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On elementary invariants of genus one knots and Seifert surfaces

Christine Lescop

TL;DR

The paper develops elementary, robust invariants for genus one knots in homology 3-spheres by leveraging Alexander forms associated with genus two handlebody exteriors. It defines two simple invariants, w_δ and w_SL, from Seifert-surface data and relates them to the Alexander polynomial and Sato–Levine-type information, establishing their invariance through a detailed analysis of Alexander forms and cobordisms. A central construction is the normalized Reidemeister torsion D(Σ) from the exterior of a genus one Seifert surface, whose degree-2 part and related series yield higher-order invariants (via W_s and δ_Σ) that connect to multivariable Alexander polynomials of two-component links. The work also proves a structure theorem for Alexander forms on genus-two rational handlebodies and shows that all genus-one Seifert surfaces for a knot yield consistent invariants, with deeper connections to known invariants (λ′, w3) and potential applications to finite-type invariants of 3-manifolds. Overall, it provides a new toolkit for distinguishing genus-one knots beyond classical invariants by encoding Seifert-surface data through Alexander forms and their induced torsions.

Abstract

This elementary article 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 two-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.

On elementary invariants of genus one knots and Seifert surfaces

TL;DR

The paper develops elementary, robust invariants for genus one knots in homology 3-spheres by leveraging Alexander forms associated with genus two handlebody exteriors. It defines two simple invariants, w_δ and w_SL, from Seifert-surface data and relates them to the Alexander polynomial and Sato–Levine-type information, establishing their invariance through a detailed analysis of Alexander forms and cobordisms. A central construction is the normalized Reidemeister torsion D(Σ) from the exterior of a genus one Seifert surface, whose degree-2 part and related series yield higher-order invariants (via W_s and δ_Σ) that connect to multivariable Alexander polynomials of two-component links. The work also proves a structure theorem for Alexander forms on genus-two rational handlebodies and shows that all genus-one Seifert surfaces for a knot yield consistent invariants, with deeper connections to known invariants (λ′, w3) and potential applications to finite-type invariants of 3-manifolds. Overall, it provides a new toolkit for distinguishing genus-one knots beyond classical invariants by encoding Seifert-surface data through Alexander forms and their induced torsions.

Abstract

This elementary article 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 two-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.
Paper Structure (24 sections, 58 theorems, 203 equations, 12 figures)

This paper contains 24 sections, 58 theorems, 203 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminary conventions and definitions
  3. An overview
  4. A first simple invariant of genus one knots
  5. A second simple invariant of genus one knots
  6. Definition of Alexander forms and Reidemeister torsions
  7. Invariants of genus one surfaces in rational homology spheres
  8. Sketch of the proof of the invariance
  9. Alexander forms of genus two homology handlebodies
  10. More numerical invariants of genus one surfaces
  11. Questions
  12. More on Alexander forms
  13. Alexander polynomials of knots from Alexander forms
  14. Normalized Alexander polynomials of links
  15. A few lemmas about the structure of Alexander forms
  16. ...and 9 more sections

Key Result

Lemma 1

Two homologous non-separating simple closed curves of a genus one oriented surface $\Sigma$ with one boundary component are isotopic in $\Sigma$.

Figures (12)

  • Figure 1: The surface $\Sigma(a,b,c)$ with its curves $\alpha$, $\beta$, and $\gamma$, $\Sigma(-1,-1,1)$, and the figure-eight knot $K(-1,3,-1)$
  • Figure 2: The handlebody $H_0=H(a,b,c)$ and the curves $A$, $B$, and $C$
  • Figure 3: A modification of the figure-eight knot $K(-1,3,-1)$
  • Figure 4: The based curves $u_{\alpha}$, $u_{\beta}$, $u_{\gamma}$, $A$, $B$, and $C$ on $\partial H_0$
  • Figure 5: The one-skeleton of $\tilde{E_0}$
  • ...and 7 more figures

Theorems & Definitions (81)

  • Lemma 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Proposition 6
  • Definition 7
  • Theorem 8
  • Remark 10
  • Proposition 11
  • Theorem 12
  • ...and 71 more