The size of a knot, and the primary parameter by which we classify it, is denoted by the half-winding number h, which we define as the number of moves, that is, the length of a knot sequence excluding T. The initial and terminal sequences dictate that the smallest knot be given by the sequence , with h=3. Practical (viz., the finite length of the tie) as well as aesthetic considerations suggest an upper bound on knot size: the Windsor knot requires much of the excess tie length and is considered large at half-winding number 8. We consequently limit our exact results to .
While the half-winding number is indicative of knot size, it usually tells us little about the shape of a knot, of which there are many for large h. The shape depends, of course, on the number of right, centre and left moves in the tie sequence. Aesthetic considerations suggest a preference for symmetric tie knots, that is, knots whose sequences possess a near equal number of right and left half-turns. It follows that, for a given half-winding number, the shape of a knot is largely characterised by the number of centre moves. We thus classify knots of equal size h by the number of centre half-turns, , where by we denote the set of all knots with half-winding number h and centre number .
Aesthetic tie knots need not possess complete symmetry -- the Half-Windsor has equal numbers of left and right moves but the remaining three knots' numbers differ by one. More generally, not all knot classes afford any completely symmetric knots. Rather than limit our attention to those classes which do, we seek the most symmetric knot, or set of knots if the symmetry is degenerate, from each class.
Whereas the centre number and the symmetry s tell us the move composition of a knot, balance relates to the distribution of these moves; it corresponds to the extent to which the moves are well mixed. A well balanced knot is tightly bound and keeps its shape. We use it as our second aesthetic constraint.
We proceed by classifying tie knots with respect to the half-winding number h and the number of centre moves ; knots with identical h and are said to belong to the same class. For a particular class, we seek the most symmetric knots and, of these, the most balanced.