  Dynamics of simplest quantum vortex knots of torus type in a superfluid at
zero temperature has been simulated within a regularized Biot-Savart law (the
torus radii $R_0$ and $r_0$ for initial vortex configuration were large in
comparison with a vortex core width $\xi$). Computations of evolution times of
knots until their significant deformation were carried out with a small step on
parameter $B_0=r_0/R_0$ for different values of parameter
$\Lambda=\log(R_0/\xi)$. It has been found that at $\Lambda\gtrsim 3$, bands of
quasi-stability appear in a region of $B_0\lesssim 0.2$, which correspond to
long knot lifetimes and to very large traveling distances --- up to several
hundreds of $R_0$. This result is new and quite unexpected, because previously
it was believed that maximal lifetime of torus knots until reconnection does
not exceed several typical periods. The opening of quasi-stable 'windows' at
increasing $\Lambda$ is due to narrowing main parametric resonances of the
dynamical system on parameter $B_0$.
