
Renzo L. RiccaUniversità degli Studi di Milano-Bicocca | UNIMIB · Department of Mathematics and its Applications
Renzo L. Ricca
PhD, University of Cambridge
Professor of Mathematical Physics
About
96
Publications
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Introduction
My work is in applied mathematics and mathematical physics of
complex, fluid systems. My current research is focused on applications
of geometric and topological methods to study morphological aspects
of vortex flows, magnetic fields, filament tangles, to determine fundamental relationships between these aspects and dynamical and energetic properties
of such systems.
More detailed information can be found at the URL website: http://www.matapp.unimib.it/~ricca/
Additional affiliations
Education
August 1989 - August 1992
October 1979 - November 1988
Publications
Publications (96)
Here we review some recent developments in the topological study of vortex knots and links. We start from fundamental properties of kinetic helicity, a conserved quantity of ideal fluid mechanics, focusing on the topological interpretation in terms of linking numbers. Then we proceed to consider the derivation from helicity of the Jones and HOMFLYP...
In these lecture notes we review the original work of Riemann on multi-valued functions, Kelvin’s application to Green’s potential theory, and the result of Gauss on the interpretation of the magnetic potential in terms of solid angle to show the relevance of these earlier results in modern topological field theory. This is done by considering some...
The last years have witnessed remarkable advances in our understanding of the emergence and consequences of topological constraints in biological and soft matter. Examples are abundant in relation to (bio)polymeric systems and range from the characterization of knots in single polymers and proteins to that of whole chromosomes and polymer melts. At...
Here we illustrate how Jones’ polynomials are derived from the kinetic helicity of vortical flows, and how they can be used to measure the topological complexity of fluid knots by numerical values. Relying on this new findings, we show how to use our adapted Jones polynomial in a new framework by introducing a knot polynomial space whose discrete p...
In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross–Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and q...
In this paper, we determine the instability effects of a phase twist superposed on a quantum vortex defect governed by the Gross–Pitaevskii equation. For this, we consider the modified form of the equation in two cases: when a uniform phase twist is present everywhere in the condensate, and when the defect is subject to a localized phase twist conf...
Using an improved numerical code we investigate the creation and evolution of quantum knots and links as defects of the Gross–Pitaevskii equation. The particular constraints put on quantum hydrodynamics make this an ideal context for application of geometric and topological methods to investigate dynamical properties. Evolutionary processes are cla...
Here we show that the Gross–Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs) admits hydrodynamic interpretation in a general Riemannian metric, and show that in this metric the momentum equation has a new term that is associated with local curvature and density distribution profile. In particular conditions of steady state a new Einst...
Line defects are one-dimensional phase singularities (forming knots and links) that arise in a variety of physical systems. In these systems, isophase surfaces (Seifert surfaces) have the phase defects as boundary, and these Seifert surfaces define a framing of the normal bundle of each link component. We define the individual helicity for each com...
Physical knots observed in various contexts – from DNA biology to vortex dynamics and condensed matter physics – are found to undergo topological simplification through iterated recombination of knot strands following a common, qualitative pattern that bears remarkable similarities across fields. Here, by interpreting evolutionary processes as geod...
In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to th...
In this paper we demonstrate that new phase defects of the Gross-Pitaevskii equation (GPE) can be produced as a Aharonov-Bohm effect resulting from pure phase twist injection on existing defects. This is a phenomenon that has physical justification in the hydrodynamic interpretation of GPE. Here we give an analytical proof of its effects by using F...
Here we show how to apply a recently introduced method based on the geometric interpretation of linear momentum of vortex lines to determine dynamical properties of a network of knots and links. To show how the method works and to prove its feasibility, we consider the evolution of quantum vortices governed by the Gross-Pitaevskii equation. Accurat...
In this paper we derive and compare numerical sequences obtained by adapted polynomials such as HOMFLYPT, Jones and Alexander-Conway for the topological cascade of vortex torus knots and links that progressively untie by a single reconnection event at a time. Two cases are considered: the alternate sequence of knots and co-oriented links (with posi...
