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In [[knot theory]], there are several competing notions of the quantity ''writhe'', or ''Wr''. In one sense, it is purely a property of an oriented [[link (knot theory)|link]] diagram and assumes [[integer]] values. In another sense, it is a quantity that describes the amount of "coiling" of a [[knot (mathematics)|mathematical knot]] (or any [[curve|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''.<ref name=bates_dnatopology_2005 />

==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. 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]].
<center>
{|
|-
| [[Image:knot-crossing-plus.svg|64px]]
|width="64px;" |
| [[Image:knot-crossing-minus.svg|64px]]
|-
| Positive<br>crossing || || Negative<br>crossing
|}
</center>

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.png|right|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 &mdash; 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 (mathematics)|knot]] is such a curve, defined mathematically as an embedding of a circle in three-dimensional  [[Euclidean space|Euclidean space, '''R'''<sup>3</sup>]]. By viewing the curve from different vantage points, one can obtain different [[knots and graphs|projections]] and draw the corresponding [[knot diagrams]]. Its ''Wr'' (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.<ref name=cimasoni_computing_2001 /> Hence, writhe in this situation can take on any [[real number]] as a possible value.<ref name=bates_dnatopology_2005 />

We can calculate Wr with an [[integral]]. Let <math>C</math> be a [[curve|smooth, simple, closed curve]] and let <math>\mathbf{r}_{1}</math> and <math>\mathbf{r}_{2}</math> be points on <math>C</math>. Then the writhe is equal to

:<math>
Wr=\frac{1}{4\pi}\int_{C}\int_{C}d\mathbf{r}_{1}\times d\mathbf{r}_{2}\cdot\frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}
</math>.<ref name=klenin_computationofwrithe_2000 />

==Numerically approximating the Gauss integral for writhe of a curve in space==

Since writhe for a curve in space is defined as a [[Multiple Integral|double integral]], we can approximate its value numerically by first representing our curve as a finite chain of <math>N</math> line segments. We then calculate<ref name=klenin_computationofwrithe_2000 />

:<math>
Wr=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j<i}^{N}\frac{\Omega_{ij}}{4\pi}</math>

where <math>\Omega_{ij}/{4\pi}</math> is the exact evaluation of the double integral over line segments <math>i</math> and <math>j</math>; note that <math>\Omega_{ij}=\Omega_{ji}</math> and <math>\Omega_{i,i+1}=\Omega_{ii}=0</math>.<ref name=klenin_computationofwrithe_2000 />

To evaluate <math>\Omega_{ij}/{4\pi}</math> for given segments numbered <math>i</math> and <math>j</math>, number the endpoints of the two segments 1, 2, 3, and 4. Let <math>r_{pq}</math> be the vector that begins at endpoint <math>p</math> and ends at endpoint <math>q</math>. Define the following quantities:<ref name=klenin_computationofwrithe_2000 />

:<math>
n_{1}=\frac{r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|},\; n_{2}=\frac{r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|},\; n_{3}=\frac{r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|},\; n_{4}=\frac{r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}
</math>

Then we calculate<ref name=klenin_computationofwrithe_2000 />

:<math>
\Omega^{*}=\arcsin\left(n_{1}n_{2}\right)+\arcsin\left(n_{2}n_{3}\right)+\arcsin\left(n_{3}n_{4}\right)+\arcsin\left(n_{4}n_{1}\right).
</math>

Finally, we compensate for the possible sign difference and divide by <math>4\pi</math> to obtain<ref name=klenin_computationofwrithe_2000 />

:<math>
\frac{\Omega}{4\pi}=\frac{\Omega^{*}}{4\pi}\text{sign}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).
</math>

This method, as well as other methods, are fully described mathematically and algorithmically in.<ref name=klenin_computationofwrithe_2000 />

==Applications in DNA topology==

[[File:Simulation of an elastic rod relieving torsional stress by forming coils.ogv|thumb|A simulation of an elastic rod relieving torsional stress by forming coils]]
[[DNA]] will coil if you twist it, 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.<ref name=bates_dnatopology_2005 />

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 in <ref name=fuller_1971 /> 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==
* [[Linking number]]
* [[Ribbon theory]]
* [[DNA supercoil]]ing

==References==

{{reflist|refs=
<ref name=bates_dnatopology_2005>{{cite book|last=Bates|first=Andrew|title=DNA Topology|year=2005|publisher=Oxford University Press|isbn=0198506554|pages=36–37|url=http://ukcatalogue.oup.com/product/9780198506553.do#.USxMzKWfOlk}}</ref>
<ref name=klenin_computationofwrithe_2000>{{cite journal|last=Klenin|first=K|coauthors=Langowski, J|title=Computation of writhe in modeling of supercoiled DNA|journal=Biopolymers|year=2000|issue=54|pages=307–317}}</ref>
<ref name=cimasoni_computing_2001>{{cite journal|last=Cimasoni|first=David|title=Computing the Writhe of a Knot|journal=Journal of Knot Theory and Its Ramifications|year=2001|volume=10|issue=387|doi=10.1142/S0218216501000913}}</ref>
<ref name=fuller_1971>{{cite journal|last=Fuller|first=F B|title=The writhing number of a space curve|journal=Proceedings of the National Academy of Sciences|year=1971|volume=68|pages=815–819}}</ref>
}}
*{{Citation|first=Colin |last= Adams|author-link=Colin Adams (mathematician)|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=American Mathematical Society|year=2004|isbn=0-8218-3678-1}}
<references />

[[Category:Knot theory]]