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

A consistent treatment of link and writhe for open rods, and their relation to end rotation

We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use these definitions to examine the extension of known results for closed rods to open rods. In particular, we formulate the analogue of the celebrated formula $Lk=Tw+Wr$ (link equals twist plus writhe) for open rods and propose an end rotation, through which the applied end moment does work, in the form of an integral over the length of the rod. The results serve to promote the variational analysis of boundary-value problems for rods undergoing large deformations.Comment: 17 pages, 4 figure

Monte Carlo implementation of supercoiled double-stranded DNA

Metropolis Monte Carlo simulation is used to investigate the elasticity of torsionally stressed double-stranded DNA, in which twist and supercoiling are incorporated as a natural result of base-stacking interaction and backbone bending constrained by hydrogen bonds formed between DNA complementary nucleotide bases. Three evident regimes are found in extension versus torsion and/or force versus extension plots: a low-force regime in which over- and underwound molecules behave similarly under stretching; an intermediate-force regime in which chirality appears for negatively and positively supercoiled DNA and extension of underwound molecule is insensitive to the supercoiling degree of the polymer; and a large-force regime in which plectonemic DNA is fully converted to extended DNA and supercoiled DNA behaves quite like a torsionless molecule. The striking coincidence between theoretic calculations and recent experimental measurement of torsionally stretched DNA [Strick et al., Science {\bf 271}, 1835 (1996), Biophys. J. {\bf 74}, 2016 (1998)] strongly suggests that the interplay between base-stacking interaction and permanent hydrogen-bond constraint takes an important role in understanding the novel properties of elasticity of supercoiled DNA polymer.Comment: 21 pages, 6 PS figures. To appear at Biophys.

Observation of a non-adiabatic geometric phase for elastic waves

We report the experimental observation of a geometric phase for elastic waves in a waveguide with helical shape. The setup reproduces the experiment by Tomita and Chiao [A. Tomita, R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937-940, 2471] that showed first evidence of a Berry phase, a geometric phase for adiabatic time evolution, in optics. Experimental evidence of a non-adiabatic geometric phase has been reported in quantum mechanics. We have performed an experiment to observe the polarization transport of classical elastic waves. In a waveguide, these waves are polarized and dispersive. Whereas the wavelength is of the same order of magnitude as the helix's radius, no frequency dependent correction is necessary to account for the theoretical prediction. This shows that in this regime, the geometric phase results directly from geometry and not from a correction to an adiabatic phase.Comment: 13 figure

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

The Shapes of Tight Composite Knots

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4\pi-4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors.Comment: Summary text file of tight knot lengths and writhing numbers stored in anc/ropelength_data.txt. All other data freely available at http:://www.jasoncantarella.com/ and through Data Conservanc