Boundary-twisted normal form and the number of elementary moves to unknot (Q713040)

Let \(M\) be a triangulated, compact 3-manifold with \(t\) tetrahedra, \(K\) a simple closed curve in the 1-skeleton of the triangulation which is an unknot, that is, it bounds a disk in \(M\). It is known that \(K\) can be isotoped in \(M\) using polygonal moves across triangles, called elementary moves, until it is the boundary of a triangle contained in a tetrahedron. \textit{J. Hass} and \textit{J. C. Lagarias} [J. Am. Math. Soc. 14, No. 2, 399--428 (2001; Zbl 0964.57005)] obtained an upper bound of \(2^{10^7t}\) for the minimum number of such elementary moves. To do so they take a double barycentric subdivision of the triangulation and then use normal surface theory. In the present paper the author considers a normal form for surfaces whose boundary is contained in the 1-skeleton of a triangulated 3-manifold; in this way the use of a retriangulation and subsequent isotopies related to a regular neighborhood \(N(K)\) of \(K\) are avoided. This not only simplifies the proof but improves the upper bound on the number of moves to \(2^{120t+14}\). A corollary of bounding the number of elementary moves is a bound on the number of Reidemeister moves needed to unknot a diagram of the trivial knot in \(S^3\) with crossing number \(n\). This result improves the bound given by Hass and Lagarias from \(2^{10^{11}n} \) to \(2^{10^5n}\).