem. Notes, Kobe Univ. vol. 9 (1981), 99–108.Google Scholar
[10]Thistlewaite, M.. The classification of knots with small crossing number, (to appear).Google Scholar
[11]Wendt, H.. Die gordische Auflösung von Knoten. Math. Zeit, 42 (1937), 680–696.CrossRefGoogle Scholar