Sossinsky:Knots. Mathematics with a twist.pdf - English

8 KNOTS
A Mathematical View of Knot Classification
Let us pose the problem in precise terms, rigorous enough to satisfy a
mathematician (readers disinclined to scientific rigor can skip this
part after glancing at the figures). First of all, the very concept of a
knot needs to be defined. We define a knot, or more precisely a representation
o[ a knot, as a closed polygonal curve in space (Figure 1.7a).
A knot is just a class of equivalent representations of knots, equivalence
being the relation of ambient isotopy, defined as follows: An elementary
isotopy is achieved either by adding a triangle (as we add ABC
in Figure 1.7b) to a segment (AB) of the polygonal curve, then replac­
ing this segment by the two other sides (AC and CB), or by doing the
opposite. Of course, the triangle must have no points in common with
the polygonal curve other than its sides. An ambient isotopy is just a
finite sequence of elementary isotopies (Figure 1.7c).
Clearly this definition corresponds to our sense of a knot as the
abstraction of a string whose ends are stuck together, and ambient
isotopy allows us to twist and move the knot in space as we do with a
real string (without tearing it). From the aesthetic point of view, it is
perhaps not satisfying to think of strings that have angles everywhere,
but it is the price to pay for defining a knot in a manner both elementary
and rigorous. I
Representing a knot as a polygonal curve has a motive other than
the ability to attach triangles to it (which presupposes that the "curve"
is made of segments); in fact, it is also a necessary condition for avoiding
"local pathologies." It is required to avoid the so-called wild knots,
which are not topologically equivalent to polygonal curves (or to a
smooth curve). Wild knots are the result of a process of infinite knot-