Writhe

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In knot theory, the writhe is a property of an oriented knot or link. More precisely, it is a property of a 2-dimensional representation of a knot or link. The writhe is the total number of positive crossings minus the total number of negative crossings.

Crossings are classed as positive or negative by assigning an orientation to the knot. That is, a direction is assigned to the knot at a point and this direction is followed all the way around the knot. If as you travel along the knot and cross over a crossing, 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.

Positive crossing		Negative crossing

The writhe of a knot is unaffected by two of the three Reidemeister moves: 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 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.

See also: linking number