Bowline draws tighter as tension is placed on the standing part of the rope. The "Birmingham bowline" has two loops; the working part is passed twice around the standing part (the "rabbit" makes two trips out of the hole and around the tree). Other two-loop bowline knots include the Spanish bowline and the bowline on the bight; these can be tied in the middle of a rope without access to the ends. A triple bowline is used to make three loops. A Cossack knot is a bowline where the running end goes around the loop-start rather than the main part and
Arf invariant of a knot Arf invariant of a knot In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If "F" is a Seifert surface of a knot, then the homology group H("F", Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot. Let formula_1 be a Seifert matrix of