can be interpreted as follows. When $n_{2d} \lesssim 1\cdot10^{13}$cm$^{-2}$, only the spin-orbit split sub-bands at K/K’ are populated, and the FS is composed only by two pockets, Fig.\[figure:kinks\]a. For $n_{2d}$ between$\sim1.5$ and $2\cdot 10^{13}$cm$^{-2}$, $E_F$ crosses the bottom of the Q$_1$ sub-band and two extra pockets appear in the FS at Q/Q’[@GePRB2013; @BrummePRB2015], Fig.\[figure:kinks\]b. The emergence of these pockets induces a Lifshitz transition, i.e. an abrupt change in the topology of the FS[@LifshitzJETP1960]. Once Q$_1$ is populated and $E_F$ is large enough ($n_{2d}\sim6\cdot 10^{13}$cm$^{-2}$), the system becomes superconducting[@LuScience2015; @YeScience2012]. For slightly larger $E_F$ ($7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$), $E_F$ crosses the bottom of Q$_2$ resulting in a second Lifshitz transition, and other two pockets emerge in the FS at Q/Q’[@BrummePRB2015], Fig.\[figure:kinks\]c.

We note that the experimentally observed kinks are at different $n_{2d}$ with respect to the theoretical ones for 3L-MoS$_2$[@BrummePRB2016]. Ref. predicted that for a $1.28\%$ in-plane tensile strain, Q$_1$ and Q$_2$ should be crossed for $n_{2d}\sim5\cdot 10^{13}$ and $\sim1\cdot 10^{14}$. Since the positions of the sub-band crossings are strongly dependent on strain[@BrummePRB2016], we estimate the strain in our devices by monitoring the frequency of the E$_{2g}^1$ mode via Raman spectroscopy.

![a) Raman spectra of the 4L-MoS$_2$ device 