History of knotting History of knotting Knots and knotting have been used and studied throughout history. For example, Chinese knotting is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song Dynasty (960-1279 AD) in China, later popularized in the Ming. Knot theory is the recent mathematical study of unknots. Knots of ancient origin include the bottle sling, bowline, cat's paw, clove hitch, cow hitch, double fisherman's knot, eskimo bowline, figure-eight knot, fisherman's knot, half hitch, kalmyk loop, one-sided overhand bend, overhand knot, overhand loop, reef knot, running bowline, single hitch, thief knot, Turk's head
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