Shoelace knot the granny (or duffer's) knot. This version will also produce asymmetrical slips; one pointing down the other up. Shoelace knot The shoelace knot, or bow knot, is commonly used for tying shoelaces and bow ties. The shoelace knot is a doubly slipped reef knot formed by joining the ends of whatever is being tied with a half hitch, folding each of the exposed ends into a loop (bight) and joining the loops with a second half hitch. The size of the loops and the length of the exposed ends are adjusted when the knot is tied. It has the stability
Arf invariant of a knot if they are related by a finite sequence of pass-moves, which are illustrated below: (no figure right now) Every knot is pass-equivalent to either the unknot or the trefoil; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent. Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above. Vaughan Jones showed that the Arf invariant can be obtained by taking the partition function of a signed planar