Bowline loop may pass around or through an object during the making of the knot. The knot tightens when loaded at (pulled by) the standing part of the line. The bowline is commonly used in sailing small craft, for example to fasten a halyard to the head of a sail or to tie a jib sheet to a clew of a jib. The bowline is well known as a rescue knot for such purposes as rescuing people who might have fallen down a hole, or off a cliff onto a ledge. This knot is particularly useful in such a situation because
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