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A 2-torsion invariant of 2-knots

Ryan Budney

TL;DR

The paper defines a $2$-torsion isotopy invariant $\mu$ of $2$-knots $S^2 \to S^4$ as the mod-$2$ degree of a quotient configuration-space map, computable from double-point data and amenable to a Polyak–Viro–style formulation via a $P_\epsilon$ paraboloid deformation. It develops a framework of intersections in configuration spaces with $\mathcal{L}^s$ and $\mathcal{L}^a$, extends to families $\mathrm{Emb}(D^j,D^n)$, and provides practical computation strategies through transversality arguments. The authors compute $\mu$ for several examples, proving $\mu(8_1)=0$ for Artin-spun knots and $\mu(\text{Fox}_{10})=1$, and discuss why many spun constructions yield zero, highlighting potential skein-type behavior for knotted surfaces. The work situates $\mu$ within the Arone–Turchin rational-homotopy framework, relates it to rational Alexander modules in codimension two, and points to broader connections with finite-type invariants and several open questions in higher-codimension knot theory.

Abstract

In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S^2 -> S^4. We give a formula for an invariant of 2-knots, taking values in Z_2 that can be computed in terms of the double-point diagram of the knot. The double-point diagram is a collection of curves and diffeomorphisms of curves, in the domain S^2, that describe the crossing data with respect to a projection, analogous to a chord diagram for a projection of a classical knot S^1 -> S^3. Our formula turns the computation of the invariant into a planar geometry problem. More generally, we describe a numerical invariant of families of knots S^j -> S^n, for all n >= j+2 and j >= 1. In the co-dimension two case n=j+2 the invariant is an isotopy invariant, and either takes values in Z or Z_2 depending on a parity issue.

A 2-torsion invariant of 2-knots

TL;DR

The paper defines a -torsion isotopy invariant of -knots as the mod- degree of a quotient configuration-space map, computable from double-point data and amenable to a Polyak–Viro–style formulation via a paraboloid deformation. It develops a framework of intersections in configuration spaces with and , extends to families , and provides practical computation strategies through transversality arguments. The authors compute for several examples, proving for Artin-spun knots and , and discuss why many spun constructions yield zero, highlighting potential skein-type behavior for knotted surfaces. The work situates within the Arone–Turchin rational-homotopy framework, relates it to rational Alexander modules in codimension two, and points to broader connections with finite-type invariants and several open questions in higher-codimension knot theory.

Abstract

In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S^2 -> S^4. We give a formula for an invariant of 2-knots, taking values in Z_2 that can be computed in terms of the double-point diagram of the knot. The double-point diagram is a collection of curves and diffeomorphisms of curves, in the domain S^2, that describe the crossing data with respect to a projection, analogous to a chord diagram for a projection of a classical knot S^1 -> S^3. Our formula turns the computation of the invariant into a planar geometry problem. More generally, we describe a numerical invariant of families of knots S^j -> S^n, for all n >= j+2 and j >= 1. In the co-dimension two case n=j+2 the invariant is an isotopy invariant, and either takes values in Z or Z_2 depending on a parity issue.

Paper Structure

This paper contains 6 sections, 10 theorems, 22 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The invariant
  3. A double-point formulation of the invariant, the $P_\epsilon$ deformation.
  4. Applying $P_\epsilon$ to the quadrisecants in the domain.
  5. Computing $\mu$ on $2$-knots
  6. The remaining cases: both $n > 4$ and $j > 1$

Key Result

Theorem 1.2

Given a smooth $2$-knot $f : S^2 \to S^4$ the mod-$2$ degree of the map $\mathcal{C}_f / {\mathbb Z }_2 \to S^2$ as an element of ${\mathbb Z }_2$ is an isotopy-invariant of $f$. As a map of the form $\mu : \pi_0 {\mathrm{Emb}}(S^2, S^4) \to {\mathbb Z }_2$ it is additive with respect to the connect

Figures (11)

  • Figure 1: Five points in standard cyclic order in $S^2$ mapped to points in non-consecutive order in $S^4$.
  • Figure 2: Parabolic quadruples, $\epsilon=0$ left. Large $\epsilon$ middle and right.
  • Figure 3: Transversality for quadrisecants intersecting double-point manifolds
  • Figure 4: The invariant $\mu$ as a count of vertical tuples.
  • Figure 5: Transversality for $\mathcal{C}_{ \hbox{$\begin{matrix} [0.5]2 \\ 1 \end{matrix}$} \hbox{$\begin{matrix} [0.5]3 \\ 4 \end{matrix}$} } \cap f_*^{-1}(\mathcal{C}_{ \hbox{$\begin{matrix} [0.5]1 \\ 3 \end{matrix}$} \hbox{$\begin{matrix} [0.5]4 \\ 2 \end{matrix}$} })$
  • ...and 6 more figures

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • ...and 10 more