  In this paper we prove the uniqueness of the saddle-shaped solution to the
semilinear nonlocal elliptic equation $(-\Delta)^\gamma u = f(u)$ in $\mathbb
R^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen-Cahn type. Moreover, we
prove that this solution is stable whenever $2m\geq 14$. As a consequence of
this result and the connection of the problem with nonlocal minimal surfaces,
we show that the Simons cone $\{(x', x'') \in \mathbb R^{m}\times \mathbb R^m \
: \ |x'| = |x''|\}$ is a stable nonlocal $(2\gamma)$-minimal surface in
dimensions $2m\geq 14$.
  Saddle-shaped solutions of the fractional Allen-Cahn equation are doubly
radial, odd with respect to the Simons cone, and vanish only in this set. It
was known that these solutions exist in all even dimensions and are unstable in
dimensions $2$, $4$ and $6$. Thus, after our result, the stability remains an
open problem only in dimensions $8$, $10$, and $12$.
  The importance of studying this type of solution is due to its relation with
the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions
are the simplest non 1D candidates to be global minimizers in high dimensions,
a property not yet established in any dimension.
