Page 302 - J. C. Turner "History and Science of Knots"

On Theories of Knots 293

Knot we obtain a Right-Hand Trefoil; but in the case of a "closed" everted
Overhand Half Knot we obtain a closed ring with a full (left-hand) screw turn
in the string (an Un-Knot in the "classical knot theory"). Hence we obtain
two vastly different knots, even in the "classical knot theory". Of course, the
"classical knot theorist" may say that the everting process should take place
after closure, but not only is such an excuse obvious nonsense (since by doing
so we delete an existing crossing and create a new one; hence if we allow
closure before the completion of braiding procedures, we may as well start
with a closed string-run in the form of a circle (their Un-Knot); it moreover
results in the loss of the relationship between an Overhand Half Knot and its
everted form.'

Schaake goes on to discuss many aspects of the construction of braids,
using examples from basic knots such as the 4-crossing knots (see the 16 grid
diagrams shown above, and the comments given upon them), the Overhand
Half Knots, composite Trefoil Knots, Strangle Knots and Constrictor Knots.
His aims are to uncover [quote] `the Facts about some REAL KNOTS'.

9. Summary

In this Chapter we have tried to present a picture of fields of knot studies which
generally are unknown to, or given only a sideways glance by, mathematical
knot theorists. We believe that if these other fields are added to that which
concerns topological knot theorists, then we shall have a more correct purview
for a balanced and useful theory of knots.* A much wider range of problems
will then present themselves to mathematicians for solution.
We began this Chapter by discussing the questions `What is a Science?
and `Is there a Knot Science?'. Let us end with the same questions. Some
might ask-perhaps with justification-Does it matter? What's in a name?
Philosophers have argued about vaguer concepts, sometimes for hundreds of
years (for example `the number of angels that can stand upon a pinhead'; we
maintain that our topic stands firmer than that one.) We might also remark
that deciding whether a subject is, or is not, a Science, is one that exercises
deeply the minds of Vice-Chancellors of Universities; because the magnitudes
of their annual subject grants depend heavily upon the decision. The Sciences
are much favoured over the Arts, for purposes of allocating grants.
an outer and an inner surface, and two lateral faces. Using grid diagrams he can define
`evert equivalency' for these braids.
*Gauss, the father of knot theory, would seem to have recognized this, when, in January
1833, he wrote that the major task for knot theorists would be to count the `different
windings' [Umschlingungen] of two closed or infinite curves, and he placed the problem in
`the boundary field between'[Grenzgebiet] Geometria Situs and Geometria Magnitudinis.