  The effect of strain in zigzag ribbons of monolayer transition-metal
dichalcogenides with induced superconductivity is studied using a minimal
3-band tight-binding model. The unstrained system shows a topological phase
with Majorana zero modes localized at the boundaries of the one-dimensional
(1D) zigzag edges. By direct inspection of the spectrum and wave functions we
examine the evolution of the topological phase as an in-plane, uniaxial
deformation is imposed. It is found that strain shifts the energy of 1D edge
states, thus causing a topological phase transition which eliminates the
Majorana modes. For realistic parameter values, we show that the effect of
strain can be changed from completely destructive -- in which case a small
built in strain is enough to destroy the topological phase -- to a situation
where strain becomes an effective tuning parameter which can be used to
manipulate Majorana zero modes. These two regimes are accessible by increasing
the value of the applied Zeeman field within realistic values. We also study
how strain effects are affected by the chemical potential, showing in
particular how unwanted effects can be minimized. Finally, as a cross-check of
the obtained results, we reveal the connection between 1D Majorana zero modes
in the zigzag edge and the multi-band Berry phase, which serves as a
topological invariant of this system.
