I'm trying to work through the The Knot Book by Colin Adams, and I seem to be consistently coming up with edges for a planar graph of the figure-eight knot where there are both positive and negative edges.
My method for determining the sign of those edges simply entails picking a direction, drawing arrows along the strand to indicate orientation and then figuring out what direction the understrand is rotated in to be lined up with the orientation of the overstrand. However when I do this I consistently obtain a result where the "top" crossings (meaning those in what appears to be a (-2)-tangle in the projection given at the end of the book) have the opposite sign of the remaining two crossings. Considering the fact that this is an alternating knot projection I am sure that there is something very wrong with my approach.
edit: I tried tackling the problem again by constructing the knot out of tangles, and that seems to have given me the correct result (and it's worked for a few other alternating knots that I've tried out).