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Difference between revisions of "Wild knot"

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A knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979801.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979802.png" /> (cf. [[Knot theory|Knot theory]]) such that there is no homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979803.png" /> onto itself under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979804.png" /> would become a closed polygonal line consisting of a finite number of segments.
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A knot $L$ in the Euclidean space $E^3$ (cf. [[Knot theory|Knot theory]]) such that there is no homeomorphism of $E^3$ onto itself under which $L$ would become a closed polygonal line consisting of a finite number of segments.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097980a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097980a.gif" />
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Figure: w097980a
 
Figure: w097980a
  
Thus, knots containing the so-called Fox–Artin arcs — certain simple arcs obtained by a [[Wild imbedding|wild imbedding]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979805.png" /> — are wild. For example, the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979806.png" /> is non-trivial for the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979807.png" /> (Fig. a); this group is trivial for the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979808.png" /> (Fig. b), but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w0979809.png" /> itself is not homeomorphic to the complement of a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097980/w09798010.png" />.
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Thus, knots containing the so-called Fox–Artin arcs — certain simple arcs obtained by a [[Wild imbedding|wild imbedding]] in $E^3$ — are wild. For example, the fundamental group $\pi_1(E^3\setminus L)$ is non-trivial for the arc $L_1$ (Fig. a); this group is trivial for the arc $L_2$ (Fig. b), but $E^3\setminus L_2$ itself is not homeomorphic to the complement of a point in $E^3$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097980b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097980b.gif" />

Revision as of 08:57, 12 April 2014

A knot $L$ in the Euclidean space $E^3$ (cf. Knot theory) such that there is no homeomorphism of $E^3$ onto itself under which $L$ 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 $E^3$ — are wild. For example, the fundamental group $\pi_1(E^3\setminus L)$ is non-trivial for the arc $L_1$ (Fig. a); this group is trivial for the arc $L_2$ (Fig. b), but $E^3\setminus L_2$ itself is not homeomorphic to the complement of a point in $E^3$.

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