It's quite amusing how some fields suddenly have a cross-over with an entirely different one. Remember the pdf I linked to a while ago, with the vegan sausages? Published under "physics of fluids".
Similarly, braids have been a topic for mathematicians as well, and that for a good while. Maybe you have seen one of the "mystery braid bracelets" at some point, usually a three-end braid made by cutting two slits in a piece of leather (so that the three ends that makes are still solidly connected at both top and bottom) and then braiding it. Yes, that works. Go try it if you've never done so before, it's fun!
Anyway, there's a whole mathematical paper written back in 1962 about braids that can be done with both sides of the ends connected. If you'd like to see it, it's available at JSTOR (where you should be able to get a free account with limited free articles per month to read it).