Unlike the classical machine that is composed of well-defined parts that interact according to well-understood rules (gears and cogs), the sliding interaction of two ropes under tension is extraordinary and interactive, with tension, topology, and the system providing the form which finally results. --Louis H. Kauffman, Knots and Physics, 1992 In this talk, we consider the geometry of tight knots in tubes of uniform circular cross-section. Using a strengthened, infinite-dimensional generalization of the Kuhn-Tucker theorem (from optimization theory), we find a natural "balance criterion", which characterizes length-critical tubes. There are a number of surprising consequences of this criterion, including an explicit solution for the unexpected shape of a "simple clasp" between two ropes. We also use our results to prove the existence of two different ropelength-critical trefoils, adding evidence for the conjecture that there are many critical configurations of an unknotted tube. This picture can be used to numerically simulate the tightening process for knots, and we'll talk about software to accomplish this simulation, with some applications in biology and physics. Finally, if time permits, we'll go a little further afield and discuss the evidence for a surprising conjecture regarding ropelength and writhing number.