Page 297 - J. C. Turner "History and Science of Knots"
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288                     History and Science of Knots

          formulae, algorithms and tables which solve many Wheelknot problems. He
          has also discovered an evolution graph for Wheelknots which is `tree-like' but
          contains cycles (i.e. it is not strictly a tree graph).
              For Schaake's work to constitute a mathematical theory, it is necessary to
          supply proofs of all the derived formulae and algorithms. In 1988, Schaake and
          I began a program for writing out the proofs, in a Research Report Series, doing
          this at the same time as producing the books and pamphlets on algorithms and
          tables. The first two Reports, [20] and [21], written in 1988, covered the work
          done on Regular Knots. At that point, several exciting mathematical projects
          arose out of the evolution theory of Regular knots; in quick succession we wrote
          books on number properties discovered from the Regular Knot Tree ([22], [24],
          and [23]). The project for grinding out proofs of braiding algorithms fell into
          abeyance; and it has not yet been restarted. We felt it more sensible to carry
          on writing up pamphlets explaining the algorithms themselves. Schaake has
          most of the proofs for his discoveries;. but it would take a further 2000 pages
          of mathematical writing to present them all. He says `Let the mathematicians
          who follow in my wake'-and he is sure that others will eventually do so-
          `supply all the proofs. Keep them busy for a while!'
              To end this subsection, we present several figures, selected from Schaake's
          books. We give them without comment, allowing them to speak for themselves.
          They illustrate some of the ideas and points we have tried to make in words;
          and they show some of the braids that Schaake has studied and `solved'.
                                                     Priacip 1 :trio6 nra
                     Gaucho Piador Knot                   and
                       (type 1) 9p/4b                 Frmcip l 4vdio6










                       Stcio6 nro oo/y            The regular 9 part-4 bight gaucho flador
                                                knot is derived from the regular 9 part-4 bight
                                                gaucho knot,  which exhibits a constant
                                                column-coding.
                                                  The principal string-run and principal
                                                coding of the regular 9 part-4 bight gaucho
                                                fiador knot is identical to the string-run and
                                                coding of the regular 9 part-4 bight gaucho
                                                knot.

                    Fig. 8. Grid diagram and String runs of a Gaucho Fiador Knot
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