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Henry Bushby Transcription Pages

VM533B94v6_266.jpg

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266.
Formula of reduction for Torsion knots with
a less than n. Call the difference by which
n exceeds a, “d”. Then
[Caption: p268]
[3 short vertical lines in the margin] Kn (n-d) a(n-1)(n-d) c = K (n-d) na[(n-1)(n-d)-d] c
(These are not fractions)
[Note: the ‘fraction’ dividing lines are doubled]
[Note: All the above is struck out by a bold oblique line from top left]
It follows that in any series Kn, if the
preceding series are known, no new knot can
be found with a less than n, i.e. after none
before the first rings formed by a
twist through 360[degrees] = n arcs. Thus in K20
[Note: in the left margin is a ‘right-facing’caret. The above two lines appear to be separated by a feint shallowly oblique line from bottom left.]
the first 19 knots have already appeared in K0 to K19.
[Note: there is a single bold vertical line in the margin highlighting the following 5 lines]
(ix.[alpha].) Whereas in S all the crossings are
alternate, in K they are in regular al-
ternating sequences of (n-1) over & (n-1) under, n being the number of original
untwisted cords in any series Kn.
(ix.[beta].) The resulting no. of cords after
twisting on which the torsion knot ap-
pears is called z as in S, and z is
the same, knot for knot, in both groups.

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