I read the following proposition in "Crossing changes" by Martin Scharlemann.
A crossing change for a knot $K:S^1\to S^3$ with crossing disk $D\subset S^3$ can be obtained by performing Dehn surgery along $\partial D$ with slope $\pm 1$. The crossing change is right handed (left handed) if the slope is $+1$ (resp. $−1$).
Here, a crossing disc for $K$ is a disk $D$ is a disk which intersects $K$ in exactly two points, with opposite orientation.
It seems Scharlemann works out a definition of crossing signs which is described by a cyclinder, and does not depend on the projections as the easiest definition of crossings. I do not really understand the exakt meaning of the proposition with respect to the classical definition of crossing signs. Let me compare. Let us focus on the case of a crossing change from left handed to right handed:
Let $K_-$ be a knot $S^1\to S^3$ with a right handed crossing. Fix a projection. We find it by looking at a projection of $K_+$ and comparing the "height" of the preimage. Define $K_-$, a knot $S^1\to S^3$ which results by changing $K_-$, i.e. change the height in the preimage, with the aim that the knot diagram, i.e. projections with crossing information, has a right handed crossing there instead.
Now we change to Scharlemanns point of view. Let $D$ be a crossing disc for $K_+$ and $\partial D$ its boundary.
Let $K^\prime_-:S^1\to \text{result of Dehn surgery on }S^3\text{ along }\partial D$ be the map which is defined to be the same map $K_-$, except that points outside of its range were replaced. Then how is this $K^\prime_-$ (which should have a crossing changed) relate to the ordinary crossing changed knot $K_+$. Note that not even the codomain is the same.