One of the first things I remember in my life, is the impression the sewing machine of my grand-mother made on me - yes, it was an old Stinger, which could somehow make knots on the fabric with the arm holding the needle remaining always at the one side of it ! Miraculous ! 
It took me some time to understand HOW this was possible, but I still remember vividly the first feelings of amazement…
It is a small step to imagine a knot-tying machine, which could tie any knot we order by having two arms able to pass the Working end(s) of one or two ropes “over” or “under”, from the one side of the knot to the other. All it needs is a mechanism which would allow the one arm / “needle” to pass the end of the rope to the other, during a tuck. I believe that an 17th or early 18th century inventor would had figured out how, and would had constructed such a machine, in a few days - at any case, in a far shorter time we need to write a descent knot-drawing computer program today !
I do not know if any knitting machine can be modified, so could tie knots with small size cords. If any member has any information about this, I would be glad if he/she shares it with us.
http://mikeshouts.com/fancy-a-robot-to-knot-up-your-tie-why-knot-does-just-that/
http://www.youtube.com/watch?v=IBY4t8XxH7E
http://www.youtube.com/watch?v=qdM48cAg0U4
http://www.youtube.com/watch?v=StKgHKiEiZw
Or ask Honda to have Asimo work on it.
I feel confident that it could be CNC programmed and articulated hand and arm could do the work.
SS
I will say nothing about tie tying, because I consider ties to be the most dumb and nowadays useless thing in the world, a disgrace for knots - while buttons, for example, invented 5.000 years before ties, is one of the most clever, and still useful. Robotic tie tying multiply an already great degree of nonsense by another great number !
I almost felt sorry for the poor robot trying to tie a bowline… Perhaps it should read some threads about bowlines in this Forum !
I think of a mostly mechanical machine, with no more intelligence than a sewing machine, and you think about a robot - it should be that great difference of age we have ! If you have to rely to 21 century technology, it would be better use a 3D printer, which are getting cheaper by the day, and fabricate the knot on some elastic material.
Mind you that we can represent all knots by diagrams on a square grid, and over/under crossings - so we will not need an articulated arm : two rods moving horizontally so they remain parallel to themselves and at right angle to each other, would do the job. The pattern of the bend could be fed into the machine without using Windows 13 or higher - 17th century punched cards, which were the brains behind the automated looms, are more than enough for the most complicated pattern.
Universal, but expensive.
String Tying Machine
http://www.youtube.com/watch?v=b9bkTllLutc
http://www.youtube.com/watch?v=TCiKIguAUhk
US Patent Class 289
KNOTS AND KNOT TYING
http://www.patentec.com/data/class/defs/289/2.html
Knot tying machine for tying spaced overhand knots
http://www.google.com/patents/US2489488
Fringe knot tying machine
http://www.google.com/patents/US3486780
Method for making a self-locking knot in a stator lacing machine
http://www.freepatentsonline.com/6944932.html
Toroid coil winding
https://www.youtube.com/watch?v=6lvm3FGTHSI
https://www.youtube.com/watch?v=XjWfHc8UcXI
Struktor
Tank you Struktor,
Although all those machines are as knots themselves should be, i.e, as simple as possible, they are one-knot machines…I want a any-one-knot machine - meaning, a machine we can feed with the tying diagram of any knot ( the square grid, the horizontal and vertical segments, and the over/under crossings ), and it will generate the knot.
There is no worse thing than to know and tie one, and one only knot, and believe it is the “best” - although that has always been - and it will always be - the holy Grail of knotting, which does not exist !
Step 1 - Vogel’s algorithm ?
Braid diagram obtained by using Vogel’s algorithm.
http://math.ict.edu.rs:8080/webMathematica/LinkSL/kn08202.htm
http://magma.maths.usyd.edu.au/~danr/site/talks/20070531.pdf
So, you say this :
Do not tie the knot itself, tie a corresponding braid. ( All knots can be transformed into braids, by implementing algorithms - like the Vogel s algorithm - although not in a unique way ). Then, go from this braid back to the parent knot.
You suppose that a machine that would be capable to weave braids would be simpler than a machine that would have to tie the knots starting from their representation as continuous paths of lines meeting each other / going over/under each other, at crossing points of a square grid. WHY ?
Let us suppose that the machine has weaved a braid corresponding to our knot. To get the knot back from this braid, we have to manipulate the lines of the braid, that is, to disentangle it as a braid and entangle it as the parent knot. How do you know that, if the braid is complicated enough, the friction forces would nt be so great that we would nt be able to disentangle the braid in the first place ?