On motivic multiple $t$ values, Saha's basis conjecture, and generators of alternating MZV's

We give an evaluation for the stuffle-regularised $t^{\ast,V}(\{2\}^a,1,\{2\}^b)$ as a polynomial in single-zeta values, $\log(2)$ and $V$. We then apply this to establish some linear independence results of certain sets of motivic multiple $t$ values. In particular, we prove the elements of Saha's conjectural basis are linearly independent, on the motivic level, and that the (suitably regularised) elements $t^\mathfrak{m}(\{1,2\}^\times)$ form a basis for both the (extended) motivic MtV's and the alternating MZV's.