Untangle

turn the crossings. haul. does it hold?

a rope through an anchor's ring with a tangle in its end: turn its crossings over, then haul; a real knot jams at the ring and brings the anchor, anything else pulls straight through. Overhand, figure-eight, figure-nine, stevedore: one more half-twist each

Whether a rope is knotted is decided at its crossings, not by how tangled it looks. Follow the rope: where it goes over, under, over, under all the way along, it holds, because any such laying is a true knot unless one of its crossings comes undone by twisting part of the rope round (Tait's conjecture, proved in 1987 by Kauffman, Murasugi and Thistlethwaite). Turn one crossing of an overhand over and the same picture pulls straight. The boards are stopper knots, tied near a rope's end so it cannot run out through a ring or a block, each one more half-twist than the last: the overhand, the figure-eight, the figure-nine and the stevedore. The card works out which knot a laying is from its Jones polynomial, computed from the crossings; the rope is simulated bead by bead and never passes through itself.

Concepts: order, knots.

An interactive you play with one thumb, on Wundr.