To Braid a Specific Knot
This topic is an attempt to present a general method of braidihg any specific single lead turkshead in the casa coded form from its definition of
Parts X Bights. The basic idea is to watch the braiding process and by crossing all previously crossed parts in a way that completes the casa coded braid pattern and observing the proper braid entry and exit for the code we are using, we can braid the complete knot by visual inspection.
Since all wraps fall parallel to the previous one, the place to watch the braid pattern is how any part to be crossed was crossed by a previous wrap or at the edge of the knot for proper braid entry or exit.
Knot Orientation
I have consistently depicted the cylinder of the knot as being vertical and the wraps being placed with a clockwise direction in the templates as a matter of convienence. This is just a matter of perspective. Many braiders prefer to work with a knot with the cylinder of the knot in a horizontal position (usually starting the standing end at the left side). Actually, since the knot is theoretically a closed braid it could be started at any point in the pattern and braided in either direction so long as the basic structure is followed from that point to terminate at the starting point. In fact there is a technique of starting a knot at the center of the thong at a bight and working both ways that is a help in some knots.
Coding
It is traditional to consider that a turkshead can have two codings. If the standing end of the knot at the start enters the braid with an under crossing so that the running end tucks under the edge of the braid when the knot is completed, we consider it regular or conventional coding. If the braid entry an the starting edge is an over crossing so that we have to go up one row to tuck under and complete the braid it has been called "sobre" coding.
In fact, for knots with an odd number of parts both codings produce the same knot. The "sobre" coded knot is just the same knot rotated 180 degrees for a different perspective, but the actual path of the thong is a mirror image of the path when started from the edge that corresponds to the conventional code.
With knots having an even number of parts, the two codings produce slightly different knots.
The main point to this is that certian knots are more convenient to braid from one code perspective than the other. A good example of this is the "quick start" knots described in another topic.
The First Wrap
In this discussion I will use the vertical orientation for the knot and wrap in a clockwise direction as usual.
The first wrap for any knot will start at the center of a bight directly below a pin on the mandrel and pass upward diagonally around the mandrel in a clockwise direction to a bight pin spaced 1/2 the number of parts in the knot from a reference position directly above the starting pin at the bottom. (This acccounts for the 1/2 bight vertical offset in the knots with an odd number of parts.) Then back down to a bight pin spaced the number of parts clockwise from the initial start at the standing end for the center of the next bight. We are using the number of parts as the increment between succesive bights at each edge of the knot. This 1/2 increment step in the first half wrap centers the wrap over the increment at the bottom for the required symetry in the braid structure of the knot. The running end may not cross the standing end in the second half wrap or may cross it one or more times dependent on the ratio of parts to bights of the selected knot. I am not strong on mathmatical anlysis for these knots but there are a couple of simple tricks to predict what the first wrap will do from the relation between the number of parts and the number of bights from the definition of the specific knot as Parts X Bights (P X B ). If we divide the Parts by the Bights as P/B= N (a whole number) + R (a remainder), N will give us the number of times that the running end will cross the standing end in the second half of the first wrap. R will be the positive bight progression in the directiion we are wrapping (clockwise). If there is no remainder R the knot is not possible because of the common divisor rule. Of course the bight increment will establish this anyway but is a check that the counting process was right. Now to determine the manner of crossing the standing end in the second half of the wrap subtract the number of bights from the number of parts. This is the row that the first crossing will happen in and if this number is still larger than the number of bights do the subtraction again for the next crossing row. do this as long as the result is larger than the number of bights (you still get a positive number as a result). The even numbered rows are under and the odd rows are overs for conventional coding and just the reverse for "sobre" coding. The panel below is a series of seven part knots to illustrate the first wrap as it changes with the number of bights.
The top strip is just a 24 bight section of 7part braid with all the edge boundaries of possible knots marked. It's not much use as is but if you save it as a file and load it in a graphics editor such as "paint" in MS-Xp you can crop a template of any 7 part knot up to 24 bights. The others are just that- templates showing the first wrap in red. A word about the lower two where there is no crossing in the first wrap. We can still get the crossing row in the second wrap by doubling the number of parts and using the same subtraction process. You might want to revue Lesson 2 for the reason for using two values for the bight progression.
We will braid each of these separately to illustrate the process of braiding by visual inspection.
7 part x 3 bight
In the first template our check of P/b+R tells us that we will have 2 crossings in the second half of the first wrap and the bight progression will be +1 or clockkwise. The succesive subtraction of bights (3) from parts (7) makes the first to be at row 4 for an under and the second to be at row 1 for on over. The actual row numbers are not really signifigant, just whether the are odd or even.
The second and third sketches are the first and second halves of the second wrap. Going up from the lower pin 2 adjacent to the existihg part we encounter an existing lead that was previously crossed at arrow 1. As it was crossed by an over from our perspective at this point, we cross under it to "complete the braid" here. Going on up we next cross a previous part at the braid exit adjacent to upper pin 1 as we go to pin 2 at the top. Any braid exit for an odd part knot is an over for conventional coding. Now in the third picture we go back down for the second half of the second wrapfrom upper pin 2. On this pass we meet three parts previously crossed by the adjacent "old lead" to the right where we again make contrary crossings to "complete the braid"
and a braid exit at the bottom where we observe the proper over crossing as before to pin 3 at the bottom.