Geometric and topological aspects associated with induction effects of field lines in the shape of torus knots/unknots are examined and discussed in detail. Knots are assumed to lie on a mathematical torus of circular cross-section and are parametrized by standard equations. The induced field is computed by direct integration of the Biot-Savart law...
In this paper we show that twist, defined in terms of rotation of the phase associated with quantum vortices and other physical defects effectively deprived of internal structure, is a property that has observable effects in terms of induced axial flow. For this we consider quantum vortices governed by the Gross-Pitaevskii equation (GPE) and perfor...
By considering steady magnetic fields in the shape of torus knots and unknots in ideal magnetohydrodynamics, we compute some fundamental geometric and physical properties to provide estimates for magnetic energy and helicity. By making use of an appropriate parametrization, we show that knots with dominant toroidal coils that are a good model for s...
By numerically solving the three-dimensional Gross-Pitaevskii equation we analyze the cascade process associated with the evolution and decay of a pair of linked vortex rings. The system decays through a series of reconnections to produce finally three unlinked, unfolded, almost planar vortex loops. Total helicity, initially zero, remains unchanged...
The process of vortex cascade through continuous reduction of topological complexity by stepwise unlinking, that has been observed experimentally in the production of vortex knots (Kleckner & Irvine, 2013), is shown to be reproduced in the branching of the skein relations of knot polynomials (Liu & Ricca, 2015) used to identify topological complexi...
Due to reconnection or recombination of neighboring strands superfluid vortex knots and DNA plasmid torus knots and links are found to undergo an almost identical cascade process, that tend to reduce topological complexity by stepwise unlinking. Here, by using the HOMFLYPT polynomial recently introduced for fluid knots, we prove that under the assu...
A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global...
Here we show that under quantum reconnection, simulated by using the three-dimensional Gross-Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that the total length of the vortex system reaches a maximum at the reconnection time, whil...
Here we show that under quantum reconnection, simulated by using the three-dimensional Gross- Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that total length of the vortex system reaches a maximum at the reconnection time, while b...
By using and extending earlier results (Liu & Ricca,
J. Phys.
A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polyno...
A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vo...
Reconnection is a fundamental event in many areas of science, from the
interaction of vortices in classical and quantum fluids, and magnetic flux
tubes in magnetohydrodynamics and plasma physics, to the recombination in
polymer physics and DNA biology. By using fundamental results in topological
fluid mechanics, the helicity of a flux tube can be c...
In this paper I present and discuss with examples new techniques based on the use of geometric and topological information to quantify dynamical information and determine new relationships between structural complexity and dynamical properties of vortex flows. New means to determine linear and angular momenta from standard diagram analysis of vorte...
By using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links, and by using ropelength data from the RIDGERUNNER tightening algorithm (Ashton et al 2011 Exp. Math. 20 57–90) we obtain the groundstate energy spectra of the first 2...
In this paper we prove that under ideal conditions the helicity of fluid knots, such as vortex filaments or magnetic flux tubes, provides a fundamentally new topological means by which we may associate a topological invariant, the Jones polynomial, that is much stronger than prior interpretations in terms of Gauss linking numbers. Our proof is base...
In this paper I extend the area interpretation of linear and angular momenta of ideal vortex filaments to complex tangles of filaments in space. A method based on the extraction of area information from diagram projections is presented to evaluate the impulse of vortex knots and links. The method relies on the estimate of the signed areas of sub-re...
By making simple, heuristic assumptions, a new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial ex...
In this paper we present an application of a new technique, based on
recent work done by Liu & Ricca (2012), to quantify structural
complexity by means of topological methods. These rely on the derivation
of the Jones polynomial from the helicity of ideal fluid flows. The
techniques discussed here can be extended and applied to real fluid
flows sub...
In this paper we examine certain geometric and topological aspects of the dynamics and energetics of vortex torus knots and un- knots. The knots are given by small-amplitude torus knot solutions in the local induction approximation (LIA). Vortex evolution is then studied in the context of the Euler equations by direct numerical integration of the B...
