There have previously been attempts to classify puzzles into subcategories and some have worked while others have failed. In my opinion, many of them have either excluded or not subclassified disentanglement puzzles enough. Here is my attempt to classify disentanglement puzzles. My system is like that of Andreas in terms of structure: it's a combination of letters and numbers.

Most disentanglement puzzles feature one of 2 main materials: wire and string. Let's focus on these for now. These are each represented by a letter: W or S. Many of those that feature string also feature wire, so this is represented by the letter C. Beads are represented by the letter B (obviously), and other Unusual materials are represented by the letter M.

This classification system is also based on the number of "features" of the puzzle. Features include rings and other things that are components of the puzzle. Any significant part of the puzzle is the body, while anything else is a feature. Let\s consider the Coiled Again. This puzzle features a string, a wire segment, and a bead. The wire segment is the body and the string with the attached bead is the feature. Other features include rings and horseshoe shapes. Not how I consider the bead/string as 1 feature due to the fact that they are directly attached to each other. I may also consider adding the type of feature, but this might be redundant.

If there is an official solution from the manufacturer, I propose that the number of steps needed to solve be included, but I am debating this currently as is may be redundant. As well I am considering adding the number of curves in each wire feature and body as well as their angle type, but this might make the signature too long.

Another thing that I think is important is that the goal of the puzzle be considered, mainly which feature needs to be removed. I have yet to find a wire disentanglement puzzle where there is no feature to be removed. Handles are H, Strings are S, Beads are B, Rings are R, and other wire shapes are W. This is fairly self-explanatory.

This unfortunately leaves a rather large gap where there would be puzzles with the same signature: puzzles with only 2 identical pieces. Fortunately, they can be classified as well. Many puzzles of this type are easy and common, and mostly they fall into two types: puzzles where the 2 pieces cross during the solution and puzzles where the 2 pieces form a cross when viewed from the end of the wire piece. These might be represented by X and x (or X and Y) respectively. I don't have specific names for any of this type of puzzle to show what I am talking about, but I hope that someone at least knows what the heck I am talking about.

I think that this is all that needs to be classified, so let's get an example. The signature for the Candy Canes puzzle would be BW1WB. I consider the lower portion to be the body because it has a bead attached. Where the pieces are identical, the number of pieces is the number of features.

I hope that this makes sense and is keeping all types of puzzles in mind. I am working on a system for cast puzzles as well. Feedback is greatly appreciated.

You are doing something really brave here. Disentanglement Puzzles are based on Knot-Theory,
but if we want to include all variations they are more advanced than Knot-Theory.

We can first have the two pieces, then three pieces, etc.

But then we need to think of other parameters, such as the dimensions of each part of a puzzle,
which in topology they simply don't count! For example, one such puzzle made of two pieces
has a specific solution. Now that solution can be diverted and changed significantly, by simply
extending one metal piece, elongating a string, increasing (or decreasing) the radius of the bead/marble.
There are more, but those are just to understand the depth of such puzzles, which may also depend
on the flexibility of the materials.

Hanayama is one of the very few companies that I know which have done some serious advances in this area,
mainly for two-to-four pieces (including the appearance of a puzzle), which is why I love their puzzles.
But I have also met people who are super experts in making custom made of those.

My suggestion is to use Knot-Plot for such puzzles, it can give you a very good idea, as well as
allow you to experiment (e.g. if a crazily complex design can be partially or fully separated).
Beware though, it is slightly different to normal windows programs, and I am certainly not equipped
enough to advise others for how to use this software (the help file is not the best one LOL):
http://www.knotplot.com/




Pantazis

@kastellorizo: After having taken a brief glance at that site I am inclined to agree that those other factors should be included (also, I have always wondered if there is a theory to solving these puzzles). At the moment, however, I am basing the signature (by signarue I do not refer to any sort of mathematical signature) on basic outward appearance alone, mainly because I do not have the knowledge of the more complex matters to add much more. Based on your recommendations, I intend to do a little bit more research into knot-theory and take a more in-depth look at that website, if only to get an idea of what might need to be added, even if I can't add it myself. I thank you for the feedback, and if there are no further immediate advances in the improvement of this proposed system I will definitely return to the idea after a little bit of time with a fresh brain.