Editing Writhe

{{Short description|Invariant of a knot diagram}}
In [[knot theory]], there are several competing notions of the quantity '''writhe''', or <math>\operatorname{Wr}</math>. 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. 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]].
{|style="margin:1em auto;"
| [[File:knot-crossing-plus.svg|64px]]
|width="64px;" |
| [[File:knot-crossing-minus.svg|64px]]
|-
| Positive<br />crossing || || Negative<br />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.

[[File:Reidemeister move 1.svg|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 — 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]], <math>\R^3</math>. By viewing the curve from different vantage points, one can obtain different [[knots and graphs|projections]] and draw the corresponding [[knot diagram]]s. Its writhe <math>\operatorname{Wr}</math> (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 />

In a paper from 1961,<ref name="Călugăreanu1961">{{cite journal|first=Gheorghe|last=Călugăreanu|author-link=Gheorghe Călugăreanu|title=Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants|lang=fr|journal=Czechoslovak Mathematical Journal |volume=11 |year=1961|issue=4|pages=588–625|mr=0149378|doi=10.21136/CMJ.1961.100486|doi-access=free}}</ref> [[Gheorghe Călugăreanu]] proved the following theorem: take a [[Ribbon (mathematics)|ribbon]] in <math>\R^3</math>, let <math>\operatorname{Lk}</math> be the [[linking number]] of its border components, and let <math>\operatorname{Tw}</math> be its total [[Twist (differential geometry)|twist]]. Then the difference <math>\operatorname{Lk}-\operatorname{Tw}</math> depends only on the core curve of the [[Ribbon (mathematics)|ribbon]],<ref name=cimasoni_computing_2001 /> and
:<math>\operatorname{Wr}=\operatorname{Lk}-\operatorname{Tw}</math>.

In a paper from 1959,<ref name="Călugăreanu1959">{{cite journal|first=Gheorghe|last=Călugăreanu|author-link=Gheorghe Călugăreanu|title=L'intégrale de Gauss et l'analyse des nœuds tridimensionnels|lang=fr|journal=Revue de Mathématiques Pure et Appliquées |volume=4 |year=1959|pages=5–20|mr=0131846|url=http://math.ubbcluj.ro/~calu/59gauss.pdf}}</ref> Călugăreanu also showed how to calculate the writhe 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 the Gauss integral

:<math>
\operatorname{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>.

== Numerically approximating the Gauss integral for writhe of a curve in space ==
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 <math>N</math> line segments. A procedure that was first derived by Michael Levitt<ref name=levitt_computationofwrithe_1986 /> for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski<ref name=klenin_computationofwrithe_2000 /> is to compute

:<math>
\operatorname{Wr}=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j<i}\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}\cdot n_{2}\right)+\arcsin\left(n_{2}\cdot n_{3}\right)+\arcsin\left(n_{3}\cdot n_{4}\right)+\arcsin\left(n_{4}\cdot 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>

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).<ref name=klenin_computationofwrithe_2000 />

[[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]]

== Applications in DNA topology ==
[[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.<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<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 ==
* [[DNA supercoil]]ing
* [[Linking number]]
* [[Ribbon (mathematics)|Ribbon theory]]
* [[Twist (differential geometry)|Twist (mathematics)]]
* [[Winding number]]

== References ==

{{reflist|refs=
<ref name=bates_dnatopology_2005>{{cite book|last=Bates|first=Andrew|title=DNA Topology|year=2005|publisher=[[Oxford University Press]]|isbn=978-0-19-850655-3|pages=36–37|url=http://ukcatalogue.oup.com/product/9780198506553.do#.USxMzKWfOlk}}</ref>
<ref name=levitt_computationofwrithe_1986>{{cite journal|last=Levitt|first=Michael|title=Protein Folding by Restrained Energy Minimization and Molecular Dynamics|journal=[[Journal of Molecular Biology]]|year=1986|volume=170|issue=3|pages=723–764|doi=10.1016/s0022-2836(83)80129-6 |pmid=6195346|citeseerx=10.1.1.26.3656}}</ref>
<ref name=klenin_computationofwrithe_2000>{{cite journal|last1=Klenin|first1=Konstantin|last2=Langowski|first2= Jörg|title=Computation of writhe in modeling of supercoiled DNA|journal=[[Biopolymers (journal)|Biopolymers]]|year=2000|volume=54|issue=5|pages=307–317|doi=10.1002/1097-0282(20001015)54:5<307::aid-bip20>3.0.co;2-y|pmid=10935971}}</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|pages=387–395|mr=1825964| doi=10.1142/S0218216501000913|arxiv=math/0406148|s2cid=15850269}}</ref>
<ref name=fuller_1971>{{cite journal|last=Fuller|first=F. Brock|title=The writhing number of a space curve|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|year=1971|volume=68|issue=4|pages=815–819|doi=10.1073/pnas.68.4.815|mr=0278197|pmid=5279522|pmc=389050|bibcode=1971PNAS...68..815B|doi-access=free}}</ref>
}}

== Further reading ==
* {{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=978-0-8218-3678-1}}

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