A river flows in from infinity in the south-west quarter of a crossroads, crosses the two roads n times without ever crossing itself, and flows off to infinity again. How many different courses can it take? Two courses count as the same if one can be slid into the other without moving a crossing off its road.
Pick a crossing count and click any river to see it large. Its itinerary lists the roads it crosses in order: the west, south, east and north arms of the crossroads.
Every column comes from one transfer-matrix computation of directed rivers that start in the south-west quarter; undirected and symmetry classes follow by Burnside's lemma. Terms past the old ends of the entries are new; four earlier values do not survive recomputation.
Squaring a quarter of the plane opens it into a half-plane, with both of its roads on the edge. There every stretch of river between two crossings becomes a plain semicircle. Taking the square root again bends the semicircles back into the quarter, so the river meets each road at a right angle.
A cut line sweeps the plane: along the west road, down the north road, around the junction, down the south road and out along the east road. At each crossing a river strand either ends at the road or starts there, and the cut keeps track of how the open strands are linked. That is Jensen's transfer matrix for meanders, bent around a crossroads.