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^{105 n}, where n is the crossing number.

Research was partially funded by the National Science Foundation (VIGRE DMS-0135345 and DMS-0636297).