Granny knot its possible use tying the necks of grain sacks. When attempting to tie a reef knot (square knot), it is easy to produce a granny knot accidentally. This is dangerous because the granny knot can slip when heavily loaded. A tightened granny knot can also jam and is often more difficult to untie than the reef knot. It is better to tie a reef knot in nearly all circumstances. One way to distinguish them is that in the reef knot each loop passes completely over, or completely under (not through) the neck of the other. The reef knot is commonly
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