[A recent algorithm](https://www.maths.ox.ac.uk/node/38304) unknots in quasipolynomial time. But I want to know what happens to the crossing number. Assuming your unknot has $n$ crossings, if I remember correctly it might be necessary to increase $n$ temporarily. But to what? $n+C$? $C\*n$? Even worse? (I don't even know if the mentioned algorithm can be translated to actual Reidemeister moves - it might not be necessary if the goal is just *recognizing* the unknot.)



Unknot recognition - how tangled does it get?
Joel Hass and Jeff Lagarias proved that one can transform any unknot diagram with $n$ crossings into the standard unknot diagram using not more than $2^{cn}$ Reidemster moves. They were able to obtain the explicit value for $c$ of $c=10^{11}$. See [here](https://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf). This was subsequently improved by Marc Lackenby to a polynomial bound in the paper in the Annals "A polynomial upper bound on Reidemeister moves" In that paper, an unknot with at most $n$ crossings is shown to have a transformation into the standard unknot diagram with at most $(236n)^{11}$ Reidemeister moves; at no stage does the diagram have more than $(7n)^2$ crossings.


