Reference+Material

toc Here's a page to share important papers, review articles, relevant news, etc.

=Some Heroes of Knotted Fields=


 * **KELVIN** All matter is knotted fields; "On vortex atoms" Proc R Soc Ed **6**(1867) 94-105
 * **MOFFATT** Knotted vortices & flows in hydrodynamics, and relation between topology and dynamically conserved quantities; "The degree of knottedness of tangled vortex lines" J Fluid Mech **35**(1969) 117-29
 * **ROBINSON & TRAUTMAN** Topological solutions of Maxwell's equations via Hopf fibration; Robinson "Null electromagnetic fields" Journal of Mathematical Physics 2 (1960) 290-1; Trautman "Solutions of the Maxwell and Yang-Mills equations associated with Hopf fibrings" Int J Theor Phys **16**(1977) 561-5
 * **BOULIGAND** Knotted defect lines in liquid crystals, and what happens to director field in knot complement; "Recherches sur les textures des états mesomorphiques. 6. Dislocations coins et signification des cloisons de Granjean-Cano dans les cholestériques" Le Journal de Physique **35**(1974) 959-81
 * **WINFREE** Knotted & linked phase singularities as organizing centres in 3D biological and chemical waves; "The Geometry of Biological Time", Springer, 1980 (2nd ed 2001); also Winfree & Strogatz "//Singular filaments organize chamical waves in three dimensions. III. Knotted waves//" Physical D **9**(1983) 333-45
 * **WITTEN** Topological Quantum Field Theories used to compute knot polynomials; "//Quantum field Theory & the Jones polynomial//" Comm Math Phys **121**(1989) 351-99
 * **FADDEEV & NIEMI** Knotted fields occur as stable toplogical solitons in Skyrme-like nonlinear field theories; "//Stable knot-like structures in classical field theory//" Nature **387** (1997) 58-61

=Holograms Tie Optical Vortices in Knots= Article from Physics Today describing work by Mark Dennis and others.

=[|Mermin's classic review on topological defects in condensed matter]= "At a minimum, homotopy theory provides //the// natural language for the description and classification of defects in a large class of ordered systems." An introduction to the parts of homotopy theory that are useful for thinking about topological defects; charmingly written and discusses many examples.

=Nematic braids, knots, and links (wiki entry from [|Ljubljana soft matter group])= Here are listed some examplary references on nematic braids, knots, and links, as listed according to their main focus: __(A) Entangled colloidal nematic__: M. Ravnik, M. Skarabot, S. Zumer, U. Tkalec, I. Poberaj, D. Babic, N. Osterman, and I. Musevic, //Entangled Nematic Colloidal Dimers and Wires,// Phys. Rev. Lett. **99**, 247801 (2007) [abstract] [PDF]. S. Čopar and S. Žumer, //Nematic Braids: Topological Invariants and Rewiring of Disclinations//, Phys. Rev. Lett. **106**, 177801 (2011) Abstract] PDF]. U. Tkalec, M. Ravnik, S. Čopar, S. Žumer and I. Muševič, //Reconfigurable Knots and Links in Chiral Nematic Colloids//, Science **333**, 62 (2011) Abstract] PDF]. __(B) Nanoparticles/entanglement__: O. Guzman, E. B. Kim, S. Grollau, N. L. Abbott, and J. J. de Pablo, //Defect Structure around Two Colloids in a Liquid Crystal//, Phys. Rev. Lett. **91**, 235507 (2003) Abstract] PDF]. T. Araki and H. Tanaka, Phys. Rev. Lett. **97**, 127801 (2006) Abstract] PDF]. __(C) Nematic braids__: M. Škarabot, M. Ravnik, S. Žumer, U. Tkalec, I. Poberaj, D. Babič, I. Muševič, //Hierarchical self-assembly of nematic colloidal superstructures//, Phys. Rev. E **77**, 061706 (2008) [[|abstract]] PDF]. M. Ravnik and S. Žumer, //Nematic colloids entangled by topological defects//, Soft Matter **5**, 269 (2009) abstract] PDF]. __(D) Related nematic topology:__ G.P. Alexander, B.G. Chen, E.A. Matsumoto, and R.D. Kamien, //Disclination Loops, Point Defects, and All That in Nematic Liquid Crystals//, Rev. Mod. Phys. **84**, 497 (2012). Abstract] PDF]. R.D. Kamien, //Knot Your Simple Defect Lines?//, Science **333**, 46 (2011). abstract] PDF]. S. Čopar, T. Porenta and S. Žumer, //Nematic disclinations as twisted ribbons//, Phys. Rev. E **84**, 051702 (2011) Abstract] PDF]. S. Čopar and S. Žumer, //Topological and geometric decomposition of nematic textures//, Phys. Rev. E **85**, 031701 (2012) Abstract] PDF]. __(E) Cholesteric structures (chiral nematic materials)__: Y. Boulignad, //Recherches sur les textures des etats mesomorphes. 6 - Dislocations coins et signification des cloisons de Grandjean-Cano dans les cholesteriques//, J. Phys. (France) **35**, 959 (1974). abstract] PDF]. R. D. Pisarski and D. L. Stein, //Surface singularities in nematics and some notes on cholesterics//, J. Physique **41**, 345 (1980). abstract] PDF]. Y. Bouligand and F. Livolant, //The organization of cholesteric spherulites//, J. Physique **45**, 1899 (1984). abstract] PDF]. J. Bezic and S. Zumer, //Structures of the cholesteric liquid crystal droplets with parallel surface anchoring//, Liq. Cryst. **11**, 593 (1992). abstract] PDF].

=Knots and links in steady solutions of the Euler equation= In this paper we prove that any (locally finite) link can be realized (up to a diffeomorphism arbitrarily close to the identity in any C^p norm) as a set of stream (or vortex) lines of a steady solution of the Euler equation in R^3. This solution is in fact analytic and of Beltrami type. If the number of links is finite, the solution can be taken to decay at infinity as |x|^{-1}.



=**Curl eigenfields,** instantaneous rotation, **and compatible metrics**= See Section 2 of the paper enclosed below

=Papers on the asymptotic link/knot invariants of divergence free vector fields (i.e. "higher" helicities)=