The full twist maps to $-I$ in the specialized reduced Burau representation at $t=-1$

The specialization at $t=-1$ of the 2-dimensional reduced Burau representation sends the full twist $\Delta^2 = (\sigma_1\sigma_2)^3$ to the central element $-1$ of $\mathrm{SL}(2,\mathbb Z)$.

With $A=\begin{pmatrix}1&-1\\0&1\end{pmatrix}$ and $B=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$ the images of the two generators, the statement is

Since $-1$ is the non-trivial central element of $\mathrm{SL}(2,\mathbb Z)$ (of order $2$), this says that the image of $\Delta^2$ has order exactly $2$, while the separate statement BurauFaithful.spec_reduced_coxeter records $(AB)^6=1$, i.e. the image of $\Delta^4=(\sigma_1\sigma_2)^6$ is trivial. Hence the kernel of the specialization contains $\Delta^4$ but not $\Delta^2$, which together with the Coxeter-Moser presentation pins the kernel down to $\langle\Delta^4\rangle$ (Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Studies 82, §3.3, pp. 129-130).

Formalization Note The elements are written as subtype elements of Matrix.SpecialLinearGroup (Fin 2) ℤ with matrix literals; the equality is a finite computation over the integers.