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.

References

• Antoine, L. (1920), 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, MR0140096 
• Fox, Ralph H.; Artin, Emil (1948), "Some wild cells and spheres in three-dimensional space", Annals of Mathematics. Second Series 49: 979–990, ISSN 0003-486X, JSTOR 1969408, MR0027512 
• McPherson, James M. (1973), "Wild arcs in three-space. I. Families of Fox–Artin arcs", Pacific Journal of Mathematics 45: 585–598, ISSN 0030-8730, MR0343276, http://projecteuclid.org/euclid.pjm/1102947540