It is shown that there exist arcs and simple closed curves in
C
3
\mathbb {C}^3
with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded, connected Runge domain of holomorphy in
C
N
\mathbb {C}^N
(
N
≥
2
N \geq 2
) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through compact, totally disconnected sets.