Formal definition of a knot diagram, including data for the crossings

Question

The definition of a knot diagram has, in some sense, two parts: the 2-dimensional projection itself, packaged with the corresponding data about where the over- and under-crossings are. It's easy enough to see what is meant by the first part, in the strictest formal sense: it's just a subset of $\mathbb{R}^2$. However, most definitions tend to hand-wave away how exactly the crossing data is "packaged". Of course, in some sense it doesn't matter, but as a student learning these things for the first time, I'd feel more comfortable having an explicit, formal definition. Does anyone know of a reference for one?