In the last two pictures we go from the lower p1n 3 in the same sequence as the last half of the second wrap, making the contrary crossings and braid exit to pin 3 at the top,a braid entry under and then back down to pin 1 at the bottom, where we take the final "tuck up" over the standing end to complete the knot. Notice that on these last passes we are "filling in the space" between adjacent parts on each side
7 part X 4 bight
This knot is similar to the 7 X 3 except the apparent bight progression is CCw instead of CW so we are now adjacent to the previous part to the right instead of to the left. The second wrap is from lower pin 4 to upper pin 3 - a braid entry and back down to lower pin3, crossing two previously crossed parts on the way.
Wrap three goes from lower pin 3. making a braid entry and crossing two parts to upper pin 2. (Notice that here we are dealing with braid entry at the edge instead of exits as above). Then back down wwith a an entry, three crossings and an exit. Since we are adjacent to the previous part, these crossihgs are contrary.
Wrap four fills the remaining space between parts "completing the braid" as we go and tucks up to finish.
7 part X 5 bight
This one is a little different. Since the P/b=n+r shows us a bight progression of +2 and a single crossing of under in the first wrap this is a good candidate for sobre coding. Why? Because it is a good general policy to make the first crossing an over with as bight progression of 2 in either direction. (More about this in the topic on Quick Start Knots.) This will make that first crossing an over.
In wrap 2 we go from lower pin 3 up to upper pin 1, crossing the first wrap on the way. Since we are going one space away from the first wrap instead of adjaceht to it we are one row of coding below and will cross over instead of the sobre exit of under. (It is important to remember the entry and exits for the coding we are using to avoid confusion.) When coming back down, since we are one space away from wrap 1, we cross in a like manner instead of contrary. The part that later fills the gap will "complete the braid". In crossing the standing end in this wrap we are again one space away from an exit so we go over and on to lower pin 5.
Wrap three passes adjacent to wrap one in a ccw direction so we can resume the use of the method for adjacent parts and use contrary crossings to finish this knot. You may notice that the first under crossing we have met is at the end of this wrap at the exit. This is in fact one of the "Quick Start Knots.
Wrap four and five are the same thing we have been doing but the alternate finish of taking the original standing back in a CCw direction lets us tuck up a sobre knot at the edge instead of one coding row up.
7 part X 6 bight
This is the classic square (one more part than bights) turkshead. The methods I have covered above will complete this one if it is new to you so I won't waste anymore space on it here.
7 part X 8 bight
We are how getting into knots with enough bights that the first wrap is a "free run" with no crossing of the standing end in the wrap. In the first template I have included the second wrap to find the first crossing. In this instance that was redundant as it is obvious from the adjacent runs but that is not always the case. You may notice that the knots with a bight progression of one in either direction build from the edges to the center of the cylinder in an orderly fashion as the braid progresses Also, since there was no crossing in the first wrap , we doubled the number of parts before subtracting the number of bights to find the code row for the first crossing in the knot.
7 part x 9 bight
Our two step analysis here shows a bight progression of -2 and a first crossing of over in the knot. This makes it a probable "qiick start" and our usual methods of completing the braid bear this out.
In the first five wraps the -2 bight progression keeps us one space away from the previous wraps we parallel.
In wrap six we begin to fill in these balnks and in the second half of this wrap we encounter the first part that we need to cross by going under at the braid entry at the top.Theu we have braided almost two thirds of this knot by wrapping over everything.
In the last three wraps we are passing adjacent to previous wraps on both sides and "complete the braid" with contrary crossings to finish the knot in the usual way.
In Summary
This long and tedious discussion is really to justify the reason I started this site in the first place. If you have a good understanding of the basic characteristics of the flat braid pattern involved, you can braid any casa coded single lead turkshead by a visual inspection of the process as you go, crossing by crossing.
Step One
Put the first wrap in place using the concept of the balanced bight increment and the successive subtraction of the bights from parts to determine the manner of crossing the standing end with the running end in the first or second wrap.
The Rest of the Knot
Place the following wraps for each progressive bight by looking at each part crossed by crossing it in a manner that logically completes the basic casa coded pattern in reference to the nearby or adjacent parts.I used seven part knots for the examples of the various situations you might encounter, but this method works for any knot.(However there is an easier way for the very narrow knots with many more bights than parts that I will discuss elsewhere as in the topic on Three Part Knots.)
Another Resource
As these methods require some thought and practice to aquire proficiency, there is help available. Mr. Tim Allwine has an excellent calculator at Knots that will give you the entire braiding sequence for any knot with either coding in a horizontal orientation from left to right. It helps greatly if you get stuck or just need to do the knot in a hurry till you master the other way.
