  Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold
with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound,
depending only on $t$, for the minimal number of elementary moves to untangle
$K$. We give a simpler proof, utilizing a normal form for surfaces whose
boundary is contained in the 1-skeleton of a triangulated 3-manifold. We also
obtain a significantly better upper bound of $2^{120t+14}$ and improve the
Hass--Lagarias upper bound on the number of Reidemeister moves needed to unknot
to $2^{10^5 n}$, where $n$ is the crossing number.
