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A geometric proof that alternating knots are non-trivial

Published online by Cambridge University Press:  24 October 2008

William Menasco  and
Morwen Thistlethwaite
William Menasco
Affiliation:
State University of New York at Buffalo, Buffalo, New York 14222, U.S.A.
Morwen Thistlethwaite
Affiliation:
University of Tennessee, Knoxville, Tennessee 37996, U.S.A.
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Extract

There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 3 , May 1991 , pp. 425 - 431
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Bankwitz, C.. Über die Torsionszahlen der alternierenden Knoten. Math. Ann. 103 (1930), 145–161.CrossRefGoogle Scholar
[2]Burde, G. and Zieschang, H.. Knots (de Gruyter, 1985).Google Scholar
[3]Crowell, R. H.. Genus of alternating link types. Ann. of Math. 69 (1959), 258–275.CrossRefGoogle Scholar
[4]Gabai, D.. Foliations and genera of links. Topology 23 (1984), 381–400.CrossRefGoogle Scholar
[5]Kauffman, L. H.. State Models and the Jones Polynomial. Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[6]Kidwell, M.. On the degree of the Brandt–Lickorish–Millet–Ho polynomial of a link. Proc. Amer. Math. Soc. 100 (1987), 755–762.CrossRefGoogle Scholar
[7]Menasco, W.. Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), 37–44.CrossRefGoogle Scholar
[8]Menasco, W.. Determining incompressibility of surfaces in alternating knot and link complements. Pacific J. Math. 117 (1985), 353–370.CrossRefGoogle Scholar
[9]Thistlethwaite, M. B.. On the Kauffman polynomial of an adequate link. Invent. Math. 93 (1988), 285–296.CrossRefGoogle Scholar