Coding of Braided Knots
There are four types of coding schemes for braided knots of the turkshead type.
They involve the coding of the columns of crossings in the knot, the coding of the rows, or both.
- 1. Casa coding. This is the form of the simple turkshead of the basket weave pattern. It is both column coded and row coded.
- 2. Gaucho coding. The gaucho pattern is column coded.
- 3. Herringbone coding. The herringbone pattern is row coded. This includes the pineapple weave patterns of nested bights.
- 4. Mixed coding. This can be a mix of pass patterns in weaves or a mix of column and row coding in the same knot. The last is rare as I have only found one example of this technique in the literature.
Knot cylinder orientation
A graphic representation is essential to any discussion of the structure of a braided knot. If you look at the published material about these knots, you will find that Bruce Grant, Ron Edwards, and Robert Woolery use a vertical cylinder and Tom Hall, Georg Schaake and Gail Hought use a horizontal cylinder.
Most of use tend to pick one and stay with it in our personal work. My preference is for a vertical perspective. It is a mental adjustment you have to make in semantics to change from one to the other.
Columns and Rows
This template (grid) is a 7 part X 6 bight casa coded simple turkshead.
I have departed from my usual vertical orientation of the knot cylinder and used a horizontal one to reconcile the semantics in general usage for rows and columns. Note that here the columns go around the knot cylinder, and the rows cross it lengthwise. With this orientation, they are seen as we usually percieve them.
When I go back to a vertical orientation for the knot cylinder, the rows will be seen as columns and the columns as rows from a semantic viewpoint.
The row coding
If you follow a row line and examine the crossing in that row, you will see that they are all the same. This constitutes the row coding of a row.In this casa, all the rows are coded the same left to right = under and right to left = over. This will not be true of all row coded patterns, just that all crossings in any row be the same.
The column coding
Now follow any blue line for a column. The same criteria applies to the crossings in a column.
The numeric relationships - These are not numbered in the template, but you can count them, bearing in mint that we split a row in laying the cylinder out flat and one half of each row appears on each side of the template.
The rows - There will be two rows for each bight in the knot. (Rows = Bights X 2) Also the number of parts is the number of strings crossed in each row.
The columns - There will be one less columns than the number of parts. (columns = parts - 1) {Rember this, it is a major factor in planning column coded knots.}
This template is for a knot with an even number of parts. It is mostly to show that the slight difference in the bight alignment across the knot between odd and even parts has no effect on the other parameters.
You might notice that a string run (half cycle) advances two rows between each column in both templates..
Column Coding
The column coded knots include several forms such as the "spanish ring knot","true gauchos" "headhunters", "fan knots" and many permutations thereof. The methods and braiding process is the same for all of them, differing only in the numbers involved. One of the common characteristics of column coding is that all bights will fall on one bight boundary with no nesting of bights. This differs from the row coding which allows the nesting of bights into two or more bight boundaries if desired. ( The pineapple knots. )
String run code sequences
The coding of each half cycle in the braiding process (one trip across the knot from edge to edge ) can be seen as composed of elements of consecutive like crossings. Thus a single crossing either over or under is a one pass element. Two consecutive crossing of a like nature is a two pass element and three such crossing would be a three pass element, etc.
The number of elements affect the order of the coding between the odd and even half cycles of each cycle.
If the number of elements in the odd half cycle is even, the sequence of the even half cycle will be the same.
If the number of elements is odd, the the even half cycle will be the inverse of the odd half cycle.
An even number of elements
These two templates are the same knot, a 9X7 gaucho weave of one string. The one on the left is with the knot horizontal and the on the right it is vertical (rotated 90 degrees CCW ).
Now look at the labeled string run (odd half cycle). The code sequence for it and all in the same direction, (upward in the right template) are identical. It is U2 - O2 - U2 - O2. Notice that in tranversing the knot, it passes through each column from the bottom to the top in sequence. Thus the essential coding for this knot is this sequence of crossings. If you used Allwines grid maker to create this template as I did, you would enter \\//\\// for the coding. If you look at the coding of the even half cycles, down from the top, you will see they are the same as the upward pass.
Now for the numbers. Since we need eight code columns for this sequence we are limited to a nine part knot. It can be braided with any number of bights that don't violate the the common divisor rule for a single string knot.The code sequence and bight increment will be the same for all of them, but the run list and bight progresion will change for each of them.
The main point here is that we have exactly the number of parts to contain the code sequence. That is the length of this sequence + one.
