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Seifert surfaces in the four-ball and composition of binary quadratic forms

Menny Aka, Peter Feller, Alison Beth Miller, Andreas Wieser

TL;DR

The article bridges number theory and low-dimensional topology by reinterpreting Gauss composition of binary quadratic forms as a geometric operation on planes in $\mathbb{Z}^4$, encoded by Klein vectors and Bhargava’s cube. This framework yields a complete characterization of when two genus-one Seifert surfaces with the same boundary can be realized as disjoint in $B^4$, via a precise discriminant and composition condition $[ax^2+xy+cy^2]^2*s_1=s_2$, and it ties to symplectic geometry of planes and the double branched covers of pushed-in surfaces. The authors develop both primitive and non-primitive form cases, provide explicit constructions, and supply concrete negative, positive, and square discriminant examples, including connections to the mixing conjecture and CM points. They also establish obstruction results for ambient isotopy in $B^4$, discuss S-equivalence, and pose numerous open problems for further exploration in both number theory and topology.

Abstract

We use composition of binary quadratic forms to systematically create pairs of Seifert surfaces that are non-isotopic in the four-ball. Our main topological result employs Gauss composition to classify the pairs of binary quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert surfaces of genus one. The main ingredient of the proof is number-theoretic and of independent interest. It establishes a new connection between the Bhargava cube and the geometric approach to Gauss composition via planes in the space of two-by-two matrices. In particular, we obtain a geometric recipe that given any two binary quadratic forms finds a Bhargava cube that gives rise to their composition.

Seifert surfaces in the four-ball and composition of binary quadratic forms

TL;DR

The article bridges number theory and low-dimensional topology by reinterpreting Gauss composition of binary quadratic forms as a geometric operation on planes in , encoded by Klein vectors and Bhargava’s cube. This framework yields a complete characterization of when two genus-one Seifert surfaces with the same boundary can be realized as disjoint in , via a precise discriminant and composition condition , and it ties to symplectic geometry of planes and the double branched covers of pushed-in surfaces. The authors develop both primitive and non-primitive form cases, provide explicit constructions, and supply concrete negative, positive, and square discriminant examples, including connections to the mixing conjecture and CM points. They also establish obstruction results for ambient isotopy in , discuss S-equivalence, and pose numerous open problems for further exploration in both number theory and topology.

Abstract

We use composition of binary quadratic forms to systematically create pairs of Seifert surfaces that are non-isotopic in the four-ball. Our main topological result employs Gauss composition to classify the pairs of binary quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert surfaces of genus one. The main ingredient of the proof is number-theoretic and of independent interest. It establishes a new connection between the Bhargava cube and the geometric approach to Gauss composition via planes in the space of two-by-two matrices. In particular, we obtain a geometric recipe that given any two binary quadratic forms finds a Bhargava cube that gives rise to their composition.
Paper Structure (29 sections, 31 theorems, 94 equations, 2 figures)

This paper contains 29 sections, 31 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Seifert surfaces with the same boundary
  3. Gauss composition via planes in $\mathbb{Z}^4$
  4. Symplectic planes and applications to topology
  5. The mixing conjecture and symplectic planes
  6. Notation
  7. The Klein map
  8. Setup and Notation
  9. Planes and the Klein map
  10. Symplectic planes and their complements
  11. Gauss composition via Klein vectors and the Bhargava cube
  12. The primitive case
  13. The general case
  14. Consequences for $L^\perp$ and $L^{\perp \perp}$
  15. Knot theory: realization of disjoint Seifert surfaces with prescribed Seifert forms
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $s_1$ and $s_2$ be $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of IBQFs. The pair $s_1$ and $s_2$ arises as the pair of Seifert forms of a disjoint genus one pair of Seifert surfaces if and only if there exist $a,c\in\mathbb{Z}$ such that $s_1$ and $s_2$ have discriminant $1-4ac$ and $

Figures (2)

  • Figure 1: Left: the surface $\Sigma_{\rm std}$ (gray) with simple closed curves $\alpha_1,\beta_1,\alpha_2,\beta_2$ (red) such that the corresponding homology classes form a basis. Right: Illustration of the positive push-off $\beta^+$ (red) of $\beta$ in $\Sigma$ (gray) needed for the definition of the Seifert form.
  • Figure 2: The two Seifert surfaces (gray) for $K$. The simple closed curves $\alpha$ and $\beta$ (red) are such that $([\alpha], [\beta])$ forms a symplectic basis of the first homology of the surfaces.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • proof : Sketch of proof of Corollary \ref{['cor:symplecticplanes']}
  • Conjecture 1.9
  • ...and 68 more