Difference between revisions of "Wild knot"
From Encyclopedia of Mathematics
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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)); | ||
| − | |||
triple horn_start=(0,-1,0.6); | triple horn_start=(0,-1,0.6); | ||
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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+ | + | 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 four_horns = surface(two_horns,left_right*two_horns,two_covers); | surface four_horns = surface(two_horns,left_right*two_horns,two_covers); | ||
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real R = horn_radius/ratio; | real R = horn_radius/ratio; | ||
| − | + | pen blackpen = currentpen+1.5; | |
| − | draw ( circle((0,1,0), 1.005R, Y ), | + | draw ( circle((0,1,0), 1.005R, Y ), blackpen ); |
| − | draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), | + | 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 ), | + | draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), blackpen ); |
draw (big_surface, yellow); | draw (big_surface, yellow); | ||
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| − | |||
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=35580
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