Page 292 - J. C. Turner "History and Science of Knots"

On Theories of Knots 283

By half-cycle we mean a complete passage from Left- *Right (or vice
versa). For example, the first half-cycle is the pass (from the Standing end,
labelled S) of the string from point 1 to point 2. There are six half-cycles
needed to complete the braid, which is completed by the last half-cycle from
point 6 to point 1, when the Working end, labelled W, arrives at point 1.
Note that every half-cycle is bisected by a string crossing; that is why the
term `2 part' is applied to the knot. And note also that at the right-hand `edge'
of the knot, the string changes direction at points 2, 4 and 6; we call these
bight points; that is why the term `3 bight' is applied to the knot. The pair
of integers (2,3), serves to specify the `whole string run' of this Turk's Head
Knot; this number pair, together with the given interweave coding, completely
specifies the Knot.
Using the diagram (c), we can trace the six half-cycles needed to complete
the Knot, and write down the following Algorithm, which specifies the braiding
process.
Algorithm
Step 1 : L--+R upwards; run 1 to 2; no crossings.
Step 2 : R--4L (right to left); run 2 to 3; no crossings.
Step 3 : L-+R (left to right); run 3 to 4; no crossings.
Step 4: R-+L; run 4 to 5; cross over 1-2.
Step 5: L-R; run 5 to 6; cross under 2-3.
Step 6 : R--+L; run 6 to 1; cross over 3-4.

If, finally, we join the working end (W) to the standing end (S) at 1, then
we shall have completed the braiding of a trefoil knot.

7. On Regular Knots

With the concepts of grid diagram and associated measures explained, we can
turn to definition of classes of braids. Schaake's first book deals with The
Regular Knot class. He defines this class (see [17]) as follows:
Definition (i)
A regular flat braid is a flat braid which has all its left-hand bights on
a single straight line, and all its right-hand bights on another, parallel
straight line. It consists of two sets of string runs, with all the members
of a set being parallel and running from one bight boundary to the other.
Figures 6 and 7 show two regular 7 part/5 bight flat braids, with grid diagrams.