In this article we present a review of some of the author's most recent results in topological magnetohydrodynamics (MHD), with an eye to possible applications to astrophysical flows and solar coronal structures. First, we briefly review basic work on magnetic helicity and linking numbers, and fundamental relations with magnetic energy and average...
Structural complexity emerges from all systems that display morpholog-ical organization. Structural complexity for an intricate tangle of fila-ments is measured by the average crossing number of the tangle. Direct relationships between total length, energy, and structural complexity of a tangle are established. These results are based on elementary...
In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and inter-section number. The reconstruction presented here is entirely based on an accurate study of Gaus...
In this paper we shall review some of the most recent developments and results on work on energy‐complexity relations and, if time will allow it, we shall provide an analytical proof of eq. (3) below, a fundamental relation between energy and complexity established by numerical experiments.
In these lectures we present for the first time a mathematical reconstruction of what might have been Gauss’ own derivation of the linking number of 1833, and we provide also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction offered here is entirely based on...
It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of
the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained
unknown. Experimental studies presented here show that this process consists of a collapse of the fi...
In this paper we determine the velocity, the energy, and estimate writhe and twist helicity contributions of vortex filaments in the shape of torus knots and unknots (as toroidal and poloidal coils) in a perfect fluid. Calculations are performed by numerical integration of the Biot-Savart law. Vortex complexity is parametrized by the winding number...
From visual inspection of complex phenomena to modern visiometrics, the quest for relating aspects of structural and morphological
complexity to hidden physical and biological laws has accompanied progress in science ever since its origin. By using concepts
and methods borrowed from differential and integral geometry, geometric and algebraic topolo...
New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the...
New results on the kinetic energy of ideal vortex filaments in the shape of
torus knots and unknots are presented. These knots are given by small-amplitude
torus knot solutions (Ricca, 1993) to the Localized Induction Approximation
(LIA) law. The kinetic energy of different knot and unknot types is calculated
and presented for comparison. These res...
With this paper we want to pay tribute to 150 years of work on topological fluid mechanics. For this, we review Helmholtz's (1858) origi- nal contribution on topological issues related to vortex motion. Some recent results on aspects of structural complexity analysis of fluid flows are pre- sented and discussed, as well as new results on topologica...
In this paper we present; and discuss ideas and new results in three different research areas of topological fluid mechanics. First, we propose a conjectured experiment to produce and observe, for the first time, vortex knotting in real fluids. Next we provide a new ropelength bound for tight, magnetic knots in ideal magnetohydrodynamics. Finally,...
In this paper we introduce and discuss new concepts useful to analyse and characterize patterns of vortex lines in fluid flows.
We define measures of tropicity to identify ‘tubeness’, ‘sheetness’ and ‘bulkiness’ of vortex lines and to measure the spreading
of field lines about preferred directions. Algebraic, geometric and topological measures base...
A geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in Euler flows. For thin filaments under the so-called localized induction approximation the components of linear momentum admit interpretation in terms of projected area. By computing the signed ar...
In this paper we introduce and analyze a set of equations to study geometric and energetic aspects associated with the kinematics of multiple folding and coiling of closed filaments for DNA modeling. By these equations we demonstrate that a high degree of coiling may be achieved at relatively low energy costs through appropriate writhe and twist di...
New kinematic equations are used to model DNA supercoiling. These equations govern the simultaneous production of multiple folding and coiling of a filament in space by diffeomorphism of a reference curve. Here we show how to use these equations to model regions of highly localized filament coiling, where mechanisms of proteic coding or viral spool...
Preliminary results on a new Stretch-Twist-Fold (STF) kinematic model for fast dynamo are presented. The evolution is prescribed by equations that govern the simultaneous stretching, writhing and coiling of a magnetic flux-tube by diffeomorphism of the initial circular configuration. Simple estimates based on minimized magnetic energy show that exp...
In this paper, we determine two quantities, of geometric and topological character, that were left undetermined in two previou results obtained by Arnold (Arnold 1974 In Proc. Summer School in Diff. Eqs. at Dilizhan, pp. 229–256.) and Moffatt (Moffatt 1990 Nature 347, 367–369) on lower bounds for the magnetic energy of knots and links in ideal flui...
