A Lower Bound for the Number of Reidemeister Moves for Unknotting

How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a split link to be disconnected. On the other hand, the absolute value of the writhe gives a lower bound of the number of Reidemeister I moves for unknotting. That of a complexity of knot diagram “cowrithe” works for Reidemeister II, III moves. We give an example of an infinite sequence of diagrams Dn of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n). However, writhe and cowrithe do not prove this. 1. An upper bound for the number of Reidemeister moves for unlinking A Reidemeister move is a local move of a link diagram as in Figure 1. Any such move does not change the link type. As Alexander and Briggs [1] and Reidemeister [7] showed that, for any pair of diagrams D1, D2 which represent the same link type, there is a finite sequence of Reidemeister moves which deforms D1 to D2. Let D be a diagram of the trivial knot. We consider sequences of Reidemeister moves which unknot D, i.e., deform D to have no crossing. Over all such sequences, we set ur(D) to be the minimal number of the moves in a sequence. Then let ur(n) denote the maximum ur(D) over all digrams of the trivial knot with n crossings. In [3], J. Hass and J. Lagarias gave an upper bound for ur(n), showing that ur(n) ≤ 2, where c = 10. (See also [2].) The author is partially supported by Grant-in-Aid for Scientific Research (No. 15740047), Ministry of Education, Science, Sports and Technology, Japan. 1