Theoretical Question about a Frictionless Rope This is kind of ridiculous to imagine, but say a person could somehow tie a frictionless rope into a knot. After making the loop and everything, and tightening the knot, technically the knot would keep on tightening until...it breaks??
 A: In the comment you talked about

the one that everyone knows where you just loop a string around itself and pass it through the loop..

I believe you refer to the overhand knot

If you just pull on both ends, it will become tight - but once the "hole" in the loop has gone down to nothing, there is nothing to make the knot continue tightening (unless you pull so hard that the rope gets thinner, at which point it can become smaller and smaller as the rope becomes longer and longer). And yes - in that case it will eventually break, and since there is curvature in the knot there will be points where the tension is higher on one side of the rope than the other.
This, incidentally, is one reason why most rope is twisted or woven: when you put tension on a twisted rope, the strands will slide over each other so each strand carries an equal share of the tension whether they are on the "inside" of the bend or the "outside". If you have a monofilament (like a fishing line), then in principle the stress on the outside is greater. This matters more as the ratio between the diameter of the wire and the bend radius becomes larger, and is what limits the strength of knots. "Good" knots will therefore attempt to take much of the tension away from the point of greatest curvature. Take for example the bowline: the point where the incoming rope is first bent, it makes a turn around two strands of the rope - this larger curvature makes the knot "strong" in the sense that it does not weaken the rope much ("only" about 30%):
For example, from http://www.thomasalspaugh.org/pub/crg/img/knot-bowline.svg

A: perhaps depends on the knot, but a reef knot would untie itself as it is held together by friction....  Many knots would just untie - maybe all.
Reef knot.

For example, people who go fishing would not be able to use the reef knot to tie thin nylon line together, but this knot works well for larger rope and is taught to sailors.
Edit after comment.
A loop to loop connection as below

Would not be able to untie itself - and this know is used for fishing line. It would get tighter and tighter until something failed - e.g. the rope broke or the force pulling it could not long pull it.
Final comment - this is actually quite a serious point for people working with light slippery fishing lines - they can slip and rub against each other so much that they heat up and melt. So books about tying fishing knots tell people to 'wet' the line before tightening a knot to stop it heating up, melting and breaking.
