It is based on two mathematics papers published by the authors in Nature[1] and Physica A while they were research fellows at Cambridge University’s Cavendish Laboratory.

Note that any knot that begins with an o move must start with the tie turned inside out around the neck.

(In the Nature paper, the lower bound was placed at a more restrictive 1:4, eliminating the knot classes containing the Kelvin, Victoria, and Grantchester; this was likely revised specifically in order to include the Victoria/Prince Albert, which has fairly extensive historical documentation.)

Fink and Mao refer to the latter, even though some knots that are slightly asymmetrical (such as the Nicky and the Windsor) appear symmetrical to the eye.

Fink and Mao describe balance as "the extent to which the moves are well-mixed", citing a tighter knot that comes loose less easily as its primary virtue.

It is calculated by a particular formula, but can be best understood by the layman as the degree to which the L, R, and C moves are evenly distributed throughout the knotting sequence, and the extent to which the L-R or R-L pattern continues uninterrupted after non-terminal centering moves (which requires a change of winding direction from counterclockwise to clockwise, or vice versa).

For the purposes of the book, when a knot has at least one variant (i.e., when two or more knots, at the greatest degree of symmetry for their class, share the same basic structure apart from one or more transposed L-R pairs), the most balanced version is given the standard designation, while the others are labeled as variants, irrespective of qualities such as being self-releasing (coming undone when the narrow end is pulled out).

However, this is not intended to mark an aesthetic preference for one variant over the other(s); as the authors note in their journal articles, "We do not attempt to distinguish between these knots and their counterparts; this much we leave to the sartorial discretion of the reader."

The Half-Windsor and Hanover variants have the advantage of being both symmetrical and self-releasing, but are less balanced than their counterparts above: The book was reviewed in Nature,[5] The Daily Telegraph, The Guardian, GQ, Physics World, and others.