{{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}} 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 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 <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=== 400px|thumb|Fox-Artin arc Example 1.1*
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 <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 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 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–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