Wild arc

Short description: Embedding of the unit interval into 3-space ambient isotopy inequivalent to a line segment

The Fox–Artin wild arc lying in [math]\displaystyle{ \mathbb{R}^3 }[/math] drawn as a knot diagram. Note that each "tail" of the arc is converging to a point.

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) found another example called the Fox-Artin arc whose complement is not simply connected.

See also

• Wild knot
• Horned sphere

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 
• 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 
• Hocking, John Gilbert; Young, Gail Sellers (1988). Topology. Dover. pp. 176–177. ISBN 0-486-65676-4. https://archive.org/details/topology00hock_0/page/176. 
• 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, http://projecteuclid.org/euclid.pjm/1102947540