Bowline on a bight advocate the use of a figure eight twisted version of the Bowline on a bight. This knot can be used to provide a toe hold in the middle of a rope; to make an emergency bosun's chair; and to create an upper rope "block" to make a crude purchase by threading the rope round an anchor point and then back up through the loop. It is sometimes used in sport climbing to tie into two anchor bolts independently. This knot is convenient when a dependable loop is required but neither end of the line is available. It's also commonly used
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