Writhe
In knot theory, there are several competing notions of the quantity writhe, or . In one sense, it is purely a property of an oriented
Writhe of link diagrams
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. 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.
Positive crossing |
Negative 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.
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.
Writhe of a closed curve
Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, . By viewing the curve from different vantage points, one can obtain different
In a paper from 1961,[3] Gheorghe Călugăreanu proved the following theorem: take a ribbon in , let be the linking number of its border components, and let be its total twist. Then the difference depends only on the core curve of the ribbon,[2] and
- .
In a paper from 1959,[4] Călugăreanu also showed how to calculate the writhe Wr with an integral. Let be a smooth, simple, closed curve and let and be points on . Then the writhe is equal to the Gauss integral
- .
Numerically approximating the Gauss integral for writhe of a curve in space
Since writhe for a curve in space is defined as a
- ,
where is the exact evaluation of the double integral over line segments and ; note that and .[6]
To evaluate for given segments numbered and , number the endpoints of the two segments 1, 2, 3, and 4. Let be the vector that begins at endpoint and ends at endpoint . Define the following quantities:[6]
Then we calculate[6]
Finally, we compensate for the possible sign difference and divide by to obtain[6]
In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).[6]
Applications in DNA topology
Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically[7] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number”.
See also
References
Further reading
- ISBN 978-0-8218-3678-1