Theorem 1.7 (necessity): a binding of a disk-like global section is unknotted with $\operatorname{sl}=-1$

Read-back. Let $H : \mathbb{R}^4 \to \mathbb{R}$ be any function; coordinates are $x = (x_0,\dots,x_3)$, $DH(x)$ is the Fréchet derivative (taken to be $0$ where not differentiable), $\Sigma = \{H = 1\}$, and

$$X_H(x) = \bigl(-\partial_1 H(x),\ \partial_0 H(x),\ -\partial_3 H(x),\ \partial_2 H(x)\bigr),\quad \partial_iH(x) = DH(x)e_i .$$

A trajectory is $y : \mathbb{R}\to\mathbb{R}^4$ with $y'(t) = X_H(y(t))$ and $H(y(t)) = 1$ for all $t$. A periodic orbit is a trajectory $x$ with a real $T>0$ and $x(t+T) = x(t)$ for all $t$ ($T$ not necessarily minimal).

Notation: $D, \mathring D, S^1 \subset \mathbb{R}^2$ are the closed unit disk, open unit disk and unit circle; $\langle a,b\rangle = \sum_i a_ib_i$, $|a| = \sqrt{\langle a,a\rangle}$; $\omega_0(u,v) = u_0v_1 - u_1v_0 + u_2v_3 - u_3v_2$; $\lambda_x(v) = \tfrac12(x_0v_1 - x_1v_0 + x_2v_3 - x_3v_2)$; $Q_1x = (-x_2, x_3, x_0, -x_1)$, $Q_2x = (x_3, x_2, -x_1, -x_0)$;

$$\xi_1(x) = |x|^{-1}\Bigl(Q_2x - \tfrac{DH(x)Q_2x}{DH(x)x}x\Bigr),\qquad \xi_2(x) = |x|^{-1}\Bigl(Q_1x - \tfrac{DH(x)Q_1x}{DH(x)x}x\Bigr).$$

Division by $0$ and $0^{-1}$ evaluate to $0$.

Hypotheses.

• $hS$ (strictly star-shaped level): $H$ is $C^\infty$ on $\mathbb{R}^4$; for every $u \neq 0$ there is exactly one $r > 0$ with $H(ru) = 1$; and $DH(x)\,x > 0$ for every $x \in \Sigma$.
• $hdc$ (dynamically convex): for every periodic orbit $(x, T)$ of $H$ (of any, not necessarily minimal, period) and every $Y : \mathbb{R}\to \mathrm{L}(\mathbb{R}^4,\mathbb{R}^4)$ with $Y(0) = \mathrm{id}$ and $Y'(t) = DX_H(x(t))\circ Y(t)$ for all $t$, the integer $\mu(M) \ge 3$, where $M(\tau)$ is the $2\times2$ matrix

$$M(\tau)_{ij} = \Bigl[c_{x(T\tau)}\bigl(\pi_{x(T\tau)}(Y(T\tau)\,\xi_j(x(0)))\bigr)\Bigr]_i,\quad \pi_x(v) = v - \tfrac{\lambda_x(v)}{\lambda_x(X_H(x))}X_H(x),\quad c_x(v) = \bigl(\omega_0(v,\xi_2(x)),\ \omega_0(\xi_1(x),v)\bigr),$$

and for a matrix path $\varphi$: $W(\varphi)$ is the set of $(\theta(1)-s)/(2\pi)$ over $s \in [0,2\pi]$ and $\theta$ continuous on $[0,1]$ with $\theta(0)=s$ and $\varphi(t)(\cos s,\sin s) \in \mathbb{R}_{>0}\,(\cos\theta(t),\sin\theta(t))$ for $t\in[0,1]$; $a = \inf W$, $b = \sup W$ (both $0$ if $W$ is empty or unbounded on that side); $\mu(\varphi) = 2(\lceil b\rceil - 1)$ if $a \le \lceil b\rceil - 1$, else $2\lceil b\rceil - 1$.

• $P$: a periodic orbit $(x, T)$ of $H$.
• $hP$ (prime): $x(t) \neq x(0)$ for all $0 < t < T$.
• $hD$ (bounds a disk-like global section): there is $e : \mathbb{R}^2 \to \mathbb{R}^4$ such that: $e$ is $C^\infty$, injective on $D$, with $De(u)$ injective for every $u\in D$; $e(D) \subseteq \Sigma$; $e(S^1) = x(\mathbb{R})$; $X_H(e(u)) \notin \operatorname{range} De(u)$ for every $u \in \mathring D$; and every trajectory $y$ with $y(0) \notin x(\mathbb{R})$ satisfies, for every $a\in\mathbb{R}$, $y(t)\in e(D)$ for some $t > a$ and $y(t')\in e(D)$ for some $t' < a$.

Conclusion (a conjunction of two statements).

1. Unknotted: there is $e : \mathbb{R}^2\to\mathbb{R}^4$ that is $C^\infty$, injective on $D$, with $De(u)$ injective for $u\in D$, with $e(D)\subseteq\Sigma$ and $e(S^1) = x(\mathbb{R})$. (These are exactly the first three conditions already contained in $hD$.)
2. Self-linking number $-1$: there is $\varepsilon_0 > 0$ such that for every $\varepsilon$ with $0 < \varepsilon < \varepsilon_0$, the linking number of the two loops

$$\gamma_1(s) = \frac{x(Ts)}{|x(Ts)|}, \qquad \gamma_2^\varepsilon(s) = \frac{x(Ts) + \varepsilon\,\xi_1(x(Ts))}{|x(Ts) + \varepsilon\,\xi_1(x(Ts))|} \qquad (s\in\mathbb{R})$$

equals $-1$, in the following sense:

• (a) there exists $N\in\mathbb{R}^4$ with $|N| = 1$ such that $\gamma_1(s)\neq N$ and $\gamma_2^\varepsilon(s)\neq N$ for all $s\in\mathbb{R}$; and
• (b) for every such $N$ (unit, avoided by both loops for all $s$), $\mathrm{Lk}_N(\gamma_1,\gamma_2^\varepsilon) = -1$, where, with $\sigma_N(y) = (1 - \langle y,N\rangle)^{-1}\bigl(y - \langle y,N\rangle N\bigr)$ and $V_N(a,b,c) = \det$ of the $4\times4$ matrix with rows $-N, a, b, c$,

$$\mathrm{Lk}_N(\gamma_1,\gamma_2) = \frac{1}{4\pi}\int_0^1\!\!\int_0^1 \frac{V_N\Bigl(\tfrac{d}{ds}\sigma_N(\gamma_1(s)),\ \tfrac{d}{dt}\sigma_N(\gamma_2(t)),\ \sigma_N(\gamma_1(s)) - \sigma_N(\gamma_2(t))\Bigr)}{\bigl|\sigma_N(\gamma_1(s)) - \sigma_N(\gamma_2(t))\bigr|^3}\,dt\,ds .$$

Edge cases and conventions. The push-off direction is the specific vector field $\xi_1$ (built from $Q_2$). In $\mathrm{Lk}_N$: a derivative at a non-differentiable point is $0$; the inner and outer integrals are $0$ if their integrands are not integrable; a zero denominator makes the integrand $0$; $\gamma_1$, $\gamma_2^\varepsilon$ evaluate to $0$ where the vector being normalized is $0$. Condition (b) ranges over all admissible unit $N$, and (a) asserts at least one exists. The first conjunct follows directly from the data in $hD$. Hypothesis $hS$(3) excludes $0$ from $\Sigma$, so $|x(Ts)| > 0$.