Why is crossing change an unknotting operation?

Question

How can we show that by changing the crossings of a knot projection from over to under and vice versa, every projection of a knot can be turned into a projection of unknot?

I know very little about knot theory, so I would really appreciate a simple proof. P.S. I found out an algorithm myself, I just don't know how to prove it.

Answer 1

Think of your 2-dimensional projection to be embedded in 3-dimensional space, say in the $x$-$y$-plane. Now walk along the projection, but shift it by increasing amounts (proportional to curve length traveled) in the $z$-direction. Shortly before arriving at the starting point, with no more crossings in the way, move the point back along the $z$-axis in a continuous way.

This is an unknot, and the projection of this unknot agrees with the original projection, except that some crossings might be flipped.

Answer 2

Pick a starting point on your knot and start walking along it. Every time you reach a new crossing that you haven't seen before, always make it the 'over' crossing with respect to the direction you're traveling. This will ensure the knot is an unknot.