Climbing harness the purpose, primarily a variation of the double bowline, but the figure-eight is usually preferred. Although it is harder to untie after a fall, it is inherently more secure, easier to tie, and easier to verify that it has been tied correctly. There are many variations of the bowline knot, and some will untie themselves when repeatedly stressed and unstressed, as is common in climbing. In addition to the weight bearing parts of the harness, there are parts of the harness that are not designed to be part of the safety system. These include the gear loops, used for carrying
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