A tie knot is initiated by bringing the wide (active) end to the left and
either over or under the narrow (passive) end,
forming the triagonal basis 
and dividing the space into right, centre and left
(R,C,L) regions, as shown in Fig. 1 (this, and all other figures, are
drawn in the frame of reference of a mirror image of the actual tie).
Once begun, the knot
is continued by wrapping the active end around the triagonal basis, as indicated,
for example, in Fig. 2 with the Four-in-Hand;
this process may be considered a sequence of half turns from one region
to another. The location and orientation of the active end
are represented by one of the six states
, where R,C and L indicate the
region from which the active end emanates and 
and 
denote the
direction of the active end as viewed from in front, viz., out of the
page (shirt) and into the page (shirt), respectively.
Figure 1: The two ways of beginning a knot, 
and 
, as viewed in
a mirror. Both give rise to the triagonal basis and divide the
space into the three regions through
which the wide (active) end can subsequently pass. For knots beginning
with , the tie must begin inside out.
The notational elements 
,
etc., initially introduced as states, may be considered moves in as
much as each represents the half-turn necessary to place the active end into the
corresponding state. This makes the successive moves 
, for
instance, impossible, and implies that 
is the inverse of 
.
It follows that no two consecutive moves may indicate the same region or
direction.
To complete a tie knot, the active end must be wrapped over the front,
i.e., either 
or 
,
then underneath to the centre, 
, and finally through (denoted T) the
front loop just made.
Figure 2: The Four-in-Hand, represented by the sequence 
.
We can now formally define a tie knot as a sequence of moves chosen
from the moveset
, initiated by or
and terminating with the subsequence 
or 
.
The complete sequence is constrained such that the move direction
oscillate between and and no two consecutive move regions
be identical. Using this notation, the ubiquitous Four-in-Hand may be
represented .