- Writhe
In
knot theory , the writhe is a property of an oriented 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. If as you travel along a link component 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 .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.
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" anisotopy 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.ee also
*
Linking number
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