OK. I posted about this once before, like five years ago(!),

https://igkt.net/sm/index.php?topic=6115.msg41317 I think I can explain the gist of it really briefly and find out if anyone is interested.

- Any THK can be "built up" in a variety of ways into larger THKs. I have worked out the math for all such buildings.
- All THKs with (leads) L>3 and (bights) B>2 can be built from smaller THKs,
*except* for THKs of the form L?(mL?2) for some integer m. Pretty sure I have a mathematical proof of this. - THKs with L≤3 or B≤2 are easy, everyone knows those.
- I have tricks for tying all L?(mL?2) THKs.

The practical upshot of which is that I can tie

**any** possible THK (by which I mean single-stranded ordinary French-braid type) in-hand without a pattern or a mandrel or anything. I've tied ones with dozens of bights, with many leads, etc etc. I took a picture of a grid I laid out of all THKs with L≤10 and B≤10, all of which I tied in-hand. (Can you guess what the trickiest one was?)

So, yeah, nothing much has

*changed* since I wrote that post back then, apart from more practice and everything, but I finally was reminded of it and thought to post again. Oh, and I've been writing a book about it, which is probably way too wordy and mostly needs pictures which I'm too lazy to get around to taking.

As usual, this represents the "finding someone who cares" stage of an interest, of which I have many (many). Not sure what comes after this stage, though; I've never really gotten past it. Anyone interested? Please let me know. Thanks!