Geometric and topological aspects associated with integrability of vortex filament motion in the Localized Induction Approximation
(LIA) context (which includes a family of local dynamical laws) are discussed. We show how to interpret integrability in relation
to the Biot-Savart law and how soliton invariants can be interpreted in terms of global g...
The kinematics of writhing and coiling of circular filaments is here analysed by new equations that govern the evolution of curves generated by epicycloids and hypocycloids. We show how efficiency of coil formation and compaction depend on writhing rates, relative bending, torsion and mean twist energy. We demonstrate that for coiling formation hyp...
In this paper, we prove that magnetic flux-tubes in inflexional configuration are in disequilibrium and evolve to an inflexion-free state. The magnetic field is defined in a tube of circular cross-section and is chosen so as to have toroidal and poloidal components contributing to the internal twist of the flux-tube. By using orthogonal curvilinear...
In this paper, we prove that magnetic flux-tubes in inflexional configuration are in disequilibrium and evolve to an inflexion-free state. The magnetic field is defined in a tube of circular cross-section and is chosen so as to have toroidal and poloidal components contributing to the internal twist of the flux-tube. By using orthogonal curvilinear...
Algebraic and topological measures based on crossing number relations provide bounds on energy and helicity of ideal fluid
flows and can be used to quantify morphological complexity of tangles of magnetic and vortex tubes. In the case of volume-preserving
flows we discuss new results useful to determine lower bounds on magnetic energy in terms of t...
We introduce and test measures of geometric and topological complexity to quantify morphological aspects of a tangle of vortex
filaments. The tangle is produced by standard numerical simulation of superfluid turbulence in Helium II. Complexity measures
such as linking number, writhing number, average crossing number and helicity are computed, and t...
In this paper we review some results on geometric and topological vortex dynamics. After some background on flow maps, topological
equivalence of frozen fields and conservation laws, we discuss geometric aspects of vortex filament motion (intrinsic equations,
connections with integrable dynamics and extension to higher dimensional manifolds) and th...
New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments. The tangle is produced by standard numerical simulation of superfluid helium turbulence, which we use as a benchmark for numerical investigation of complex systems. We find that the measures u...
New measures of algebraic, geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments. The tangle is produced by standard numerical simulation of superfluid helium turbulence, which we use as a benchmark for numerical investigation of complex systems. We find that the measures u...
Leading experts present a unique, invaluable introduction to the study of the geometry and typology of fluid flows. From basic motions on curves and surfaces to the recent developments in knots and links, the reader is gradually led to explore the fascinating world of geometric and topological fluid mechanics.
Geodesics and chaotic orbits, magnetic...
In this paper we address the problem of measuring structural complexity of generic tangles of vortex lines in a fluid domain, by using a combination of geometric and topological techniques. To this end new concepts based on the idea of structural 'tropicity' are introduced to determine 'tubeness', "sheetness" and 'bulkiness' of a vortex tangle and...
In this paper we show how new techniques of topological fluid mechanics and physical knot theory can be applied to estimate magnetic energy levels in solar physics. In particular, we show that magnetic energy stored in complex configurations of plasma loops present on the Sun can be quantified by geometric and topological information. These studies...
For the first time since Lord Kelvin's original conjectures of 1875 we address and study the time evolution of vortex knots in the context of the Euler equations. The vortex knot is given by a thin vortex filament in the shape of a torus knot T(p,q) (p > 1, q > 1; p, q co-prime integers). The time evolution is studied numerically by using the Biot-...
A full investigation of the energy spectrum of a twisted flexible string under elastic relaxation is presented and discussed in detail for the first time. New polynomial expressions for critical energy states are derived and the whole spectrum of critical states (minima, maxima and inflexion points) of the elastic energy is found and discussed in r...
In this paper we review classical and new results in topological fluid mechanics based on applications of first principles of ideal fluid mechanics and knot theory to vortex and magnetic knots. After some brief historical remarks on the first original contributions to topological fluid mechanics, we review basic concepts of topological fluid mechan...
