New Results

(i) Classical vortex dynamics and fluid knots

topological cascade of fluid knots in terms of geodesic flows in polynomial space (75, 2020; 81, 2023; 83, 2024)
influence of winding number on the evolution of vortex torus knots (71, 2019)
proof of the monotonic cascade of physical torus knots and links under anti-parallel reconnection (61, 2015; 62, 2016)
proof of writhe conservation under anti-parallel reconnection of fluid knots (58, 2015)
derivation and application of the adapted HOMFLYPT polynomial to fluid knots(59, 2015; 66, 2018, 83, 2024)
derivation and application of the adapted Jones polynomial to fluid knots (49, 2012; 50, 2012; 53, 2013; 56, 2014, 83, 2024)
kinetic energy computation of vortex knots and unknots (41, 2009; 44, 2010; 54, 2013)
proof of the influence of torsion on vortex filament motion (8, 1994)
torus knot soliton solutions to the non-linear Schrödinger equation (7, 1993)
derivation of the self-linking number from the kinetic helicity of a vortex knot (4, 1992)
geometric interpretation of soliton invariants under LIA (3, 1991; 5, 1992; 11, 1995)

correction of defect velocity by Dirac delta distributional approach (84, 2024)
analytical proof of the zero helicity condition for quantum vortex defects (80, 2022)
instability effects for a defect subject to global and localized twist (79, 2022)
topological proof of zero helicity of Seifert framed defects (76, 2021)
derivation of hydrodynamic equations of a defect subject to a twist phase (73, 2020)
proof of production of secondary defect due to twist superposition (68, 2018; 70, 2019)
linear and angular momentum computation by geometric methods (37, 2008; 52, 2013; 72, 2019)
proof of relaxation of twist helicity of reconnecting quantum defects (65, 2017)
geometric and topological measures of vortex tangles (21, 2000; 23, 2001; 25, 2001; 27, 2002)
proof of helicity conservation under quantum reconnection of vortex rings (60, 2015)
evolution of vortex torus knots (16, 1998; 19, 1998; 20, 1999)

proof of winding number influence on the energy and helicity of magnetic braids (64, 2017; 67, 2018)
computation of the groundstate energy spectra of magnetic knots and links (42, 2009; 57, 2014; 69, 2018)
crossing number lower bounds to magnetic energy (26, 2002; 35, 2008; 51, 2013)
application of a twist and fold model to kinematic dynamo (33, 2007)
inflexional instability of magnetic knots with applications to solar flares (15, 1997; 30, 2005)
Hammock configuration of magnetic flux tube from writhe helicity (9, 1994; 22, 2000)
topological decomposition of magnetic helicity in terms of writhe and twist (6, 1992)

comprehensive study of geometric properties of torus knots and unknots (63, 2016)
parametric equations for the twist and fold kinematics of space curves (31, 2006; 32, 2006)
application of twist and fold kinematics to filament spooling and packing (34, 2007; 36, 2008)
new measures of structural complexity for dynamical systems (29, 2005; 38, 2009; 39, 2009; 47, 2011; 48, 2012; 55, 2014)
progress in knot theory (28, 2005)
energy spectrum of elastic string under twist energy relaxation (10, 1995; 14, 1996)

quantum knot dynamics driven by Seifert surfaces of minimal area (78, 2022)
topological change of a soap film surface by a twisted fold singularity (43, 2010)

extension of Gross-Pitaevskii equation and hydrodynamic description in general Riemannian manifolds (77, 2021)
derivation of intrinsic kinematic equations of a vortex defect in general odd-dimensional Riemannian manifolds (1, 1991)

derivation of the linking number formula from Gauss’s own work and proof of the independent derivation by Maxwell (45, 2011; 46, 2011)
progress in topological fluid mechanics (13, 1996; 17, 1998; 18, 1998; 24, 2001; 40, 2009; 74, 2020)
re-discovery of Levi-Civita’s work on vortex dynamics and asymptotic potential theory of thin tubes (12, 1996)
re-discovery of Călugăreanu’s work on the self-linking number formula (6, 1992)
re-discovery of Da Rios’ work on the intrinsic kinematics of space curves and vortex filament motion (2, 1991)