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A recent question http://mathoverflow.net/questions/77570/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?

flag	asked Oct 11 2011 at 4:22 algori 16k●1●29●94

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front.math.ucdavis.edu/1104.1882 – Agol Oct 11 2011 at 4:32

Dear Ian -- thanks a lot! if you choose to post this as an answer, I'll accept it. However, the upper bound they give is absolutely huge, and I am wondering if there is a smaller one for the maximal number of crossings the sequence of diagrams must pass through. – algori Oct 11 2011 at 5:02

Not an answer, but related: arxiv.org/pdf/math/0501490 – Scott Carter Oct 11 2011 at 13:41

I haven't read this paper of Suh's but there's a stated lower bound than the Hass and Lagarias one: front.math.ucdavis.edu/1010.4101 – Ryan Budney Oct 11 2011 at 16:41

Hass and Nowik show that the best upper bound you can hope for is quadratic in the number of crossings: arxiv.org/abs/0711.2350 – bb Oct 21 2011 at 22:45

Agol · Accepted Answer · 2011-10-11 16:39:16Z UTC

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.

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

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