Sossinsky:Knots. Mathematics with a twist.pdf - English
Sossinsky:Knots. Mathematics with a twist.pdf - English
Sossinsky:Knots. Mathematics with a twist.pdf - English
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10 KNOTS<br />
here are a few remarks on their "wild" kin (<strong>with</strong> a few drawings in<br />
cluded).<br />
Digression: Wild <strong>Knots</strong>, Spatiallntuition, and Blindness<br />
The examples of wild knots shown up to now possess a single isolated<br />
pathoIogicaI point, toward which a succession of smaIler and smaIler<br />
knots converge. Wild knots <strong>with</strong> severai points of the same type can<br />
easily be constructed. But one can go further: Figure 1.9 shows a wild<br />
knot that has an infinite (even uncountabIe, for those who know the<br />
expression) set of pathoIogicaI points.<br />
This set of wild points is in fact the famous Cantor continuum, the<br />
set of points in the segment [O, 1] that remain after one successiveIy<br />
eliminates the centraI subintervai (1/3, 2/3), then the (smaller) centraI<br />
subintervais (1/9, 2/9) and (7/9, 8/9) of the two remaining segments,<br />
O .l l.<br />
9 9<br />
.l<br />
3<br />
l.<br />
3<br />
L<br />
9<br />
.a<br />
9<br />
-=== == == =: :.- -=::: :I I •<br />
Figure 1.9. A wild knot converging to Cantor's continuum.
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