Number of the Reidemeister moves needed to transform one diagram into another one

Lackenby updated his arXiv paper today (12Dec2014) in which he proves a polynomial bound on the moves to uncross the unknot. I am uncertain if this also implies a polynomial bound on moving from one knot to another.

Marc Lackenby. "A polynomial upper bound on Reidemeister moves." 2014. (62 pages.) (arXiv abs link.)

"We prove that any diagram of the unknot with $c$ crossings may be reduced to the trivial diagram using at most $(236 c)^{11}$ Reidemeister moves. Moreover, every diagram in this sequence has at most $(7 c)^2$ crossings."

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(19Feb2021). See update at "Are there any very hard unknots?". for Lackenby's Feb 2021 "Unknot recognition in quasi-polynomial time."

Coward and Lackenby have an upper bound on the number of Reidemeister moves, which is a tower of exponentials. The existence of some such bound is not surprising, since Waldhausen had proven that the knot isotopy problem was solvable, so some computable upper bound exists.

Suppose you had a much better upper bound on the number of crossings of diagrams in the sequence of moves than their bound. Then since the number of diagrams with $c$ crossings is no more than say $k^{k^c}$ for some $k$, one would get a much better bound on the number of reidemeister moves to get between two diagrams. So I think one would need a new idea to get such an estimate.