Number of the Reidemeister moves needed to transform one diagram into another one A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. Lagarias show that one can transform any unknot diagram with $n$ crossings into the standard unknot diagram using not more than $2^{cn}$ Reidemeister moves, with $c=10^{11}$.
[As an aside: this is quite a large bound, so the first thing that comes to mind when one looks at it is a computer falling apart with all its atoms decaying long before it manages to untie a diagram with a single crossing. As far as I understand, for those diagrams the algorithm works faster, but still it is probably impractical for untying knots that can't be untied by trial and error.]
It seems plausible that the methods of Hass and Lagarias can be adapted to give a similar explicit upper bound for the number of the Reidemeister moves needed to transform two diagrams representing isotopic links into one another. I would like to ask whether this is indeed the case, and if so, whether there is a reference for that.
A related question: given a nonnegative integer $n$, is it possible to estimate from above the minimal $m$ such that any two link diagrams with $\leq n$ crossings that represent isotopic links can be connected by a sequence of diagrams with $\leq m$ crossings such that each is obtained from the preceding one by a Reidemeister move?
 A: 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.
A: 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."


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