I like the pictures from chapter 0 of Hatcher for that purpose.
And also Louis Kauffman made some nice pictures for the Fox calculus.
And Thurston's 3-manifolds book has good pictures as well.
"Normal maths works with squares, triangles, perfect circles; it counts beans. In topology we're freer to work with more interesting shapes." (cue picture from Dr Seuss)
This is basically what I did in http://tmblr.co/ZdCxIydt_Owi, drawing on the familiar philosophical distinction between reductivism vs holism to work in a cohomology element (as an argument for holism: some things can't be decomposed into atoms without losing some essential property that only shows up in how they're put together).
A romantic notion I felt when I was first reading Hatcher is that the "counting rocks" and "straight-sight cannonball" problems that seem more typical of a more boring mathematics, get replaced by "feminine" "weaving-based" mathematics. (Women were the ones tasked with purling, picking, and wrapping in most of history.) So this is a nice tie-in for non-mathematicians, although I'm not sure AT actually is more "feminine" (whatever that means) it's an attention-grabber.
You can also talk about topological spaces vs metric spaces by using highways
versus airplane connections
taking a ring road versus trying to not back-track, versus taking small streets with different speed limits, could be thought of as metrics -- whereas if you're at the airport, then it's a question more of does the city you're in actually connect to where you're going or not, rather than 'physical' distance. You might end up flying past your destination (or at least extra distance) before taking a second flight/network hop, so the "network distance" is two (2 nbds), compare that to "ground distance". (Maybe for example Colorado Springs does not receive flights from L.A., so one has to go further east to Denver first ... if my geography is wrong then some pattern like what I'm suggesting does exist somewhere.)
As far as applications, I like the suggestions of Ghrist & Carlsson's work. I would add Ghrist's robots-not-bumping-into-each-other in warehouses, anyone can easily envisage as a braid the robots not bumping into each other over time. (See Ghrist's EAT book ch 1 for the relevant picture.)
But I would make a further point, which is that the goal is not to rehash the applications already developed, but rather why the novel approaches might lead to something new. In other words, "How has this already been applied?" and "How might this generate applications in the future?" are two separate questions.
To that end I think it's fine, as long as you use clear non-technical language, to talk in generalities about how the "style" of topological thinking differs from maths they may be more familiar with. You could give some examples of "wiggly" or "non-rigid" situations and say that the ambition is that with enough development of AT, these situations could be handled.
Justin Curry made a pitch like this in http://arxiv.org/abs/1303.3255, saying essentially "Linear algebra was amazing for applications; I think my thing could be similarly productive, from a different starting point."
On the other hand, we probably all know examples of scientists who thought their research would be used for totally different purposes than what it ended up in. My favourite example is the inventor of NMR (worth a Nobel prize) thought NMR would be used for "calibrating industrial magnets". Instead it was used to look inside people's brains. (But that was someone else's work, to apply it to that.)
On the other other hand, probably no politicians slapped Purcell & Bloch on the wrist when they said they were working on technology to calibrate industrial magnets. That can be envisaged by a politician and sounds science-y and maybe-important (just not as important as what it ended up being). Some mathematicians have also been known to name their ideas for marketing purposes, for example "dynamic programming" and "information entropy". (There are funny stories going along with the naming of each of these.)
If I had a piece of paper, my 10-second elevator pitch for algebraic topology would be:
- Most data-analysis methods assume data is "Square" (meaning Euclidean--independent dimensions, flat, etc)
- But how do we know that? What if the data is shaped like a peace sign?
- We don't want to be caught off-guard by assumptions we weren't necessarily trying to make, so do AT to be aware of other possibilities the world might hit you with.