It is well known that any knot diagram can be unknotted by a sequence of crossing changes (i.e., changing an overcrossing with an undercrossing or vice versa) and of Reidemeister moves. More precisely, one can first perform a certain number of crossing exchanges to modify the given knot diagram into a diagram representing the unknot (the minimal number of such changes is known as the unknotting number of the knot diagram), and then one can perform a sequence of Reidemeister moves to actually unknot the diagram. Yet, during this process the number of crossings in not necessarily decreasing. A classical example is the knot diagram at the bottom of page 4 [here](http://www.math.cornell.edu/files/outreach/mec2014_knots/worksheets.pdf "here"), where the knot diagram actually represents an unknot, and so one can effectively unknot it by a sequence of Reidemeister moves, but the sequence of Reidemeister moves needed to unknot will increase the number of crossings at some point. Yet, if one can use both Reidemeister moves and crossing changes, then it is easy to see how to build a sequence of unknotting moves always decreasing (or better, never increasing) the number of crossings.


So my question is: it is always true that given a knot diagram $\Gamma$ there exists a sequence of Reidemeister moves and crossing changes that transforms $\Gamma$ in the standard diagram for the unknot in such a way that the number of crossings is decreasing (or better, non-increasing) along the process?


The reason for this question is, apart from knowing it for the sake of itself, an attempt to rigorously understand a line in [Link polynomials and a graphical calculus](http://www.worldscientific.com/doi/abs/10.1142/S0218216592000069) by Louis Kauffman and Pierre Vogel, where they say "Since any 4-valent planar graph can be undone by a series of moves of the type -shadow of a Reidemeister move" (page 78). Namely, although this line is pretty clear at an intuitive level, my feeling is that in order to make their argument completely rigorous one should know that number of crossing decreasing sequences (possibly involving crossing changes) always exist.



Unknotting knot diagrams by Reidemeister moves and crossing changes
***Edit: This answer has been edited to correct a mistake graciously pointed out by Ian Agol in the comments below.***


The answer to you question is yes. It is possible to take a sequence of crossing changes and Reidemeister moves to unknot a diagram in a non-decreasing manner. 


Here is an algorithm to accomplish this. For a fixed (finite crossing) diagram D of a knot K (with at least one crossing), pick a crossing, number it 0 and choose one of the two arcs in the under crossing. Label that arc a0. Follow a0 to the next crossing label that 1 and the next arc a1. Continue the process until either you get back to crossing 0 via the over crossing or you get back to a crossing already numbered. In the second case, start at that crossing and label it 0 choose the under crossing that is part of the path and label that a0 and start the process over. 


***Added case (selecting an 'innermost loop'):***


This path is a loop which bounds a disk in the plane. Shade in this plane and check to see if there are arcs of the diagram contained in the disk which begin and end one the boundary loop. If such an arc exists, check the arc for loops, and repeat the process until you have found an `innermost loop', i.e. a loop that 1) bounds a disk which contains no loops and 2) such that no subpath of the loop is itself a loop. For each loop there is a well defined crossing to begin and end the loop determined by the diagram. Call that crossing 0.


Thus we can assume that we have a path that begins and ends at crossing 0 which is innermost. Make crossing changes so that this path only contains over crossings (except at crossing 0). Then use Reidemeister II and III moves to consolidate this path to a small neighborhood of crossing 0 such that no crossing of the diagram are in this neighborhood. The innermost property assures that Reidemeister II moves will decrease the number of crossings between the loop and the rest of the altered diagram. Finally, use a Reidemeister I to remove crossing 0.


If crossings remain, apply the method above until there are no more crossings. 


Of course, it well known that the crossing changes are required to run such an algorithm. For example, there is a 10 crossing diagram of the unknot aptly dubbed "the Culprit", Figure 2 of:


*Allison Henrich and Louis H. Kauffman*, MR 3193721 [**Unknotting unknots**](http://dx.doi.org/10.4169/amer.math.monthly.121.05.379), *Amer. Math. Monthly* **121** (2014), no. 5, 379--390.


which cannot be simplified in a non-decreasing fashion using only Reidemeister moves. 


