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Wild knot

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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.

How to Cite This Entry:
Wild knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wild_knot&oldid=11381
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article