Insights on the homogeneous $3$-local representations of the twin groups

We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $V{T}_n$, and the welded twin group $W{T}_n$, for all $n\ge 4$. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of $T_n$ to be irreducible in the case $n=4$. The obtained results provide insights into the algebraic structure of these three groups.