Wild arc

This article is about a mathematical object. For animal rehabilitation, see Wild Animal Rehabilitation Center.

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.

Fox-Artin arc Example 1.1

Antoine (1920) found the first example of a wild arc. Fox & Artin (1948) found another example, called the Fox-Artin arc, whose complement is not simply connected.

Fox-Artin arcs

Two very similar wild arcs appear in the Fox & Artin (1948) article. Example 1.1 (page 981) is most generally referred to as the Fox-Artin wild arc. The crossings have the regular sequence over/over/under/over/under/under when following the curve from left to right.

The left end-point 0 of the closed unit interval ${\displaystyle [0,1]}$ is mapped by the arc to the left limit point of the curve, and 1 is mapped to the right limit point. The range of the arc lies in the Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ or the 3-sphere ${\displaystyle S^{3}}$.

Fox-Artin arc variant

Fox-Artin arc Example 1.1*

Example 1.1* has the crossing sequence over/under/over/under/over/under. According to Fox & Artin (1948), page 982: "This is just the chain stitch of knitting extended indefinitely in both directions."

This arc cannot be continuously deformed to produce Example 1.1 in ${\displaystyle \mathbb {R} ^{3}}$ or ${\displaystyle S^{3}}$, despite its similar appearance.

The Fox–Artin wild arc (Example 1.1*) lying in ${\displaystyle \mathbb {R} ^{3}}$ drawn as a knot diagram. Note that each "tail" of the arc is converging to a point.

Also shown here is an alternative style of diagram for the arc in Example 1.1*.

See also

• Wild knot
• Alexander 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 (in French), 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