View source for Wild arc

{{Short description|Embedding of the unit interval into 3-space ambient isotopy inequivalent to a line segment}}
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[[File:Wild3.png|400px|thumb|Fox-Artin arc Example 1.1]]

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 [[Path_(topology)#Arc|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 (set theory)|complement]] is not [[simply connected]].

==Fox-Artin arcs==

Two very similar wild arcs appear in the {{harvtxt|Fox|Artin|1948}} article. Example&nbsp;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 <math>[0,1]</math> 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]] <math>\mathbb{R}^3</math> or the [[3-sphere]] <math>S^3</math>.

===Fox-Artin arc variant===
[[File:Wild1.png|400px|thumb|Fox-Artin arc Example 1.1*]]

Example&nbsp;1.1* has the crossing sequence over/under/over/under/over/under. According to {{harvtxt|Fox|Artin|1948}}, page&nbsp;982: "This is just the [[chain stitch]] of [[knitting]] extended indefinitely in both directions."

This arc cannot be continuously deformed to produce Example&nbsp;1.1 in <math>\mathbb{R}^3</math> or <math>S^3</math>, despite its similar appearance.

[[Image:Fox-Artin (large).png|thumb|The Fox–Artin wild arc (Example&nbsp;1.1*) lying in <math>\mathbb{R}^3</math> drawn as a [[knot diagram]]. 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&nbsp;1.1*.

==See also==
*[[Wild knot]]
*[[Alexander horned sphere]]

==Further reading==
*{{citation|first=L.|last=Antoine|title=Sur la possibilité d'étendre l'homéomorphie de deux figures à leurs voisinages|journal=C. R. Acad. Sci. Paris|year=1920|volume=171|page=661|lang=fr}}
*{{Citation | last1=Fox | first1=Ralph H. | last2=Harrold | first2=O. G. | title=Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) | publisher=[[Prentice Hall]] | mr=0140096  | year=1962 | chapter=The Wilder arcs | pages=184–187}}
*{{Citation | last1=Fox | first1=Ralph H. |author1-link=Ralph Fox| last2=Artin | first2=Emil | author2-link=Emil Artin | title=Some wild cells and spheres in three-dimensional space | jstor=1969408 | mr=0027512  | year=1948 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=49 | issue=4 | pages=979–990 | doi=10.2307/1969408}}
* {{cite book|first1=John Gilbert|last1=Hocking|first2=Gail Sellers|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=0-486-65676-4|pages=[https://archive.org/details/topology00hock_0/page/176 176–177]|url-access=registration|url=https://archive.org/details/topology00hock_0/page/176}}
*{{Citation | last1=McPherson | first1=James M. | title=Wild arcs in three-space. I. Families of Fox&ndash;Artin arcs | url=http://projecteuclid.org/euclid.pjm/1102947540 | mr=0343276  | year=1973 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=45 | issue=2 | pages=585–598 | doi=10.2140/pjm.1973.45.585| doi-access=free }}

{{Topology}}

[[Category:Geometric topology]]