China
Africa
Explore chapters and articles related to this topic
Rigid-Body Motion
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
When r is sufficiently small, it is useful to represent the linking number into the sum of two quantities: the writhe (or writhing number), denoted as Wr, and the twist (or twisting number), denoted as Tw.It has been shown [10, 50, 49, 63] that the linking number of xand x+ is Lw(x,x+rv)=Wr(x)+Tw(x,v)
Essential properties of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
and w(D), the writhe of D, is a quantity easily read off from D. (See Chapter 18 for details.) For general link diagrams, the Jones polynomial can be derived in a similar way from Kauffman's signed Tutte polynomial[680], an analogue of the Tutte polynomial defined for graphs with signs on the edges.
Hopfions, heliknotons, skyrmions, torons and both abelian and nonabelian vortices in chiral liquid crystals
Published in Liquid Crystals Reviews, 2022
Jin-Sheng Wu, Ivan I. Smalyukh
Although exceptionally useful in classifying topologically distinct field configurations, homotopy theory does not provide the means for exploring the entirety of topological complexity of fields in soft matter even in cases when defects and solitons are embedded within a bulk of an ordered medium like the LC [30,76,78–80]. For example, a closed loop of a half-integer π1(2/2) = 2 disclination is equivalent to a point defect π2(2/2) = in the far-field, but its hedgehog charge (topological charge of a point defect) depends on how this disclination is closed on itself, its local structure, twisting, knotting and possible linking with other defect loops [30,76,79,80]. Knowledge of this relation cannot be predicted solely by the homotopy theory but can be understood by invoking the analysis of the disclination’s structure along the loop, its twist and writhe [79,80]. In other words, the homotopy theory identifies what the complex order parameter fields can be comprised of, but not how to obtain field configurations with desired π2(2/2) hedgehog number by looping and knotting π1(2/2) vortex lines or how to construct solitons with desired π3(2/2) Hopf index by looping and knotting the 2D π2(2/2) solitons. Moreover, when LCs interact with surfaces due to various boundary conditions, the topology of structures of these fields interplays with that of surfaces, which can be rather nontrivial and are a subject of ongoing studies [81–92].