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)