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Difference between revisions of "Wild knot"

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(picture remake)
m (picture remake: a bit cleaner code)
Line 15: Line 15:
 
currentprojection = perspective((900,-350,-650));
 
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));
 
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));
 
  
 
triple horn_start=(0,-1,0.6);
 
triple horn_start=(0,-1,0.6);
Line 33: Line 32:
 
surface two_covers = surface(cover_left,left_right*cover_left);
 
surface two_covers = surface(cover_left,left_right*cover_left);
  
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;
+
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 horn = tube( horn_axis, scale(horn_radius)*unitCircle );
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 
 
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
 
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
Line 46: Line 44:
  
 
real R = horn_radius/ratio;
 
real R = horn_radius/ratio;
 
+
pen blackpen = currentpen+1.5;
draw ( circle((0,1,0), 1.005R, Y ), currentpen+2 );
+
draw ( circle((0,1,0), 1.005R, Y ), blackpen );
draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
+
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 ), currentpen+2 );
+
draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), blackpen );
  
 
draw (big_surface, yellow);
 
draw (big_surface, yellow);
 
pen blackpen = currentpen+1.5;
 
  
 
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
 
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );

Revision as of 20:35, 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=35596
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article