Wild knot

A knot in the Euclidean space (cf. Knot theory) such that there is no homeomorphism of onto itself under which would become a closed polygonal line consisting of a finite number of segments.

Figure: w097980a

Thus, knots containing the so-called Fox–Artin arcs — certain simple arcs obtained by a wild imbedding in — are wild. For example, the fundamental group is non-trivial for the arc (Fig. a); this group is trivial for the arc (Fig. b), but itself is not homeomorphic to the complement of a point in .

Figure: w097980b

For references see Wild sphere.