of framed knots. It so happens [@BF] that this invariant coincides with the so-called ‘writhing number’ of the curve, obtained by taking the number of positive crossings and subtracting the number of negative crossings. Thus, we have constructed an invariant analogous to the *linking number* of two disjoint space curves.

There is also another combinatorial description of our invariant, which is to my mind more appealing [@Poh], [@Aic]. Let us imagine the knot as a roller-coaster, with us sitting in a car facing forwards. At every point, the tracks face away from the knot in the direction of the normal vector, and as the car travels along the rails, our head is always pointing “up”. Let us also assume that our head is locked in place, such that we can only look straight ahead (in the direction of the tangent).

The roller coaster starts up, and we start moving along the track. The car rises and falls, twists and loops, swooshing along. Every now and again, we may see another portion of track coming up directly into our field of vision— Pohl calls such points ‘cross-tangents’. We count these with appropriate sign, depending on the orientation of the tracks (which 