Elevator Talk on Topology 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: 


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*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.
 A: Having thought for some decades about how to convey the substance of "the underlying mathematics" of second-generation mathematics and beyond... sadly, I think it is not really possible (without lying, etc).
That is, even the genuine basics of contemporary mathematics in 2018 are several steps abstracted/evolved from the otherwise-intuitive physical mathematics that (I might claim) most human beings understand instinctively. E.g., just by using visual cortex and being able to throw and catch a ball, and/or not crash cars much at all, considering the high volume.
I might claim that a large part of the disconnect between contemporary mathematics and the general (intellectually/scientifically inclined) human population's context is due to the nearly-compulsive mathematical style of backforming concepts from examples and applications. Apart from perhaps conspiracy theories, this is not what most people do. E.g., to compare a doughnut and a coffee cup? Srsly? Cute, at best, and what's the point? No, that trope is worse than unconvincing... it is convincing of the frivolity and goofiness of mathematicians. And, of course, topology was never motivated by the homotopy equivalence of coffee cup and doughnut. This truly is fake news.
By this point, I have come to think that the best way to justify/explain the point/efficacy of various bits of "pure mathematics" for non-mathematicians is to explain the outcomes. Even though I am sympathetic to Lockhart's Lament, it doesn't "sell" to lots of people, and, in fact, I think accidentally parodizes (for a good purpose, I know) professional mathematicians' motivations. Just as G. H. Hardy's "Apology" did, I think.
So, seriously, don't turn math into some weird steampunk fetishism... (No offense intended to those people and their hobby!) All the "gears and stuff" in math are not just decorative. They are functional. And not just Rube Goldberg's version of "functional".
So, yes, sadly, many of the technical details do not lend themselves... Not everything can be sold in an elevator pitch to laypersons... 
(So, to be clear, I no longer try to explain the "genuine" underlying mathematics, but only the manifestations "on the surface". People understand those phenomena much better, because they fit into the standard phenomenology of human culture.)
A: 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).
