  In this paper, we discuss the possibility of achieving tunable topologically
protected edge modes through the application of uniaxial deformation in an
auxetic metamaterial. The proposed structure consists of a thin slab with
oriented cuts in a hexagonal lattice, where topologically protected band gaps
are opened by introducing a controlled variation in the cut lengths. Numerical
simulations demonstrate the existence of topologically protected and
scatter-free wave propagation in the structure at the interface between two
sub-domains with modified cells, in distinct frequency ranges. This only occurs
in the presence of auxeticity. In addition, exploiting geometrical
nonlinearity, the application of a uniaxial strain can be used to close the
topological band gaps or to modify their frequency range, i.e., to weaken the
localization effects or to shift the frequency at which they occur. The spatial
and temporal variation of the applied strain field can thus be used for the
dynamic tuning of metamaterial topological waveguiding properties, with
applications in mechanical devices for logic operations and computations.