An odd number of elements
This is the 8 X 7 knot with column coding. The column code sequence is U1 - O2 - U1 - O2 - U1 for a total of seven columns to fit the eight part structure.(notice the odd number of elements in this sequence)
I won't even to try to give it a name, as the P x B and code sequence fully describes it.
If you inspect the template in the example above for the 9 X 7 knot you will see that each return path from top to bottom is a duplicate of the upward run. Now look at this one. The return (even half cycle) is the inverse of the upward run. This is the result of the column code sequence with an odd number of elements.
Designing a column coded knot
To document a complete tutorial for a knot we need two things.
1. A picture of the knot in the form of a grid (or template if you prefer, in this context they are the same thing ). The best tool for this purpose is Tim Alwine's grid maker program. The alternative is to hand code it in a graphic editor such as "Paint" in windows or "Paintbrush" on a Mac.
2. A run list. For the column coded knots, we will use an adaption of the algorithm diagram from the work of Georg Schaake as described in Tom Hall's book "Introduction to Turkshead Knots".
The single string case
This is where the grid maker program shines. If you have the Mozilla Firefox browser and have downloaded the "grid maker" from khww.net you are in business. It will produce both the grid and the run list in any configuration you might desire. I will discuss the permutations of the alogtithm diagrams in the next section if you code your own grids.
A hint here. I have better luck selecting and copying each from the browser and pasting them into "Paint for the image and "Word Pad" for the run list rather than trying to save them directly as a file.The copy of the image on my system (Windows XP) gives a black background and deletes the pin numbers,but this is easy to fix in Paint. .
Knots with a GCD greater than one
The Grid
This is the three steps to create a two string composite knot (GCD=2) from a larger single string (GCD=1) grid copied from the Grid Maker. Since the Grid Maker will only create single string knots we used a 10 X 9 knot to get the braid structure for 10 parts to extract the 8 bights from.
In fact you would do everything on the first pasting and end up with the final result.
By the way the final product done here in two colors is one version of the barber pole color pattern. We will discuss color patterns later in this topic.
The Run List
There is only one essential number we need to construct the algorithm diagram for any knot. This is the negative bight progression value. There are two ways to get this value. The easy way is by a visual inspection of the grid for the knot. The hard way is to calculate it fron the Parts and Bights of the knot.
The easy way
Using the 10 X 8 grid from the example above, I have highlighted the first cycle of the composite knot. Notice that it starts at bight pin 1 and terminates at bight pin 3.
The negative bight progression is the number of bight positions between these two points as viewed in the direction (CCW) opposite that of the string run (CW). Since the bight position 1 is split between both edges of the grid, we count back from the left edge for this count. In this case the value is 6 (the blue numbers).
An algorithm diagram
Step by step
Analyzing the knot
Here we define the knot characteristics by Parts and Bights of the composite knot we are building and get the value of V for the cyclis bight sequence .Notice that I have listed two methods for this. They both are based on modular math. If you have a mod calculator that will handle negative numbers, enter the negative Parts for this. If not then calculate it as in the second choice (P/B = N + R. The best way to do this division is by subtraction. The number of times you have to subtract the bights from the parts to get a remainder less than the parts is N and that remainder is R. What the value of N represents is the number of times the running end will cross the standing end in the first cycle of the braid. This will suffice for this example and I will go into the case where the number of bights exceeds the number of parts a little later.
Next, determine the Parts and Bights of the base knots that will be interwoven to produce the final composite knot.
Then do a simple test of the column coding sequence. We will use this to enter the coding line on the chart later.
Creating the cyclic bight sequence and scan line
Now we create the cyclic bight sequence, The first line (in red), is created by laying down a line of red dots equal to the number of bights in the composite knot.
Begining at the leftmost, sequencially enter the bights of the base knot in increments of the value of V (6 in this example). Notice the blank spaces left by this. They leave space for the interweave in the scan line we will create in the next step.
The scan line is created by moving one space to the right and laying down a series of dots (blue here) equal to the code columns of the composite knot (P - 1 = 9 ).
Now begin with the second element (a blank here) and fill the scan line with the cyclic bight sequence, looping back to the start to finish out the length.
** If you do this for a single string knot there won't be any blank spaces in either of these lines. **
Creating the chart
This is the chart we will enter the information we have generate into so we can use it effectively. I do these on 1/4 inch graph paper available from Walmart in a ring binder or planning pads from an ofice supply as "planning pads".
