Bell-ringing(ish) cardigan

Hey folks!
Well again I've failed to update this for way too long, despite having a couple of projects I should've mentioned. So without further ado - the bell-ringing cardigan. Sort of.

The idea for this cardigan grew out of a discussion of bellringing with Mair, and how this might be expressed in knitted cables. As I understand it, a bellringing pattern is produced by repeated applications of a pair of permutations (subject to some constraints), and the aim is to run through every ordering of a collection of bells(usually four, five or six). Some quick calculations suggest that a full 'peal' with five bells would involve 120 changes and would run roughly the full height of a jumper.

This project is a baby cardigan along similar lines: if you use just three cables, any pair of transpositions will run through every ordering of the three cables. This pair of transpositions can be chosen in six different ways, so the idea is for this cardigan is to have six different cables to represent each of these choices. The cables are picked out in intarsia to make the rearranging clearer as well as to make the whole thing pretty and colourful.
The 'ish' is because not all of these transpositions is allowable in the bell-ringing problem - the position of a bell in the sequence can only move by one place for physical reasons, but we haven't included this constraint here - some of the patterns involve swapping cable 1 with cable 3. This leads to some fabricky issues too, since this is quite a big cable and distorts the fabric a bit. Some experimenting shows that this isn't too bad provided the cable rows are appropriately spaced though.
One other problem is that these patterns don't distinguish "over" and "under" crossings, so these have to be more or less made up. To make the braids (more or less) alternating(so that if a cable went 'over' on the previous cable row it will go 'under' on the next) narrows it down a bit, and I seemed to come up with the right number of solutions, but it struck me as a little arbitrary and I'm not sure if there are others. Quite possibly there are exactly six though, and I just haven't spotted the reason for it yet.

So there we go - colourful little baby cardigan with intarsia cables, group theory and possibly some bell-ringing. I actually did most of the knitting over christmas and just need to get around to sewing it all together, so hopefully it'll be all done and ready to post pretty soon!
Happy knitting,
Hugh.