Why the Braid Group of Order 4 is the Best Braid Group

 See also: A New Kind of Algebra

Consider the braid group of order 1. It has a single element, the identity element "1", representing section of a single strand. If you stick them together... well, 1*1 = 1, and one untwisted strand stuck onto another untwisted strand just gets you... an untwisted strand. Completely trivial and uninteresting.

Consider the braid group of order 2. It has two generators (basic members which cannot be created by combining other members, but which can combine to create the rest) in addition to the identity element: a twist, and an untwist. This is already an infinite group, because you can make an unlimited number of new unreducible things just by adding more twists! But, it's still a pretty uninteresting one. The only elements are just powers of a single generator--because any sequence that contains both a twist and an untwist simplifies to one that doesn't.

Consider the braid group of order 3. It has four generators, because you can put twists between either of the two pairs of adjacent strands. And finally, you get some non-trivial structure, because twists on different sets of strands don't cancel each other out!

But now, consider the braid group of order 4. It has six generators, and commutative relations! Now you can actually get some interesting algebra going on. And if you continue on to the braid group of order 5? Nothing new. After 4, you just get more things that commute with each other. No truly new structure ever appears again.

Furthermore, if we select elements of B4 with specific interesting properties, we get an interesting number...

Suppose we take a subset of B4 such that all elements obey the rules of Celtic knotwork: i.e., any given strand must alternate between over-crosses and under-crosses; no single strand can cross over another or under another twice in a row. All six generators follow this rule trivially, because they are all single crossings. However, we can characterize the remaining members of this set rather simply: they are all of the braids which contain only sequences of the following pairs of generators:

1. aa
2. bb
3. cc
4. /a/a
5. /b/b
6. /c/c/
7. a/b
8. b/a
9. /ab
10. /ba
11. b/c
12. c/b
13. /bc
14. /cb
15. ac (=ca)
16. a/c (=/ca)
17. c/a (=/ac)
18. /a/c (=/c/a)

So, the entire set can be reduced down to 24 basic elements.

Now, imagine that we print the graphical representations of these elements out of cards, which we can physically manipulate. In addition to lining them up to concatenate them (equivalent to group multiplication), physical cards permit a new operation which is not part of the normal braid algebra: 180-degree rotation. Rotation has the following effect on each of the basic elements:

• r(a)    = c     r(c)    = a
• r(/a)   = /c    r(/c)   = /a
• r(b)    = b
• r(/b)   = /b
• r(aa)   = cc    r(cc)   = aa
• r(/a/a) = /c/c  r(/c/c) = /a/a
• r(bb)   = bb
• r(/b/b) = /b/b
• r(a/b)  = /bc   r(/bc)  = a/b
• r(b/a)  = /cb   r(/cb)  = b/a
• r(/ab)  = b/c   r(b/c)  = /ab
• r(/ba)  = c/b   r(c/b)  = /ba
• r(ac)   = ac
• r(a/c)  = c/a   r(c/a)  = a/c
• r(/a/c) = /a/c

So, suppose that we wanted to build a deck which could provide any of these 24 elements with the minimum number of cards. For any rotational pair, we only need to print a single card, since we can get either member of the pair depending on which way we lay it down. That eliminates 9 elements, getting us down to 15. (It's on odd number because one pair of inverses rotate into each other: a/c ~ c/a.)

If you remove double twists, because they just aren't as cool, that gets you 11 basic cards--which is few enough to fit into the space of a 13-card suit in a regular deck! Thus, you can do Celtic braid algebra on 4 strands by assigning a simple braid value to each of a set of regular playing cards, with two left over--and you get four copies per deck, so you can actually do interesting things!

Go up to order 5, and the number of basic Celtic sequences is considerably larger, so even by throwing out double-twists, you can't get it to fit into a normal card deck nearly as nicely.