Unknotted loop

The simplest of such functions is the Alexander polynomial A(f).i3 For a closed, unknotted loop (the unknot U, homeomorphically equivalent to a circle) the polynomial is trivial = 1. The simplest knot is the trefoil knot... 

In the strict sense neither the link L of Figure 3.3 nor the simple loop U, informally called the "unknot", is a knot, nevertheless, the generic term "knot" is often used for them. Links are formed by more than one rope, which ropes may or may not be knotted, whereas the unknot U is not knotted. 

This procedure converts the molecular space curve into a closed curve. The resulting simple loop (unknot U) or knot is denoted by (recall that for sake of simplicity, the unknot U is also referred to as a knot). We shall analyze the resulting knot on two levels ... 

Thus, unknots are exponentially rare in long loops, almost all long loops are knotted. 

The above operations on the protein backbone produce a loop in most cases. The simple loop (called the "unknot") is the trivial knot, and it has V(Ko)=l as Jones polynomial. Nevertheless, one can derive nontrivial knots when constructing a family of loops from the original curve, by introducing a sequence of formal switches in the original overcrossing pattern. 

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