Annular braids
An annular braid is a braid that wraps around a cylinder, rather like the ribbons on a Maypole. Some of the work on annular braids draws them by picturing the Maypole as a strand, something like this:
Annular braid s4 s5 s6 in the style of braidlab
Annular braid s4 s5 s6 on an unrolled cylinder
Here's a larger example:
An annular braid on an unrolled cylinder
- Introduce a new parameter "annular", so that "annular=1" tells the parser that it's making an annular braid.
- Adjust the parse logic so that, for an n-strand
braid, the crossing s_n is identified as the "around the
cylinder" crossing and given special treatment
(to make this work, I also had to futz with the strand counting logic) - Add in a new generator pair for the "twist" move where all the strands
simultaneously shift one place left or right. Andrew's parse
logic allows the user to use any character they want for
their crossing generator name, so that s_1, or b_1
or a_1 are all read as the same braid. This means that
identifying a new special pair risked turning some user's
existing braids into twists.
- However, I set it up so that the parser only looks for twists if "annular=1" is set, which it evidently wouldn't be on existing braids.
- I chose the "special" pair to be "[, ]", which I thought would be extremely unlikely to have been used for anyone's generators anyway. I think they work well for twists as they give a reasonable sense of "pushing" the braid to the left or right. (They are a bit confusing for a code syntax highlighter).
- Add in the "sides" of the cylinder, so that we can see
the wraparound generators going off one side of the
cylinder and on the other. Since it was quite easy to allow
the user to specify their own style for the cylinder sides
(any style that can be applied to a tikz path draw), I did
that -- by default we have dashed lines, but a user could specify dots, zigzags, or even rows of little dancers:
Annular braid examples, drawn with different styles for the identified edges of the cylinder (any TikZ path styling works).
I was left with one unresolved issue: on a 2-strand braid,
the parser, which breaks a crossing down into a list of
strands to swap, cannot tell the difference between the
crossing s_1 ("swap 1,2") and the crossing s_2 ("swap 1,2").
Here is an annular braid which has both:
Annular braid s1 s2 s1, shown on a cylinder
A user who wants to draw a two-strand annular braid like this is probably best off drawing a three-strand braid with an invisible strand (i.e. the strand colour is the page colour and it is the understrand on all its crossings). I was mostly working in matlab/octave, so for myself, I wrote a quick matlab script that can "double up" a strand to convert a two-strand braid into a three-strand braid to achieve this. Here's an example:
A three-strand "workaround" braid for the problematic two-strand braid, shown on the left with the "fake" strand in light gray and on the right with the fake strand white.
Circular braids
Moving away from annular braids, and looking at braids as closures of links, we sometimes find that we have munged a link into a sort of doughnut shape, ready to cut it open and unfurl in a braid (the usual Artin braid). It may from time to time be appropriate, either for clarity or for reasons of space, to draw the braid in this circular form:
A five-component link; an isotopic version of the link that is close to closed braid form, and the closed braid shown in a circle. The unrolled braid would be very long.
To achieve these circular braids, I went back to the braids library. This time, I left Andrew's parsing logic alone, and cut out the drawing logic to replace it with logic to draw arcs and crossings. In this case, rather than add an option to the usual braid pic command, I renamed the library and the pic command, so that I could use regular braids and circular braids as separate libraries in the same tikz picture:
The closed braid "circular" form of a braid, next to the unrolled braid, drawn as circbraid and braid with the same braid specification.
Unlike my annular braids library, I didn't make any effort to make sure all of Andrew's existing options worked. Circular braids do not allow for the specification style a_{1,4}, where non adjacent strands are crossed by building a sequence of crossings; they do not allow for "floors" to be marked, and they do not have all the coordinates that Andrew makes available in a usual braid. The strand style specifications do still work.
The (crossing) width parameter also still works; I replaced crossing height with a parameter I call "innerradius" which specifies how far from the centre of the circle the first (innermost) strand should be. Here are some examples of different radii and crossing widths:
The same braid shown three times with different innerradius parameters
The braid from above, now with the crossing width reducing as the inner radius is increased
Future project: annular braids on cylinders?
Another way to draw an annular braid is to make use of TikZ's projection options:
Annular braids shown in isometric view on a cylinder. Two styles are shown: "pipe" view in which the cylinder is not shown, and "line" view in which the braids are shown on a cylinder; some opacity is suggested by the lines changing from solid to dashed for the "back" of the cylinder.
I drew the above manually, using isometric view in Tikz.
A line segment is given by e.g.:
\draw plot[domain=180:360,samples=145] ({cos(\x+45)},{-sin(\x+45)},{2*(\x-180)/180});
- domain=180:360 this is the section of the circle that the strand travels for this crossing
- (\x-180)/180 we have two strands here, so a strand travels a 180 degree section on a crossing
- 2*(...) we set the crossing height to 2
With this logic, the order of the \draw commands defines the over-/under-crossing effect. To achieve this programmatically, we need to draw the strands in the right order: the "over" (outer) strands that are currently moving between circle positions 180 and 360 degrees, then the "under" (inner) strands in this part of the circle, then the "under" (inner) strands on the front half of the cylinder, and finally the "over" (outer) strands on the front half of the cylinder.
There are a couple more little fiddles: to draw the "pipe" view version, we set the line style to achieve the 3D effect, and it then works best if the lines are overlapped by a few degrees, to avoid awkward junctions at the line end. To draw the "line" view version, we need to add the "dashed" style on the back of the cylinder. If the number of strands is odd, there may be crossings that happen on the boundary between "front" and "back" of the cylinder (from this perspective), which will require a little extra work to deal with.
In principle, we could take the braids library and chop out the drawing code again, and replace with this logic. However, Andrew's library doesn't draw the strands bit by bit like this. It builds up a path command for each strand, which consists of a sequences of lines/curves and jumps (moves). The strands are then simply drawn one after another.
It's certainly not unthinkable that we could use Andrew's parse logic and then build suitable drawing logic to achieve these cylindrical braids. Anyone?