Wild arc

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

{{About|a mathematical object|animal rehabilitation|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.

{{harvtxt|Antoine|1920}} found the first example of a wild arc. {{harvtxt|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 {{harvtxt|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 $[0,1]$ 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=

File:Wild1.png

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

Image:Fox-Artin (large).png. Note that each "tail" of the arc is converging to a point.|400px]]

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

Wild arc

Fox-Artin arcs

=Fox-Artin arc variant=

See also

Further reading