6 research outputs found
On comparing the writhe of a smooth curve to the writhe of an inscribed polygon
We find bounds on the difference between the writhing number of a smooth
curve, and the writhing number of a polygon inscribed within. The proof is
based on an extension of Fuller's difference of writhe formula to the case of
polygonal curves. The results establish error bounds useful in the computation
of writhe.Comment: 16 pages, 5 figure
Writhe formulas and antipodal points in plectonemic DNA configurations
The linking and writhing numbers are key quantities when characterizing the
structure of a piece of supercoiled DNA. Defined as double integrals over the
shape of the double-helix, these numbers are not always straightforward to
compute, though a simplified formula exists. We examine the range of
applicability of this widely-used simplified formula, and show that it cannot
be employed for plectonemic DNA. We show that inapplicability is due to a
hypothesis of Fuller theorem that is not met. The hypothesis seems to have been
overlooked in many works.Comment: 20 pages, 7 figures, 47 reference
The theory and applications of writhing
Writhe measures the extent to which a curve is kinked and coiled about itself in
space. It has generally been expressed as a double integral. This measure can be
interpreted as the average number of signed crossings seen by each viewer, over all
possible viewpoints of the curve. This simple geometrical interpretation is used to
describe the established properties of the writhe, as applied to closed spacecurves.
These descriptions differ from previous work as they do not require the construction
of an artificial ribbon structure.
A major feature of this thesis concerns the evaluation of the writhe along a preferred
direction. A directional measure termed the polar writhe will be developed
which can be applied to generic curves (open or closed) . This single integral expression
is shown to be equivalent to the double integral writhe measure for closed
curves. However for open curves the two measures are shown to differ. Further, it
is shown that the polar writhe has distinct advantages when analysing curves with
a strong directional bias.
The thesis then discusses in detail the properties of both the writhe and the polar
writhe measures for open curves. The use of artificial closures for both measures
is examined. In the case of the writhe a new closure is defined that allows the
evaluation of the writhe using single integral expression via the theorems of Fuller.
This closure is unique in that it can be applied to open curves whose end points are
in general position. A simple expression for calculating the non-local polar writhe is
derived which generalises a closed curve expression defined in (Berger Prior J. Phys.
A: Math. Gen. 39, 8321-8348, (2006)). A quantitative study on the effect of the
choice of evaluation direction of the polar writhe is conducted.
The polar writhe formulation is applied to a simple linear force-free magnetic
field model where the field lines form loops above a boundary plane. Loops with
a sufficient amount of kinking are generally seen to form S or inverse S (Z) shaped
structures. Such field lines structures are commonly observed in the Sun’s corona.
A popular measure of the field line morphology is the magnetic helicity. We use
the polar writhe, the correct form for the writhe helicity in the coronal region, to
challenge some popular assumptions of the field. Firstly, the writhe of field lines
of significant aspect ratio (the apex height divided by the foot point width) can
often have the opposite sign to that assumed in a recent review paper by Green et
al (Solar Phys., 365-391, (2007)). Secondly, we demonstrate the possibility of field
lines forming apparent Z shaped structures which are in fact constructed from a pair
of S shapes and have a writhe sign expected of an S shaped structure. Such field
lines could be misinterpreted without full knowledge of the line’s three dimensional
structure. Thirdly, we show that much of the interesting morphological behaviour
occurs for field lines located next to separatrices