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Difference between revisions of "Knotted sphere"

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A non-trivial [[Two-dimensional knot|two-dimensional knot]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556201.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556202.png" />; a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556203.png" /> which cannot be obtained by rotation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556204.png" /> of a knotted arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556205.png" /> situated in the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556206.png" /> around the plane bounding the half-space. The fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055620/k0556207.png" /> of a knotted sphere is not a knot group (cf. [[Knot and link groups|Knot and link groups]]).
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A non-trivial [[Two-dimensional knot|two-dimensional knot]] in the $4$-dimensional Euclidean space $E^4$; a sphere $S^2$ which cannot be obtained by rotation in $E^4$ of a knotted arc $k$ situated in the half-space $E_+^3$ around the plane bounding the half-space. The fundamental group $\pi(E^4\setminus S^2)$ of a knotted sphere is not a knot group (cf. [[Knot and link groups|Knot and link groups]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.H. Crowell,  R.H. Fox,  "Introduction to knot theory" , Ginn  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.H. Crowell,  R.H. Fox,  "Introduction to knot theory" , Ginn  (1963)</TD></TR></table>

Latest revision as of 08:37, 12 April 2014

A non-trivial two-dimensional knot in the $4$-dimensional Euclidean space $E^4$; a sphere $S^2$ which cannot be obtained by rotation in $E^4$ of a knotted arc $k$ situated in the half-space $E_+^3$ around the plane bounding the half-space. The fundamental group $\pi(E^4\setminus S^2)$ of a knotted sphere is not a knot group (cf. Knot and link groups).

References

[1] R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963)
How to Cite This Entry:
Knotted sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knotted_sphere&oldid=12342
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article