Editing Writhe (section)

== Writhe of link diagrams ==
In [[knot theory]], the writhe is a property of an oriented [[link (knot theory)|link]] diagram. The writhe is the total number of positive crossings minus the total number of negative crossings.

A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the [[right-hand rule]].
{|style="margin:1em auto;"
| [[File:knot-crossing-plus.svg|64px]]
|width="64px;" |
| [[File:knot-crossing-minus.svg|64px]]
|-
| Positive<br />crossing || || Negative<br />crossing
|}

For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams.

[[File:Reidemeister move 1.svg|thumb|150px|A Type I [[Reidemeister move]] changes the ''writhe'' by 1]]
The writhe of a knot is unaffected by two of the three [[Reidemeister move]]s: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is ''not'' an [[Knot invariant|isotopy invariant]] of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all.