I am interested in what others do when trying to give an elevator talk on their research interests, particularly on trying to explain what topology is. I am particularly interested in giving an elevator talk to someone whose knowledge does not exceed the standard high school mathematics curriculum.
Currently I do one of two things:
I like to talk about Topological Data Analysis as a cool application of what topology does. I can talk about how topology can be used to recover the shape of molecules and how important that can be to chemistry and biology. This works but I would like to have an explanation of what topology is, not just what it can do.
I make a feeble attempt at describing topology as the study of shape, perhaps mentioning the old coffee mug and donut equivalence.
I am hoping someone has come up with a better alternative to 2.
I realize that this is only one topic in topology, but knot theory is particularly tangible. I suggest taking a look at Bill Thurston's Lectures "Knots to Narnia" (there are a few on youtube). His take on topology is quite "hands on". Moreover, the knot theory has an incredibly rich set of theorems and results that shape topology in general (the topic is not an isolated example).