When I first came to the IGKT Website, I expected to find a complete database of all known rational knots and a means of filtering through them to find the knot which had piqued my interest. I was disappointed that I did not find such a resource, but perhaps I should not have been dissappointed with the IGKT, because I could not find such a resource anywhere, perhaps because to date such a repository and locational tool probably did not exist.

As I thought about how you might search such a repository (when one eventually does exist) I was struck with the fact that knots are so three dimensionally complex, we would probably be stuck with having to find a person with the experience and mental caliber of Budworth or Lehmanian proportions, send them the knot and beg their time to find its 'name' for you. After a while it struck me that knots could be roughly categorised by making some measure of their complexity - how many times the cords crossed to make the knot. My 'Overs Index' was born and quickly dashed with a "Read Budworth's 'Knots in Crime' - he did this years ago" And sure enough, so he had - 120 knots categorised out across 24 groups based on the number of 'crossings' the knot exhibited when laid out flat and as simply as possible.

I started to build on Geoffrey's 120 knots but quickly came to realise that the 80 : 20 rule (or worse) was going to get to work here as most of the knots fell ever more often into certain popular 'pots'. The Crossings count was a useful segregator but was no where near sensitive enough to be anything other than a crude segregation.

Further thought led me to realise that probably the next important distinction between knots was how 'complete' their crossings were. The most 'complete' or as I came to think of it 'saturated' a knot could not possibly be more 'tangled' - every over crossing was followed by an under crossing - they were fully saturated, but a pile of cord which might have the same number of crossings would effectively have zero saturation because as you pulled on the ends, nothing held on to anything else and the whole thing just unravelled. This way two knots belonging to the 'Six' crossing pot say, the Reef and the Granny, could now be split into two sub groups, the Reef was 6:10 while the Granny was 6:12. But it was still fairly poor. Using this method the 20 knots in Budworth's V1 class could only be resolved into five sub classes and did only marginally better with the 19 knots in his VIII class which could be resolved into eight subclasses but we still had over half the knots in just two of those sub classes.

Allied to the limited ability of the Overs Index to be able to locate any particular knot was the almost universal call of 'It's just too complicated - the man in the street can't do it'. So, although the Overs Index was potentially better than nothing, in reality it was a useless as nothing because virtually no one could count the crossings and the saturations.... and so it languished until FCB4 came into being.

If you have read the post, then you will be familar with Frank Brown's cypher method for exchanging a knot diagram and the little drawing program that has been written to allow knot diagrams to be easily drawn and recreated from their cypher files. While writing the program, it struck me that it would be relatively straightforward to add in a function to count the crossings and work out the saturation - at least this would remove one of the hurdles to using the OI - just draw out your knot and the OI gets worked out for you -- handy.

As I worked on how to calculate the OI, I started by creating a 'picture' of the path of the cord by writing down a '1' if the cord went over and writing down a '0' when it went under. From this pattern, the 'Crossings' was simply the number of 1's and the saturation was twice this number minus the number of consecutive events. Taking the following diagram of the overhand knot, starting with the WP symbol, first the cord goes over (so a '1'), then under (so a '0')etc. its full sequence is 101010 giving an OI of 3:6

You have probably seen it already, but it took me a couple of days to realise that the sequence was just a binary number that reflects the passage of the cord through the knot - it IS the knot. As nobody can remember binary numbers, I converted it to decimal and when I saw what 101010 was in decimal I new - sure as my name is not Zaphod Beedlebrock - that I was onto something significant.

THE BINARY SIGNATURE OF THE SIMPLEST KNOT - THE OVERHAND KNOT - IS

**42** AKA THE ULTIMATE ANSWER TO THE ULTIMATE QUESTION !!!!

Naturally I immediately started looking at other knot diagrams and working out their binary signatures. In the following diagram, the cord and the bent cord have no crossings and have a signature of zero. Make a loop and it has a signature of 2. Two piled loops has a signature of 56, but with an OI of 3:2 it is so unsaturated that it just falls to pieces a non knot. But change one little crossing shown in green in the red diagram and the unsaturated pile turns to a fully saturated OH 3:6 with a signature of 42.

I checked the 20 knots in Budworth's Group VI Unfortunately eight of them were all fully saturated so they scored 101010101010 or 2730 in decimal, however the remainder were all different variations except for a small group - the Reef, Thief, Strap and Phoebe counting in at 2898.

By the time I had got up to the 19 knots in group VIII there were 17 discrete groups -- almost complete seperation.

How easy is it to get this index number - well, if you have downloaded the FCB4 utility, then if you can draw out the diagram, you can either work it out by hand, or email me the cypher file and I will send the result back to you. Alternatively, in a couple of weeks I should have managed to write the code to do the calculations automatically, and then it will be as simple as drawing the diagram.

Here is one I did earlier - the Sliding Grip Hitch - OI 12:19 binary index 10,048,725

Imagine one day, when the database is built, draw the diagram and the utility could link you to the wiki database and take you directly to the page that features the knot(s) with that binary signature.

"One day my son, meat will come in little boxes" -- and "one day you will just have to draw a knot to find out all about it"

Derek