when the object is trapped in the middle of the two polymers, the symmetry line $\theta=\pi/2$ is a line of equilibrium points, stable in the angular direction and neutral in the radial one. Hence the object is brought back to the symmetry line and force flow lines are circles $r=cst$ heading towards $\theta=\pi/2$. A remarkable property holds: $$\label{11}
{\cal P}_m(z,\bar z)= \frac{16}{5} {\cal P}^{(1)}_{l}(z,\bar z) {\cal P}^{(1)}_r(z,\bar z)\ ,$$ where ${\cal P}^{(1)}_{l/r}$ is the (Schramm) probability for a single self-avoiding polymer to pass left/right of the point. Hence, if the point is in the middle, the fluctuation force is [*the same*]{} as for two independent polymers, i.e. mutual avoidance does not change the result, as can be checked on (\[middlef\]). This is [*not true*]{} if the polymers are on the same side of the object.

Let us consider again the geometry of Fig. \[f:2\] with now both polymers attached at $x=-L$, and both passing through a hole at $x= L$, see top right of figure \[f:6\]. An object trapped in the middle acquires a free energy: $${\cal F}_m=
-k T \log \left(\frac{16 r^2 L^2 \sin ^2(\theta )}{5 \left(r^4-2 \cos (2 \theta ) r^2L^2+L^4\right)}\right)\ .$$ The equipotential lines are given on 