Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)

Question

Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\forall x$ s.t. $f'(x)\in \mathbb{Q}$ and $x\in (0,a)$, the torus $T^2$ in the fiber of $(x,f(x))$ is foliated by Reeb orbits which "wind $v_1$ and $v_2$ times", in the respective directions on the torus induced by the angle coordinates coming from the Arnold-Liouville Theorem. Take any one of these orbits and call it $o_{v_1,v_2}$ Likewise, we have that the singular fibers over the points $(0,b)$ and $(a,0)$ (which are topologically $S^1$) are also Reeb orbits, call them $\gamma_1,\gamma_2$. The claim I would like to prove is: $$link(\gamma_1,\gamma_2)=1$$ $$link(\gamma_1,o_{v_1,v_2})=-v_2$$ $$link(\gamma_2,o_{v_1,v_2})=v_1$$ $$link(o_{v_1,v_2},o_{w_1,w_2})=min(-v_1w_2,-v_2w_1)$$

The first case can be deduced from the fact that, by taking stereographic projection of $S^3$ into $\mathbb{R}^3$ on a point of $\gamma_2$, we get that $\gamma_2,\gamma_1$ turn into the z-ax and a circle in the xy-plane around the z-ax (by istopy think of it as the circle $x^2+y^2=1$), respectively. From this we get the result.

Now the second case I'm almost sure that comes by perturbing isotopically the orbit $\gamma_1$ to the embedded loop in the torus above $(x,f(x))$ given by the direction $(0,1)$ and the minus comes from orientation considerations (this perturbation process can certainly be done trough all $x\in (0,a)$ because there is an easy identification of these tori, to make the leap form (0,a) to [0,a) we take the projection of the torus above $(x,f(x))$ into its first angle coordinates for $x$ small enough, and notice that this can be effectively done by considering the family of tori foliating $\mathbb{R}^3$ and degenerating at the image of the orbits $\gamma_1$ and $\gamma_2$ by the stereographic projection.

Now the last case is the one puzzling me, because by the same logic we could translate one orbit in a torus isotopically to one orbit into another torus containing the other orbit, and then in my mind, the linking number should be the sum $-v_1w_2-v_2w_1$.

I appreciate any help with this case, and in the case of any of the other arguments being wrong I would like a correction.

To give you some context, this has been extracted from the paper https://arxiv.org/pdf/1310.6647.pdf page 24. Also, the angle coordinates and the toric domain is defined as the preimage of the region described at the beggining by the map $$p:\mathbb{C}^2\mapsto \mathbb{R}^2$$ $$p(z_1,z_2)=(\pi|z_1|^2,\pi|z_2|^2)$$.

Answer 1

With the usual notion of linking number, the puzzling equality cannot be valid for every pair of disjoint embedded curves: reversing the orientation of one of the two curves, say $o_{v_1, v_2} \to o_{-v_1, -v_2}$, should merely change the sign of the linking number, which is not how the minimum between $-v_1 w_2$ and $-v_2 w_1$ generally changes upon such reversion. I have not read the paper you referred to, but since the authors seem to restrict attention to some specific orbits, it may be that the puzzling equation can be made sense of in their specific context.

In the generality of your question, the correct expression for the linking number is given in the proposition below. For $x \in [0, a]$, let $$T(x) = p^{-1}(x, f(x)) = \{(z_1, z_2) \in \mathbb{C}^2 : \pi|z_1|^2 = x, \pi|z_2|^2 = f(x) \} .$$ To fix ideas, for every $x \in [0,a]$, let $o_{v_1, v_2}$ be the embedded circle in $T(x)$ parametrized by the map $t \mapsto \left( \sqrt{\frac{x}{\pi}} \, e^{2\pi i v_1 t}, \sqrt{\frac{x}{\pi}} \, e^{2\pi i v_2 t} \right)$, where $t \in \mathbb{R}/\mathbb{Z}$.

Proposition. If $o_{v_1, v_2}$ lies in $T(V)$ and $o_{w_1, w_2}$ lies in $T(W)$ where $0 \le V < W \le 1$, then up to an overall sign depending on conventions, $$ \mathrm{link}(o_{v_1, v_2}, o_{w_1, w_2}) = v_2 w_1 . $$

Proof. For $t \in [0,1]$, consider the curves $o_{v_1, v_2}$ in $T((1-t)V)$ and $o_{w_1, w_2}$ in $T(ta + (1-t)W)$. These define two regular homotopies, the first sending $o_{v_1, v_2} \subset T(V)$ to $v_2 \gamma_2$ in $T(0)$ and the second sending $o_{w_1, w_2} \subset T(W)$ to $w_1 \gamma_1$ in $T(a)$. Since every curve $o_{v_1, v_2}$ is disjoint from every curve $o_{w_1, w_2}$, the linking number is preserved by these homotopies. Hence

$$ \mathrm{link}(o_{v_1, v_2}, o_{w_1, w_2}) = \mathrm{link}(v_2 \gamma_2, w_1 \gamma_1) = v_2 w_1 \, \mathrm{link}(\gamma_2, \gamma_1) = \pm v_2 w_1 \, . \qquad \square$$

Remark. This result is coherent with the fact that when $V=W$, the algebraic intersection number of $o_{v_1, v_2}$ with $o_{w_1, w_2}$ is $v_1 w_2 - v_2 w_1$.