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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.

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 $[0,1]$ 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 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.

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

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

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+{{Short description|Embedding of the unit interval into 3-space ambient isotopy inequivalent to a line segment}}
-[[Image:Fox-Artin (large).png|thumb|The Fox-Artin wild arc lying in '''R'''<sup>3</sup> Note that each "tail" of the arc is converging to a point.|400px]]
+{{About|a mathematical object|animal rehabilitation|Wild Animal Rehabilitation Center}}
+[[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.
-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,  and {{harvtxt|Fox|Artin|1948}} found another example called the '''Fox&ndash;Artin arc''' whose [[Complement (set theory)|complement]] is not [[simply connected]].
+{{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}}
+*{{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|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=49 | pages=979–990}}
+*{{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|ref=harv|first1=John Gilbert|last1=Hocking|first2=Gail Sellers|last2=Young|title=Topology|year=1988|origyear=1961|publisher=Dover|isbn=0-486-65676-4|pages=176–177}}
+* {{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 | pages=585–598 | doi=10.2140/pjm.1973.45.585}}
+*{{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]]

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