Knot-tying with four-piece fixtures

We present a class of fixtures that can be disassembled into four pieces to extract the loosely-tied knot. We prove that a fixture can be designed for any particular knot such that the knot can be extracted using only simple pure translations of the four fixture sections. We explore some of the issues raised by our experimental work with these fixtures, which show that simple knots can be tied extremely quickly (less than half a second) and reliably (99% repeatability) using four-piece fixtures.