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On Elementary Invariants of Genus One Knots and Seifert Surfaces

In: Essays on Topology

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  • Christine Lescop

    (Institut Fourier, CNRS, Université Grenoble Alpes)

Abstract

This elementary chapter introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the Alexander form. The Alexander form of a 3-manifold E with boundary contains all Reidemeister torsions of link exteriors obtained by attaching 2-handles along the boundary of E. It is a useful tool for studying Alexander polynomials and Reidemeister torsions. We extract invariants of genus one Seifert surfaces from the Alexander form of their exteriors.

Suggested Citation

  • Christine Lescop, 2025. "On Elementary Invariants of Genus One Knots and Seifert Surfaces," Springer Books, in: Louis Funar & Athanase Papadopoulos (ed.), Essays on Topology, chapter 0, pages 437-494, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-81414-3_20
    DOI: 10.1007/978-3-031-81414-3_20
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