In the first column on the left, after leaving three blank spaces for the scan and coding information, I set up pairs of lines for each bight of the composite knot showing the up and down directions of the odd and even half cycles.
The second column is for the pin numbers from the grid if you use them.
Next, leave enough columns blank to hold the code column sequence ( 9 for this example).
Then I divide the chart into sections for the first base knot and the interweave. These are highlighted in red and blue here.
Next another column for pin #s.
The last column is just a replication of the bight numbers from the base knot in each of the base and interweave sections to use with the scan line to enter the coding in the next step.
** Again, for the single string case there will only be one knot section, as there is no interweave in a single string knot **
Framing the chart
This step I call framing the chart for want of a better term. It is a method of getting the first crossing of each even half cycle placed so we can complete the chart by inference as explained in the next step.
The first step is to place the cyclic bight count and scan line in the first two rows as indicated at the right. In reality I just do it for the first time in this space, as it just fits there and saves some labor.
Notice that I filled the blank spaces in the scan line with an A. We will use this in a minute.
Next enter the second line of the code sequence test from the analysis below the scan line.
The method is to read it from right to left as you enter it from left to right, thus reversing it. In this example this is not readily apparent but it is important.
Now the scanning of the chart can begin.
In the last column of the chart, begin at the first bight number and proceed to the left to the column with the same number in the scan line and enter the code below that number in that space. Repeat this with the remaining bight numbers in the section for the base knot.
Here we begin the section devoted to the interweave
The first line at the odd half cycle of bight 5 needs special attention. Since it is an odd half cycle and our code under the scan line is for even half cycles, and this code sequence has an odd number of elements we need to invert the codes. Also, since this is the interweave, we use the A in the scan line for the columns.
Now repeat the entry of the base knot coding to incorporate it into the interweave pattern.
Completing the chart
We can now complete the chart by inference from the braid characteristics of the coding sequence of succeeding half cycles.
The first of these is that each odd half cycle code sequence is identical to the preceeding even half cycle in length and any increase in length will occur on an even half cycle.
The second factor is that the actual coding of the half cycle sequences is inverted between those with an odd number of code elements and identical for those with an even number of code elements.
Thus we can, in any column in the chart, complete the code entries.
As before, we will work with the two knot sections of the base and the interweave separately, but we will use the same procedure in each.
Since we are dealing with half cycle code sequences with an odd number of elements we can just go to all columns in each section with blank spaces below them and complete them with an alternating set of codes to complete them.
The last step is to fill in the pin # columns. The easy way to do this is by a visual inspection of the grid. Just follow each half cycle and note the starting and ending pin number(b-ottom to t-op or t to b as appropiate.
In the sad case that you did not creat a grid, they can be calculated with the concept of bight progression and some more modular math.
The positive and negative bight progression I have always used in these dscussions is actually just as they are percieved as seen on a bight numbered grid. The real physical progression for a full cycle is equal an increment around the knot cylinder equal to the number of parts. Since a cycle is symetrical, it is 1/2 the parts for a half cycle. On the chart we begin the free run (first half cycle at pin 1 on the bottom of the grid. Here the increment is 10 so this half cycle will terminate 5 pins further at the top for a pin # of 6. Bring this number back to the left for the start of half cycle 2(the second half of bight 1).Now add 5 to this to get its end point. The result is 11. Since this is greater than the modulus of the 8 bight count around the knot, we subtract 8 (11 - 8) for the actual pin number. Repeat this process for the first base knot(4 bights).
Now for the interweave knot. We have moved to pin 2 at the bottom of the grid to start this new string so start the first half cycle of bight 5 with tne number 2 and repeat the process for this section of the composite knot.
A composite view
Another Example
See the discussion below.
This is essentially the knot from the previous example with one very minor change. I inserted a two pass section in the code sequence . This was used to generate the new grid.
You will see that this made some major changes in the coding of the chart.
The first is in the code sequence in the scan line.
Also, since we now have an even number of code elements in each half cycle, we complete the chart by just replicating each column instead of the alternate entries as before.
Also notice the color pattern from the grid. The two pass section changed the direction of the stripes of the "barber pole". Thus it becomes a "broken barber pole".
One reason I chose this particular knot for these examples was to make the point that the base knots don't necessarily have a casa coded weave pattern. In the first example, they are actually Mathew Walkers (I think), and here I don't really know what the should be called.
