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. found the first example of a wild arc, and  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 article. Example 1.1 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 $$[0,1]$$ 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 $$\mathbb{R}^3$$ or the 3-sphere $$S^3$$.

Fox-Artin arc variant


Example 1.1* has the crossing sequence over/under/over/under/over/under. According to, 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 $$\mathbb{R}^3$$ or $$S^3$$, despite its similar appearance.



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