One of the reasons knots have given mathematicians fits is that the same knot can appear in very different guises. Tug here, tug there, and soon a knot will become unrecognizable, but remain fundamentally unchanged. To allow a knotted string to wiggle around without danger of untying, mathematicians seal its two ends together, making it a knotted circle. The first question mathematicians have to answer is simply, when are two knots really, secretly the same?
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