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The first-order genus of a knot

Published online by Cambridge University Press:  01 January 2009

PETER D. HORN
PETER D. HORN*
Affiliation:
Mathematics Department, Rice University, 6100 S. Main St., Houston, TX 77005-1892, U.S.A. e-mail: phorn@rice.edu
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Abstract

We introduce a geometric invariant of knots in S3, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 135 - 149
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Brittenham, M. and Jensen, J. Canonical genus and Whitehead doubles of pretzel knots. Available at http://arxiv.org/abs/math/0608765 (2006).Google Scholar
[2]Cochran, T.Noncommutative knot theory. Algebr. & Geome. Topo. 4 (2004), 347–398.CrossRefGoogle Scholar
[3]Cochran, T., Orr, K. and Teichner, P.Knot concordance, Whitney towers and L 2-signatures. Anna. of Math. 157 (2003), 433–519.CrossRefGoogle Scholar
[4]Cochran, T. and Teichner, P.Knot concordance and von Neumann rho-invariants. Duke Math. J. 137 (2) (2007), 337–379.CrossRefGoogle Scholar
[5]Hempel, J. 3-Manifolds. Ann. of Math. Stud. no. 86 (Princeton University Press, 1976).Google Scholar
[6]Kawauchi, A.Classification of pretzel knots. Kobe J. Math. 2 (1984), 11–22.Google Scholar
[7]Lickorish, W. B. R.Prime knots and tangles. Trans. Amer. Math. Soc. 267 (1) (1981), 321–332.CrossRefGoogle Scholar
[8]Minsky, Y.The classification of punctured-torus groups. Ann. Math. 149 (1999), 559–626.CrossRefGoogle Scholar
[9]Rolfsen, D.Knots and Links (Publish or Perish, Berkeley, 1976).Google Scholar
[10]Schubert, H.Knoten und Vollringe. Acta Math. 90 (1953), 131–286.CrossRefGoogle Scholar
[11]Teichner, P.What is a grope? Notices Amer. Math. Soc. 51 (8) (2004), 892–893.Google Scholar
[12]Whitten, W.Isotopy types of knot spanning surfaces. Topology 12 (1973), 373–380.CrossRefGoogle Scholar