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We just use a bic lighter and hold both pieces over the flame. When they are both hot (or melting) we just shove both ends into each other. When cool, we hit it on a belt sander, and repeat the entire the process until the seam is smooth. Just try not to inhale that magic black smoke…
Our first tries were with fire, didn’t like the fumes, burnt rubber smell, and the joints were less uniform. This really is simple. Used it last year and on our Lunacy robot without fail. Pull the trigger on the soldering gun until both sides melt, and slide out the gun. Let it cool for a few minutes, and trim off the part that mushrooms with side cutters.
Yes, but how would MacGyver do it in a sealed room with only polycord and a bic? 
No, I agree, the less magic black smoke the better. You will live much longer doing it that way… 
I found an easier way to do it.
We did it a bit differently. Took a 1"x1"x.125"thick aluminum U-channel about 4 inches long. Then i hot glued two aluminum angles (1"x1"x.125" thick) about an inch long each into the channel with half an inch in between so the aluminum doesnt take away the heat and cause the belt to cool rapidly. heat the first quarter inch of each side of the belt over a heat gun, not a flame or soldering iron. The heat gun lets it cool easier. put one side of the belt in each angle piece, stick them together. hold for two minutes, the water cool under running water, as a bucket will force you to bend the connection. trim in with wire cutters and take it to the belt sander. dont hold the belt directly on the table for the sander, because it can rip it through the gap between the sander and the table and break the sander, get somebody hurt, and ruin the belt.
Good Luck Teams 
Any reason you guys are melting the polycord together instead of just tieing it?
My team has decided to go with tieing due to concerns about the melted together joint being weak and our inability to identify any benefit to melting it
Unless you are doing something wrong then your joints should not break. I’d be more concerned with my knot untying or messing up the movement of the ball/rollers.
The way we have done polycord welding was with a heat gun (or flame but you have to keep the cords in a magic spot so they don’t burn and so that they aren’t too far away, heat gun preferred). Once you see the two cords turn liquidy on the end we pushed them together and held them for a minute or two until they cooled. Once they have set for a while we would take a knife and shave off the loose ends. We did this in 2009 and I can’t remember a single joint breaking it is wicked simple and so easy I don’t know why you would consider tying them.
Take a small segment and trying melting them together as a test before resorting to tying. A little too sketchy with polycord for me and I have never seen that done before.
Teams are melting it because that is the proper method for joining/welding it.
Welded joint is as strong as the cord itself. We pull test every joint, and have never had a failure in competition.
If your team does not want to join it with heat, I suggest using the hollow polycord with connectors or find another belt system to use.
For out polycord welding we milled out a simple assembly like above.
Two 1" blocks of aluminum are attached to a rail system on a base. The tops are notched to cradle the polycord.
On one of the sliders there is a hole just large enough to put the polycord through. Each end of the cord is slid through and cut against the face of aluminum with a razor blade so that it’s perfectly flat.
The hot knife is then held between the seated ends. Once hot the rail cradles are slid together and form a clean and even “weld”!
I can upload some photos upon request.
I’d be interested in some pics. Thanks!
I’ll see if I can get some up by tomorrow or Monday night. 
Photos are here: