When we say that a certain knot is "simpler" than another, what exactly do we mean ? There are four different things that one can take into account, in order to measure this elusive "simplicity" ;
1.
Rope length. We can claim that a knot that consumes less material, is simpler. However,
when are we supposed to measure this rope length ? Right after the initial set-and-dress phase, or when the knot is tightened to its strength limit?
2.
Number of tucks. Most knots are tied by driving the working end through loops, a number of times in a row. Knots that require a smaller number of tucks are often easier to tie, so we can say that they are simpler. However, it is easier to pass the working end through the same loop two or three times in a row, than through two or three different loops - unless those two different loops are one and the same two or three coiled "neck' ! So, this measure of knot s simplicity has its own twists, too.
3a.
Topology (genus) of the knot. A bend formed by two interlinked overhand knots can be considered simpler than one based on two double overhand knots, and/or two fig. 8 knots. However, one can argue that a double overhand knot is in fact simpler than a fig. 8 knot, although it is formed by twice the number of tucks...
3b.
Topology (crossing number) of the knot. In a lose, flattened knot, there is a certain number of points where the strands cross each other, go over or under each other. A small number of crossing points mean an easier to represent in 2D knot...but the final form of the knot has often little relation with the initial form ( as it happens in the case of the Carrick bend, for example).
3c.
Topology (
winding number). In a bend or a hitch, if we count the number of times the working end turns around the other link s strand or around the pole/line, we can have yet another measure of simplicity. A knot that utilizes friction effectively, without having too many turns and twists, should probably be thought as a simple knot.
4.
Total curvature of the rope s path. Who ordered
this ?
I think that this is perhaps the best way to measure the simplicity of a knot - because it is a measure of
how much this knot is convoluted in 3D space, how many "turns" the working end has taken, to arrive at the tail, starting from the standing end.
{ We can think of the total curvature as a number that is the sum of all the angles formed alongside a curved path. To be more precise, for a space, 3D curved path - like the one followed by the rope inside the knot s nub - we first have to consider a straight line tangent to this path, and the 2D curved surface that is formed by this line, as it walks alongside a point that moves on the path (starting from the standing end and arriving at the tail). This surface is called "developable", because it is a "ruled" surface, it is "made" by straight lines the one next to the other - a curved surface that can be flattened on a 2D plane nevertheless. In short, a surface that can be made by a sheet of paper or metal, like a cylinder or a cone, but more freely formed. Now, if we flatten this surface and place it on a plane, we can measure its 2D total curvature, which is the total curvature of the initial un-developed space (3D) curve. }
