All the Turk's Heads, again

[clsn]

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!

[clsn]

Ugh, sorry.  I didn't realize the Unicode characters I was using in my notation would turn into question marks.  Even the preview showed them okay.  So in lines 2 and 4, that should be Lx(mL +- 2), and in line 3 those are less-than-or-equal signs, in case those didn't show up.

I guess this is mostly what I said five years ago, but I think it's laid out a little clearer.

[KnotMe]

I think I remember reading your first message but I don't remember why I didn't chime in then (on the other hand, if I did see it then then surely I would have subscribed to your YouTube, as I just did).  In any case, count me among those who are interested.  I would love to add your book to my THK collection.  As to your claim of being able to tie any THK (alas, I don't know from French braiding unless it's my hair, or sobre or most of the things Mabel says that goes right over my head.    ), then how about the attached?  I got that from https://bbs.zhongguojie.org/thread-224477-1-1.html.

[clsn]

All I mean by "French Braid" is over-one-under-one braid, as opposed to over-two-under-one or any other variant.  Simple braiding.  Just to keep clear on the limits of what I'm considering in the broader set of knots sometimes called THKs.  I guess this is sort of a purist's definition (though I do intend for the book to include simple instructions for tying a M?bius strip THK, which comes out quite nicely.  I hope that o-umlaut comes out nicely.)

Basically, here's the math for building:  Given a knot that is LxB, find the number n < B such that nL divided by B leaves a remainder of 1 and m = B - n.  Then, for any non-negative number F, you can make the knots (L + 2(nL)/B + 2FL) x (2nB + 2FB) and (L + 2(mL+1)/B + 2FL) x (2mB + 2FB), if I'm reading my notes right.  Might have to confirm, it's been a while since I did the math.  At any rate, that's a bunch a knots and covers all of them except for the Lx(kL +- 2) cases, which I have special tricks for.

[alana]

i didn't (couldn't) fact check your work, but great to see, good on you.
i'm a fan of thk / ring/ cylindrical knots, in single or double strand form

i learnt some of the base knots from spirograph-looking 'line-bight' patterns in flat circle format. 

to expand them to bigger knots it became impractical and then i've 'expanded' to the hand mandrel system of tying (but it's amateur hour)

i notice some of the larger under-over knots in practice didn't look as appealing as they are in paper,
they have their own niche or purpose,
but pretty they ain't.

so i'm a fan of two-pass patterns