We have investigated numerically the motion and the stability of quantized vortex knots.
We have investigated numerically the motion and the stability of quantized vortex knots.
In this paper we present new results concerning the evolution and stability of vortex knots in the context of the Euler equations. For the first time, since Lord Kelvin’s original conjecture of 1875, we have direct numerical evidence of stability of vortex filaments in the shape of torus knots. The results are based on the analytical solutions of R...
This paper presents new results concerning evolution and inflexional instability of twisted magnetic flux tubes in the solar corona. Inflexional configurations, attained when the curvature of the tube axis vanishes, are generally present in coronal magnetic structures and are invariably associated with the early stages of kink formation. New equati...
The use of topological ideas in physics and fluid mechanics dates back to the very origin of topology as an independent science. In a brief note in 1833 Karl Gauss, while lamenting the lack of progress in the “geometry of position” (or Geometria Situs, as topology was then known I, gives a remarkable example of the relationship between topology and...
In this paper we present for the first time a detailed account of the work of L.S. Da Rios and T. Levi-Civita on what is believed to be one of the first major contributions to three-dimensional vortex filament dynamics. Their work spanned a period of almost 30 years, from 1906 to 1933, and despite many publications remained almost unnoticed through...
Following our recent investigation [1] on the energy spectrum of closed, twisted, flexible strings under elastic relaxation, we compare and analyse further the minimum energy states attained by relaxed strings in helical and supercoiled configuartion. We show that for very high values of specific linking difference (high superhelicity) the writhing...
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In this paper we analyse in detail, and for the first time, the rôle of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement...
In this paper is shown how to interpret the nonlinear dynamics of a class of one-dimensional physical systems exhibiting soliton behavior in terms of Killing fields for the associated dynamical laws acting as generators of torus knots. Soliton equations are related to dynamical laws associated with the intrinsic kinematics of space curves and torus...
The helicity of a localized solenoidal vector field (i.e. the integrated
scalar product of the field and its vector potential) is known to be a
conserved quantity under `frozen field' distortion of the ambient
medium. In this paper we present a number of results concerning the
helicity of linked and knotted flux tubes, particularly as regards the
t...
In the context of the localized induction approximation (LIA) for the motion of a thin vortex filament in a perfect fluid, the present work deals with certain conserved quantities that emerge from the Betchov–Da Rios equations. Here, by showing that these invariants belong to a countable family of polynomial invariants for the related nonlinear Sch...
The helicity H associated with a knotted vortex filament is considered. The filament is first constructed starting from a circular tube, in three stages involving injection of (integer) twist, deformation and switching of crossings. This produces a vortex tube in the form of an arbitrary knot K; each vortex line in the tube is a (trivial) satellite...
The remarkable story of the discovery of a set of equations at least
three times this century shows once again that independent discoveries
can occur and exist for some time.
This paper presents the natural algebraic-geometrical generalization of the Da Rios-Betchov intrinsic equations governing curvature and torsion of an isolated vortex string moving in an unbounded, perfect fluid flow. The filament is embedded in a manifold that can be assumed to be homeomorphic to an odd-dimensional Euclidean space, and whose connec...
Interest in the study of invariant quantities is generally motivated by the need to interpret and to understand their meaning and their fundamental role in the theory. The invariants we shall consider in this paper emerge in two contexts. In the context of the localized induction approximation (LIA) for the motion of an inextensible vortex filament...
The largest chromosome in the human genome contains about 18 mm of DNA that is believed to exist as one giant molecule, that gets packed into a structure by a factor of 10.000. In order to achieve such high level of condensation the DNA filamentary structure folds and supercoils its double-stranded genomes to near-crystalline density. In recent yea...
In this paper we present and discuss a synthetic method to evaluate linear and angular momentum of a tangle of vortex filaments in ideal fluid. Vortex filaments are assumed to be thin and move according to the Localized Induction Approximation (LIA) law. In the LIA context the components of linear momentum of a single vortex filament admit interpre...