So Denison (the college I recently graduated from) has a professor who's pretty big into Knot Theory (he's got multiple papers published in the field, and has held the UnKnot Conference at Denison). He had this neat way of representing knots with mosaics. Well, they (he and his students) were using real tiles and everything for this, and so I made a little program to make it so that they didn't have to. I'm going to release it here in case anyone is interested at all about Knot Mosaics and Knot Theory. I know it's a stretch, but, what the hell, might as well.
http://merthsoft.calcg.org/Knot.zip

You could also use this program to design your own Celtic knots, if you're so inclined. Or just mess around, I like doing that.

Oh, and because everyone loves a screen shot, here's a trefoil in a trefoil (this program supports multiple tilesets, this is one I made called "Free Hand", so it's supposed to look like that. There's a more consistent looking one called "Standard"):

Hey, that's actually quite fun! I really like using it on my Tablet PC, in particular, as it's trivial to tap a tile then tap around the map. I could see this being expandable to be a great tool for designing maps, even for TI ASM games or something.

I haven't really looked at knot theory too closely, but I got an intro to it sitting in on a couple talks at CURM this spring. Looks like a fun little program though Does your program calculate any invariants on a given knot?

Extending it to a map editor shouldn't be too hard.

As for calculations, it doesn't right now, I just build around the specifications. I don't think I'll be the one maintaining it for knots, since I sent it to a CS department, but if they need any help I'll be willing. I had some ideas, but I just wanted to get what they wanted done. Well, almost. The changing tilesets on the fly was my own doing.

I enjoyed the Random option, Merth; it gave me a few ideas for knots to build myself. It would be nifty if that could alternatively generate random legal variants on a knot, as elfprince13 said.

*Bump* Moved to the "Your Projects" section.

Any thoughts on the generating-random-valid-knots idea that was tossed around here, Merth?

I just bought a copy of KnotPlot, but I'll have to take a look at this for my class. Maybe share it with everyone since it's a seminar format.

KnotPlot is definitely more powerful. The tool I made was because the researches I was dealing with were specifically dealing with knot tilings, and so the 3D stuff wasn't quite what they wanted.

We haven't talked about knot tilings yet, but it should give me something fun to talk about.

Also, KnotPlot is awesome.

Do you know if you will talk about them (they're also called know mosaics). Just in case, here's some slides about them:
http://personal.denison.edu/~ludwigl/UnKnot2012/Slides_files/Ludwig.pdf
If it's something you're interested in and you don't get to talk about them, or want to learn more, you can contact Lew Ludwig at Denison, he can tell you stuff about knots, man.

And, yes, KnotPlot is pretty fantastic.