Knot point constructed on the standing end (such as a loop or overhand knot) into which the working end is easily hitched, using a round turn. An example of this is the bowline. "Constricting" knots often rely on friction to cinch down tight on loose bundles; an example is the Miller's knot. Knots may belong to more than one category. Trick knots are knots that are used as part of a magic trick, a joke, or a puzzle. They are useful for these purposes because they have a deceptive appearance, being easier or more difficult to tie or untie than their
Arf invariant of a knot graph associated to a knot diagram. This approach to the Arf invariant is by Raymond Robertello. Let be the Alexander polynomial of the knot. Then the Arf invariant is the residue of modulo 2, where "r" = 0 for "n" odd, and "r" = 1 for "n" even. Kunio Murasugi proved that the Arf invariant is zero if and only if Δ(−1) formula_8 ±1 modulo 8. From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a slice knot formula_9 factors as formula_10 for some polynomial formula_11 with integer coefficients, we know that the determinant formula_12 of a