Consider the plight of a gardener struggling with a recalcitrant tangle of garden hose. Sometimes, no amount of pulling or twisting unsnarls the coils. At other times, the tangles readily come apart, and the hose emerges unknotted.
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Mathematicians also tussle with knots, but their task has an additional constraint. Unlike a knotted piece of rope, a mathematical knot has no free ends. In this context, a knot is a one-dimensional curve that winds through itself in three-dimensional space, finally catching its tail to form a closed loop.