Salter's question on the image of the Burau representation of $B_4$

Abstract: In 1974, Birman posed the problem of identifying the conditions under which a matrix with Laurent polynomial entries lies in the image of the Burau representation. Building on this, Salter, in 2021, refined the inquiry to ask whether the central quotient of the Burau image group coincides with the central quotient of a specific subgroup of the unitary group. Assuming the faithfulness of the $B_{4}$ Burau, we solve Salter's question negatively in the case $n=4$ constructing counterexamples. Additionally, we offer two remarks on the faithfulness of the $B_{4}$ Burau. First, we establish that the restriction to the centralizer of a standard generator in $B_{4}$ is faithful modulo $p$ for every prime $p$, extending both Smythe's result in 1979 and Moran's result in 1991. Second, we present a building-theoretic criterion for the faithfulness.