Page 221 - J. C. Turner "History and Science of Knots"

212 History and Science of Knots

branches of topology. One result, which he gave without proof (after the
quotation just given) is the following integral:

(x' - x)(dydz' - dzdy') + (y' - y)(dzdx' - dxdz' ) + ( z' - z)(dxdy' - dydx')
= 4m7r
ff ((x' - x)2 + (y' - y)2 + (z' - z)2)1
where m is the number of `windings around' (Umschlingnngen), and the inte-
gral extends over both curves.
Another note of special interest, recorded in December 1844, gave numer-
ous forms which closed curves with four knots can exhibit.
These mere snippets represent Gauss' known researches relating to knots;
further mention of his knot work may be found in Stickel [811. One can only
surmize what further thoughts this genius may have had, and what results
gained, on the nature and properties of knots.

2 3

Fig. 5. A Knot by Otto Boeddicker
By contrast with all his other fields of interest, Gauss was not a very active
researcher in topology. It has been alleged that Schniirlein, a pupil of Gauss,
carried on intensive research with his help, on the application of higher analysis
to topology; but no one has been able to verify this. On the other hand, Otto
Boeddicker's work from 1876 is with certainty an independent continuation of
Gauss' work. In his inaugural dissertation Boeddicker discusses at considerable