The Tangled Task of Distinguishing Knots

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.

Trefoil knot.

Different views of the figure-eight knot. Robert Scharein

Braid (left) and its closed form (right).