Topology of linked defects in condensed matter

Monastyrsky, Michael; Retakh, Vladimir

doi:10.1007/bf01211760

Cited by 44 publications

(25 citation statements)

References 19 publications

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“…It may also be linked by another disclination, which is measured by the fundamental group of the field encircled by the disclination loop. Disclination loop can be wrapped in a torus and classified by two π 1 winding numbers (small and large cycles of the torus) and the π 2 topological charge [18][19][20][21]. These pieces of information are connected by the action of the first homotopy group π 1 on the second homotopy group π 2 , which causes ambiguity in the π 2 topological charge [22].…”

Section: Overviewmentioning

confidence: 99%

Quaternions and hybrid nematic disclinations

Čopar

Žumer

2013

Proc. R. Soc. A.

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Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number − 1 2 . The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.

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Section: Overviewmentioning

confidence: 99%

Quaternions and hybrid nematic disclinations

Čopar

Žumer

2013

Proc. R. Soc. A.

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“…An example of Brunnian 4-component link is given in Figure 4 (see [73,84], and also [38]). It will be again used in the next Section for the demonstration of the differential geometric construction described therein.…”

Section: Brunnian Links and The Borromean Algorithmmentioning

confidence: 99%

“…Their Brunnian 4-link (also appearing in [73], see [84] and Figures 4 and 7) can be again represented as a closed coloured braid , with one of the strands (the first one, say) winding around the others, depicted as straight line segments ( [38]). Now, observing that, in general 182 M. Spera Vol.…”

Section: Remarksmentioning

confidence: 99%

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A Survey on the Differential and Symplectic Geometry of Linking Numbers

Spera

2006

Milan j. math.

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“…Previous integral formulas for Milnor's triple linking number and attempts to define a higher order helicity can be found in the work of Massey,16,38 Monastyrsky and Retakh, 39 Berger,40,41 Guadagnini, Martellini, and Mintchev, 42 Evans and Berger, 43 Akhmetiev and Ruzmaiken, 44,45 Arnold and Khesin, 34 Laurence and Stredulinsky, 46 Leal, 47 Hornig and Mayer, 48 Rivière, 49 Khesin, 50 Bodecker and Hornig, 51 Auckly and Kapitanski, 13 Akhmetiev,52 and Leal and Pineda. 53 The principal sources for these formulas are Massey triple products in cohomology, quantum field theory in general, and Chern-Simons theory in particular.…”

Section: Other Integral Formulasmentioning

confidence: 99%

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

DeTurck

Gluck

Komendarczyk

et al. 2013

Journal of Mathematical Physics

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To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space. C 2013 American Institute of Physics.

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Topology of linked defects in condensed matter

Cited by 44 publications

References 19 publications

Quaternions and hybrid nematic disclinations

Quaternions and hybrid nematic disclinations

A Survey on the Differential and Symplectic Geometry of Linking Numbers

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

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