Removable sets for Sobolev spaces with Muckenhoupt $A_1$-weight

Let $\Omega$ be an open set in $R^n$, $n\ge2$, and $E$ be a relatively closed subset of $\Omega$. In this paper we obtain a criterion of equality $L^1_{1,\omega}(\Omega\setminus E)=L^1_{1,\omega}(\Omega)$ in terms of $E$ as an $NC_{1,\omega}$-set in $\Omega$ with $A_1$-weight $\omega$. In addition, we establish exact characterizations of $NC_{1,\omega}$-sets in terms of $NED_{1,\omega}$-sets and of the $(1,\omega)$-girth condition. In the case $\omega\equiv1$, these results complete the studies of Vodop'yanov and Gol'dstein on removable sets for $L^1_p(\Omega)$, $p\in(1,+\infty)$.