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Difference between revisions of "Wild knot"

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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.
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/w097980a.gif" />
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<center><asy>
 +
settings.render = 0;
  
Figure: w097980a
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unitsize(100);
 +
 
 +
import three;
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import tube;
 +
 
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import graph;
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path unitCircle = Circle((0,0),1,35);
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 +
currentprojection = perspective((900,-350,-650));
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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));
 +
// currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0.5,-0.5,0.5),(0.5,0.5,0.75));
 +
 
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triple horn_start=(0,-1,0.6);
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triple horn_end=(0,0.4,0.2);
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real horn_radius=0.2;
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real ratio=horn_end.z/(-horn_start.y);    // fractal levels ratio
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transform3 implode_right = shift(horn_end) * scale3(ratio) * rotate(-90,X) * shift(-horn_start.y*Y);
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transform3 left_right = reflect(O,X,Z)*rotate(90,Y);
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path[] cover_with_holes = scale(horn_radius/ratio)*unitCircle^^
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  shift((horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle)^^
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  shift((-horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle);
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surface cover = surface(cover_with_holes,ZXplane);
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surface cover_left = shift((horn_start.x,horn_start.y,0))*cover;
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surface two_covers = surface(cover_left,left_right*cover_left);
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path3 horn_axis = horn_start..horn_start+(0,0.01,0)..(0,0,0.7)..(0,0.2,0.6)..horn_end+(0,0,0.01)..horn_end;
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surface horn = tube( horn_axis, scale(horn_radius)*unitCircle );
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surface two_horns = surface(horn,reflect(O,X,Y)*horn);
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surface two_horns = surface(horn,reflect(O,X,Y)*horn);
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surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
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surface four_small_horns = implode_right*four_horns;
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surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
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surface big_surface = surface(four_horns,eight_small_horns);
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real R = horn_radius/ratio;
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draw ( circle((0,1,0), 1.005R, Y ), currentpen+2 );
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draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
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draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
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draw (big_surface, yellow);
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pen blackpen = currentpen+1.5;
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draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
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draw ( circle(horn_start, 0.98horn_radius, Y ), blackpen );
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draw ( circle((horn_start.x,horn_start.y,-horn_start.z), 0.98horn_radius, Y ), blackpen );
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real phi=0.9;  // adjust to the projection
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triple u = (cos(phi),0,sin(phi));
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draw( R*u-Y -- R*u+Y, blackpen );
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draw( -R*u-Y -- -R*u+Y, blackpen );
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</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 $E^3$.

Revision as of 18:56, 12 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=35580
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article