Sum of writhe and winding number of a knot is odd

Question

In Trace's paper "On the Reidemeister moves of a classical knot", Trace defines both writhe (or framing as he calls it) and winding number of a knot and uses these definitions to prove that two knot diagrams of the same knot are equivalent modulo Reidemeister moves $2$ and $3$ if and only if they have the same winding number and writhe. He then argues that the sum of these two quantities is always odd. I've been trying to see this but so far I'm unsuccessful. I can obviously see it given a knot diagram but I'm unable to see it in general. Could you please help me?

Answer 1

Each time you do a Reidemeister I move, it changes both the winding number and writhe by one, so the sum of winding number and writhe changes by $-2$, $0$, or $2$, depending on the signs. So, if you allow all the Reidemeister moves, the sum of the winding number and writhe is constant modulo $2$.

If you change the type of a crossing, the winding number remains the same, but the writhe changes by $\pm 2$, so this leaves the sum of the winding number and writhe constant modulo $2$. Every knot can be unknotted by a sequence of crossing changes, and since the sum of the winding number and the writhe of the standard unknot diagram is $\pm 1$, we have that that the sum of the winding number and writhe is odd.