Editing Linking number (section)

==Definition==
Any two closed curves in space, if allowed to pass through themselves but not each other, can be [[homotopy|moved]] into exactly one of the following standard positions.  This determines the linking number:
{| border=0 cellpadding=5 style="margin:auto; text-align:center;"
|- style="vertical-align:center;"
| ⋯
| [[Image:Linking Number -2.svg|140px]]
| [[Image:Linking Number -1.svg|140px]]
| [[Image:Linking Number 0.svg|140px]]
|
|
|- style="vertical-align:center;"
|
| linking number −2
| linking number −1
| linking number 0
|
|
|- style="vertical-align:center;"
|
|
| [[Image:Linking Number 1.svg|140px]]
| [[Image:Linking Number 2.svg|140px]]
| [[Image:Linking Number 3.svg|140px]]
| ⋯
|- style="vertical-align:center;"
|
|
| linking number 1
| linking number 2
| linking number 3
|
|}
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as [[regular homotopy]], which further requires that each curve be an [[immersion (mathematics)|''immersion'']], not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an [[h-principle|''h''-principle]] (homotopy-principle), meaning that geometry reduces to topology.

=== Proof ===
This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
* A single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is ''homotopic'' is clear, since 3-space is contractible and thus all maps into it are homotopic, though the fact that this can be done through immersions requires some geometric argument.
* The complement of a standard circle is [[homeomorphic]] to a solid torus with a point removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary), or the complement can be analyzed directly.
* The [[fundamental group]] of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the [[Seifert–Van Kampen theorem]] (either adding the point at infinity to get a solid torus, or adding the circle to get 3-space, allows one to compute the fundamental group of the desired space).
* Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number.
* It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument.