Lesson Five

Templates from a Mat Grid

In this lesson we will examine the use of a mat grid to extract a template for any possible knot. The grid I use is an adaptation of the one on T.J. Bartruff's site that was created with the "Paint" bit mapped graphic editor in Microsoft Windows. It is scaled to fit the tools on the tool bar to make editing easy. I think that I have learned more about the simple srtucture of the braided knots from this graphic tool in the last few years than from 65 years of braiding the darned things.

The Grid

The grid is a graph of an under one - over one flat braid mat that can be woven or braided with one string. A segment of this graph can be extracted so that when rolled into a cylinder, it will represent this braid pattern in a cylindrical form. This is the definition of the simple turkshead braided knot and is the template that we can use to analyze any such knot.

Vertical bight alignment
The template must have the proper bight alignment across the knot for odd parts or even parts as appropiate.
The starting point
This should be from the center of a bight at the lower right corner of the template. (This bight is completed on the left lower corner by the final wrap of the knot.) The point directly above the start at the top will be at the center of another bight for even part knots and at the cross between bights for odd part knots. This choice of the start automatically established the proper bight alignment
The path of the running end across the template
To braid any knot the path of the running end is from the start - upward to the left across the template to a bight at the top and back down to the left. This follows the convention of a clockwise wrap around the knot. When you reach the right edge of the templat you have completed one circuit around the cylinder and then jump to the same level on the right side and continue downward to the left.

XXThe big grid at the left is a 12 X 12. It is casa coded but having an equal number of parts and bights it is not a possible knot in itself but its structure incorporates all the possible knots with twelve or less parts and bights.
It has three knots outlined as examples of the process used to extract a template for any knot. The first at the immediate right is a 5 part X 4 bight knot coded with the conventionally coded perspective. The additions to the coding are in red for additions and green for erasures to produce a clean template. This minor editing makes the template a little easier to use.
The one in the center is the same knot done one bight boundary higher to get a sobre coded perspective.
The third at the left is an 11 part by 3 bight example of a long knot. Notice that the first wrap makes three complete turns around the cylinder to the violet X and then goes on to a bight two spaces beyond the start for the expected eleven pin bight increment.

The Five Bight Knots

XX This is a segment of the mat grid five bights wide and fourteen parts long. All the five bight knots with fourteen or less parts can be derived from this.
The bight boundaries of each knot are marked off and labeled to the right of the figure at the left.
The 4p X 5b is done at the bottom of the larger grid and edited a little to make it plainer. The next two are a 6p X 5b and a 7p x 5b that have been set aside and "cleaned up" but they were both taken from the larger grid at the marked bight boundaries. The run sequence for the first two wraps of each, first wrap in green and second in blue, is given over the template. Note that the first half wrap of both go to the same pin number at the top but the bight alignment of the 7p knot has shifted one half bight to the left for the half bight increase to center the first wrap over it's bight increment.
XX Here are the 8, 9, and 11 part templates from the same 5 bight grid segment.
Notice the shift in vertical bight alignment between the 8 and 9 part knots.
The running end makes two full turns around the knot in the first half wrap and crosses itself twice in the second half of the first wrap in the 11 part template at the near right. This conforms to the expression P/B=N+R ( 11/5=2+1 ).

In Summary

A thoughtful look at these examples will explain how the flat braid pattern of the number of parts determines the characteristics of the cylindrical braid of the basic turkshead knot. The bight increment by the number of parts, and apparent bight progression by the ratio of parts to bights, and the vertical bight alignment by whether the parts are odd or even.

Copyright Sidney Wood 06/12/07