I always liked knot coloring and how it related to knot groups and classifications of knots. The property of if a knot is n-colorable or not is an property of the knot (so it doesn't change under any movement that doesn't cross the strands of the knot). The colors of a presentation of a knot's strands can be assigned labels which are related to each other according to whoever moves over/under a strand, which can lead to a generalized way to label the strands of a knot in a way that assigns (almost uniquely) a group to each knot (usually defined in terms of the fundamental group of $\mathbb{R}^3-K$, but you can get at it through a particular presentation, too.)
An example with pictures, http://en.wikipedia.org/wiki/Fox_n-coloring
There are also associated group-like structures, http://unapologetic.wordpress.com/2007/05/02/more-knot-coloring/
