  We obtain a new lower bound of 0.06576 for the 1-entanglement critical
probability (in dimension 3), and prove that the critical point for the
existence of a sphere surrounding the origin and intersecting only closed bonds
in $\mathbb{Z}^d$ is greater than $\frac1{8(d-1)}$, $d \geq 3$. This
substantially improves the previous lower bounds and gives the correct order of
magnitude for large $d$.
