Writhe

Inknot theory,there are several competing notions of the quantitywrithe,or$\operatorname {Wr}$.In one sense, it is purely a property of an orientedlinkdiagram and assumesintegervalues. In another sense, it is a quantity that describes the amount of "coiling" of amathematical knot(or anyclosed simple curve) in three-dimensional space and assumesreal numbersas 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.

Writhe of link diagrams

Inknot theory,the writhe is a property of an orientedlinkdiagram. 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 theright-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.

A Type IReidemeister movechanges thewritheby 1

The writhe of a knot is unaffected by two of the threeReidemeister 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 isnotanisotopy invariantof 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, aknotis such a curve, defined mathematically as an embedding of a circle in three-dimensionalEuclidean space,$\mathbb {R} ^{3}$.By viewing the curve from different vantage points, one can obtain differentprojectionsand draw the correspondingknot diagrams.Its writhe$\operatorname {Wr}$(in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.Hence, writhe in this situation can take on anyreal numberas a possible value.

In a paper from 1961,Gheorghe Călugăreanuproved the following theorem: take aribbonin$\mathbb {R} ^{3}$,let$\operatorname {Lk}$be thelinking numberof its border components, and let$\operatorname {Tw}$be its totaltwist.Then the difference$\operatorname {Lk} -\operatorname {Tw}$depends only on the core curve of theribbon,and

$\operatorname {Wr} =\operatorname {Lk} -\operatorname {Tw}$.

In a paper from 1959,Călugăreanu also showed how to calculate the writhe Wr with anintegral.Let$C$be asmooth, simple, closed curveand let$\mathbf {r} _{1}$and$\mathbf {r} _{2}$be points on$C$.Then the writhe is equal to the Gauss integral

$\operatorname {Wr} ={\frac {1}{4\pi }}\int _{C}\int _{C}d\mathbf {r} _{1}\times d\mathbf {r} _{2}\cdot {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\left|\mathbf {r} _{1}-\mathbf {r} _{2}\right|^{3}}}$.

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

Since writhe for a curve in space is defined as adouble integral,we can approximate its value numerically by first representing our curve as a finite chain of$N$line segments. A procedure that was first derived by Michael Levittfor the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowskiis to compute

$\operatorname {Wr} =\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\Omega _{ij}}{4\pi }}=2\sum _{i=2}^{N}\sum _{j<i}{\frac {\Omega _{ij}}{4\pi }}$,

where$\Omega _{ij}/{4\pi }$is the exact evaluation of the double integral over line segments$i$and$j$;note that$\Omega _{ij}=\Omega _{ji}$and$\Omega _{i,i+1}=\Omega _{ii}=0$.

To evaluate$\Omega _{ij}/{4\pi }$for given segments numbered$i$and$j$,number the endpoints of the two segments 1, 2, 3, and 4. Let$r_{pq}$be the vector that begins at endpoint$p$and ends at endpoint$q$.Define the following quantities:

$n_{1}={\frac {r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|}},\;n_{2}={\frac {r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|}},\;n_{3}={\frac {r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|}},\;n_{4}={\frac {r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}}$

Then we calculate

$\Omega ^{*}=\arcsin \left(n_{1}\cdot n_{2}\right)+\arcsin \left(n_{2}\cdot n_{3}\right)+\arcsin \left(n_{3}\cdot n_{4}\right)+\arcsin \left(n_{4}\cdot n_{1}\right).$

Finally, we compensate for the possible sign difference and divide by$4\pi$to obtain

${\frac {\Omega }{4\pi }}={\frac {\Omega ^{*}}{4\pi }}{\text{sign}}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).$

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).

A simulation of an elastic rod relieving torsional stress by forming coils

Applications in DNA topology

DNAwill coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity ofwritheto 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 asDNA supercoilingand is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.

Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematicallyhow 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 supercoiling
• Linking number
• Ribbon theory
• Twist (mathematics)
• Winding number

Further reading

• Adams, Colin(2004),The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots,American Mathematical Society,ISBN978-0-8218-3678-1