A band with half-twists

Drag k and the strip gets k half-twists before its ends are joined. Before the join its front is gold and its back ice blue; when k is odd, gold meets blue along a seam: walk once round along the strip and the front has become the back, so it has one side and one edge. When k is even it has two sides and two edges, linked k/2 times (not at all when k = 0). For odd k the single edge is a plain loop at k = 1, a trefoil at k = 3, and beyond that the torus knot T(2, k), going twice round the axis and k times round the tube. How many sides and edges depends only on whether k is odd or even; how many times the edges link, and whether they are knotted, shows only in space. Now pick a cut. Down the middle, even k gives two bands linked k/2 times; odd k leaves a single band, twice as long, with two sides and two edges. Those edges link 2k times: at k = 1 the band has four half-twists, and from k = 3 on the band is knotted as well. A third of the width in, the cut for odd k goes round twice before it closes, leaving a thin one-sided band and a long two-sided one, linked k times; for even k it closes after one lap, so both edges are cut a third of the way in, giving three bands, each pair linked k/2 times. The linking in the corner comes from Gauss's integral along the curves.