What is a star configuration for a projection of a knot?

Question

What is a star configuration for a projection of knot? This is a terminology used in Peter Cromwell's Knots and Links in the section crossing number of the chapter link diagrams

Answer 1

It appears "star configuration" is more of a hint than a definition. If the number of segments $n$ is odd, then take $n$ equally spaced points on a circle, and connect the $i$th point to the $(i+\frac{n-1}{2})$th point. This results in a single circuit, so is a knot projection. For example, here is $n=7$:

Each segment crosses $2(\frac{n-1}{2}-1)=n-3$ segments (the only segments you care about are the ones incident to the $\frac{n-1}{2}-1$ intermediate points when going from the $i$th point to the $(i+\frac{n-1}{2}$)th point). There are $n$ segments, so since we are double counting, there are $\frac{1}{2}n(n-3)$ crossings in total.