The connectedness of the invertibles question for arbitrary nest has been reduced to the case of the lower triangular operators with respect to a fixed orthonormal basis $e_n$ for $n \geq 1$. For each $f ∈ H^∞$, let $T_f$ be the Toeplitz operator. In this paper we prove that $T_f$ can be connected to the identity through a path in the invertible group of the lower triangular operators if $f$ satisfies certain conditions.