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Difference between revisions of "Wild knot"

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m (picture remake: a bit cleaner code)
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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.
 
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.
  
<center><asy>
+
{{:Wild knot/Fig1}}
settings.render = 0;
 
 
 
unitsize(100);
 
 
 
import three;
 
import tube;
 
 
 
import graph;
 
path unitCircle = Circle((0,0),1,35);
 
 
 
currentprojection = perspective((900,-350,-650));
 
currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0,-0.5,0.5),(0.5,0.5,0.75));
 
 
 
triple horn_start=(0,-1,0.6);
 
triple horn_end=(0,0.4,0.2);
 
real horn_radius=0.2;
 
 
 
real ratio=horn_end.z/(-horn_start.y);    // fractal levels ratio
 
 
 
transform3 implode_right = shift(horn_end) * scale3(ratio) * rotate(-90,X) * shift(-horn_start.y*Y);
 
transform3 left_right = reflect(O,X,Z)*rotate(90,Y);
 
 
 
path[] cover_with_holes = scale(horn_radius/ratio)*unitCircle^^
 
  shift((horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle)^^
 
  shift((-horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle);
 
surface cover = surface(cover_with_holes,ZXplane);
 
surface cover_left = shift((horn_start.x,horn_start.y,0))*cover;
 
surface two_covers = surface(cover_left,left_right*cover_left);
 
 
 
path3 horn_axis = horn_start..horn_start+0.01Y..(0,0,0.7)..(0,0.2,0.6)..horn_end+0.02Z..horn_end+0.01Z;
 
 
 
surface horn = tube( horn_axis, scale(horn_radius)*unitCircle );
 
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
 
 
 
surface four_small_horns = implode_right*four_horns;
 
surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
 
 
 
surface big_surface = surface(four_horns,eight_small_horns);
 
 
 
real R = horn_radius/ratio;
 
pen blackpen = currentpen+1.5;
 
draw ( circle((0,1,0), 1.005R, Y ), blackpen );
 
draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), blackpen );
 
draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), blackpen );
 
 
 
draw (big_surface, yellow);
 
 
 
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
 
draw ( circle(horn_start, 0.98horn_radius, Y ), blackpen );
 
draw ( circle((horn_start.x,horn_start.y,-horn_start.z), 0.98horn_radius, Y ), blackpen );
 
 
 
real phi=0.9;  // adjust to the projection
 
triple u = (cos(phi),0,sin(phi));
 
draw( R*u-Y -- R*u+Y, blackpen );
 
draw( -R*u-Y -- -R*u+Y, blackpen );
 
 
 
</asy></center>
 
  
 
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 $E^3$.
 
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 $E^3$.

Latest revision as of 12:00, 13 December 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.

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=35622
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article