Topology and Manifolds

So for a Senior project, my partner and I were assigned the topic "Mathematical Curiosities", which is quite a broad scope. We have, however, since narrowed it down (with some work on the properties of Möbius Strips) to the field of Topology, specifically manifolds and their properties. We can't, however, think of any good actual project ideas. I want to actually do something as opposed to simply exploring. I was wondering if you guys might have any suggestions? For instance, the string theories are starting to tie topology into physics and we were curious as to whether or not we could work that angle (since I'm going to major in physics in college). Anything would be welcome, though.