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 and improve the Hass–Lagarias upper bound on the number of Reidemeister moves needed to unknot to 2 5n, where n is the crossing number.