Wild arc

In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. Antoine (1920) found the first example of a wild arc, and Fox & Artin (1948) found another example called the Fox–Artin arc whose complement is not simply connected.

The Fox–Artin wild arc lying in R3 Note that each "tail" of the arc is converging to a point.

See also

Further reading
  • Antoine, L. (1920), "Sur la possibilité d'étendre l'homéomorphie de deux figures à leurs voisinages", C. R. Acad. Sci. Paris, 171: 661
  • Fox, Ralph H.; Harrold, O. G. (1962), "The Wilder arcs", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice Hall, pp. 184–187, MR 0140096
  • Fox, Ralph H.; Artin, Emil (1948), "Some wild cells and spheres in three-dimensional space", Annals of Mathematics, Second Series, 49 (4): 979–990, doi:10.2307/1969408, ISSN 0003-486X, JSTOR 1969408, MR 0027512
  • Hocking, John Gilbert; Young, Gail Sellers (1988) [1961]. Topology. Dover. pp. 176–177. ISBN 0-486-65676-4.
  • McPherson, James M. (1973), "Wild arcs in three-space. I. Families of Fox–Artin arcs", Pacific Journal of Mathematics, 45 (2): 585–598, doi:10.2140/pjm.1973.45.585, ISSN 0030-8730, MR 0343276
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