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 Reidmeister 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 Reidmeister 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, where the knot diagram actually represents an unknot, and so one can effectively unknot it by a sequence of Reidmeister moves, but the sequence of Reidmeister moves needed to unknot will increase the number of crossings at some point. Yet, if one can use both Reidmeister 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 Reidmeister moves and crossing changes that transforms $\Gamma$ in the standard diagram for teh 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 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 Reidmeister 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.

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, 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 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.