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 link diagram and assumes integer values.

In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot (or any closed simple curve) in three-dimensional space and assumes real numbers as values.

In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.

[1] In knot theory, the writhe is a property of an oriented link diagram.

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.

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 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,

(in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.

[2] Hence, writhe in this situation can take on any real number as a possible value.

[1] In a paper from 1961,[3] Gheorghe Călugăreanu proved the following theorem: take a ribbon in

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.

Then the writhe is equal to the Gauss integral Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of

A procedure that was first derived by Michael Levitt[5] for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski[6] is to compute where

is the exact evaluation of the double integral over line segments

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] DNA will coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress.

In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.

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”.