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{{Short description|Embedding of the unit interval into 3-space ambient isotopy inequivalent to a line segment}}
{{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}}
{{About|a mathematical object|animal rehabilitation|Wild Animal Rehabilitation Center}}
[[File:Wild3.png|400px|thumb|Fox-Artin arc Example 1.1]]
[[Image:Fox-Artin (large).png|thumb|The Fox–Artin wild arc 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]]


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, and {{harvtxt|Fox|Artin|1948}} found another example called the '''Fox-Artin arc''' whose [[Complement (set theory)|complement]] is not [[simply connected]].
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==
==Fox-Artin arcs==

[[File:Wild3.png|400px|thumb|Fox-Artin arc Example 1.1]]
Two very similar wild arcs appear in the {{harvtxt|Fox|Artin|1948}}. Example&nbsp;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.
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>.
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*]]
[[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."
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==
==See also==

Latest revision as of 15:51, 22 September 2024

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.

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

[edit]

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 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 or the 3-sphere .

Fox-Artin arc variant

[edit]
Fox-Artin arc Example 1.1*

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 or , despite its similar appearance.

The Fox–Artin wild arc (Example 1.1*) lying in drawn as a knot diagram. Note that each "tail" of the arc is converging to a point.

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

See also

[edit]

Further reading

[edit]
  • 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