Knot cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is an organization dedicated to the Promotion of Knot tying. Truckers in need of securing a load may use a trucker's hitch, gaining mechanical advantage. Knots can save spelunkers from being buried under rock. Many knots can also be used as makeshift tools, for example, the bowline can be used as a rescue loop, and the munter hitch can be used for belaying. The diamond hitch was widely used to tie packages on to donkeys and mules. In hazardous environments such as mountains, knots are
Arf invariant of a knot the knot, constructed from a set of curves on a Seifert surface of genus "g" which represent a basis for the first homology of the surface. This means that "V" is a 2"g" × 2"g" matrix with the property that "V" − "V" is a symplectic matrix. The "Arf invariant" of the knot is the residue of Specifically, if formula_3, is a symplectic basis for the intersection form on the Seifert surface, then where formula_5 denotes the positive pushoff of "a". This approach to the Arf invariant is due to Louis Kauffman. We define two knots to be pass equivalent