Boundary-twisted normal form and the number of elementary moves to unknot

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.