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We establish strong constraints on the kernel of the (reduced) Buraurepresentation $\beta_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitlydetermine the entries of the Burau matrices of braids in $B_4$, and this is animportant step toward demonstrating that $\beta_4$ is faithful (a longstandingquestion posed in the 1930s). The theory is based on a novel combinatorialinterpretation of $\beta_4\left(g\right)$, in terms of the Garside normal formof $g\in B_4$ and a new product decomposition of positive braids. We developcancellation results for words in matrix groups to show that if $\sigma$ is ageneric positive braid in $B_4$ and if $t\neq 2$ is a prime number, then theleading coefficients in at least one row of the matrix$\beta_4\left(\sigma\right)$ are non-zero modulo $t$. We exploit thesecancellation results to deduce that the Burau representation of $B_4$ isfaithful almost everywhere.

A strong characterization of the entries of the Burau matrices of  $4$-braids: The Burau representation of the braid group $B_4$ is faithful  almost everywhere