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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.)