TITLE: Unknot recognition - how tangled does it get?
QUESTION [15 upvotes]: A recent algorithm 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.)

REPLY [23 votes]: 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. 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.