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The first-order genus of a knot

  • PETER D. HORN (a1)
Abstract
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.

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[2] T. Cochran Noncommutative knot theory. Algebr. & Geome. Topo. 4 (2004), 347398.

[3] T. Cochran , K. Orr and P. Teichner Knot concordance, Whitney towers and L2-signatures. Anna. of Math. 157 (2003), 433519.

[4] T. Cochran and P. Teichner Knot concordance and von Neumann rho-invariants. Duke Math. J. 137 (2) (2007), 337379.

[7] W. B. R. Lickorish Prime knots and tangles. Trans. Amer. Math. Soc. 267 (1) (1981), 321332.

[8] Y. Minsky The classification of punctured-torus groups. Ann. Math. 149 (1999), 559626.

[10] H. Schubert Knoten und Vollringe. Acta Math. 90 (1953), 131286.

[12] W. Whitten Isotopy types of knot spanning surfaces. Topology 12 (1973), 373380.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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