Log in to Scripto | Recent changes | View item | View file

## Henry Bushby Transcription Pages

### VM533B94v6_266.jpg

« previous page | next page » |

You don't have permission to transcribe this page.

## Current Page Transcription [history]

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] K_{n} ^{(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 K_{n}, 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 K_{20}

[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 K_{0} to K_{19}.

[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 K_{n}.

(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.