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. Fox & Artin (1948) found another example, called the Fox-Artin arc, whose complement is not simply connected.

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 ${\displaystyle S^{3}}$.

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.

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