One of my favourite applications of knot theory is to classify all aesthetic tie knots, using topology to uncover unknown sartorial gems. It is a great piece of research from Cambridge, a short and accessible paper called “Tie knots, random walks and topology” by Thomas M.A. Fink and Yong Mao published in Physica A 276 (2000) 109–121.
It is available on the authors’ home page:
http://www.tcm.phy.cam.ac.uk/~tmf20/