Page 224 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 215

Briefly, a pair of polynomials are computed from the diagram, one in the
`variable' 6, and the other in the `variable' A. The exponents on the terms of
these polynomials correspond to the numbers of sides surrounding the regions
in the diagram: thus, for example, suppose that there are 5 regions, each
having 3 sides and bearing the symbol 6; then the term 563 will appear in the
6-polynomial of the Complexions-Symbol.
A full example will clarify the matter. The extraction of the Complexions-
Symbol from a diagram of a specific 7-crossing knot is shown below (Fig. 7).
Note that all the regions in the diagram are monotypical.

Fig. 7. A 7-crossing knot, with labelled regions
In this case there are four 6-regions, of which three are 3-sided and the
unbounded region adjoins five sides. There are five A-regions, of which two
pairs are respectively 2- and 3-sided, while the remaining one is 4-sided. The
pair of polynomials for the knot would be shown by Listing thus:

65 + 363
A4+2A3+2.12
In general, a Complexions-Symbol has the form

f ao6n + a16i-1 + ... + an-161
boAn + bran-1 + ... + bn-1A1

where the coefficients ai and bi, with 0 < i < n - 1, indicate the numbers of
the various kinds of region occurring in the diagram.
Listing noted that if a term with exponent unity existed in either of the
two polynomials, that term would derive from one or more simple twists in the
diagram. These could be removed by `untwisting'; and so such term(s) could be
dropped from the Complexions-Symbol. Note that Euler's polyhedral formula