Page 294 - J. C. Turner "History and Science of Knots"

On Theories of Knots 285

Definition (iii)
A Turk's Head Knot is a Regular Knot whose super-imposed coding
alternates overs and unders throughout.
The description `Turk's Head' is used freely in the literature, for many
kinds of knot, without precise definition. For example, in Chapter 17 of Ash-
ley's Book of Knots [1], a Turk's Head Knot is said to be a `tubular knot that
is usually made around a cylindrical object, such as a rope, a stanchion, or a
rail.' Ashley goes on to say: `It is one of the varieties of the BINDING KNOT
and serves a great diversity of practical purposes, but it is perhaps even more
often used for decoration only; for which reason it is usually classed with `fancy
knots'.' In his chapter he describes the construction of over a hundred such
knots, many of which would not be called `Turk's Head' by Schaake (either
because they are not Regular Knots, or because they do not have the strictly
alternating coding pattern).
Schaake knows, of course, the necessity to define one's terms precisely,
as one builds up a consistent mathematical theory. So, in his first book on
Braiding [17], he lays down the definitions of grid diagram, string run, coding,
Regular Knot, Turk's Head Knot. All of his theories are grounded upon the
first three terms; then `Regular Knot' defines a vast (infinite) class of single
string cylindrical knots with reference to their string run grid diagrams only.
Superimposing the alternating coding upon these gives the infinite class of
Turk's Head Knots.

8. Development of Schaake 's Braiding Theory

8.1. Aims and Achievements
The principal concerns of Schaake's Theory, for any given class of Knot which
he defines, are first to discover geometrical and algebraic relationships between
the parameters (e.g. the numbers p and b, representing numbers of parts and
bights in a knot) involved in the string run grid diagrams for members of the
class. Then, if a coding (weaving pattern) is given, the concern is to find an al-
gorithm which enables a braider to construct a braid with that coding. Schaake
solves these problems as generally as possible-not for particular braids, but
for infinite classes of them. Recurrence equations with boundary conditions
are determined for a general knot within a class, and methods, diagrams and
tables are supplied which enable any braider to formulate the algorithm by
which he or she may arrive at the desired braid.
A third concern is to determine how braids within a given class are inter-
related. What does it mean to say `This braid is an enlargement of that one',
or `This braid evolves from that one, but not from this other one'? Again,
suppose that knots in a certain class are determined by an integer vector