## Video Recordings and Invited Lectures

### Knots and Applications 3: An Elementary Introduction (in Italian)

February 17, 2017.

Third lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 1:52:37.

## Knots and Applications 2: An Elementary Introduction (in Italian)

February 17, 2017.

Third lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 1:52:37.

Summary

Lecture 3 (of 3): (i) Laboratory From Gauss coding to modern tabulation: Alexander-Briggs notation; braid words, Gauss and Dowker-Thistlethwaite code; Jones polynomial; ropelength, tight knots and ideal shapes. (ii) KnotAtlas (Bar Natan, 2000, 2004): online database of knots and invariants. (iii) LinKnot (Jablan-Razdanovic, 2006): online knot theory software. (iv) KnotPlot (Scharein, 2011): visualization and mathematical exploration software; tangle calculator; dynamical systems interface; mathematical experimentation.

### Knots and Applications 2: An Elementary Introduction (in Italian)

February 10, 2017.

Second lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 2:17:25.

## Knots and Applications 2: An Elementary Introduction (in Italian)

February 10, 2017.

Second lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 2:17:25.

Summary

Lecture 2 (of 3): (i) When topology meets geometry: self-linking of a ribbon = writhe + twist; compactification, relaxation and DNA topology. (ii) From knots to braid presentations: bridge index and closed braids; braid index and Seifert surfaces; plasma loops on the Sun (iii) The polynomial era: from Alexander to Jones and modern times; Jones’ skein relations for knot polynomials; examples of computation; (iv) Topics of current research: vortex dynamics: HOMFLYPT best quantifier of topological complexity.

- Kauffman, L.H. (2001)
*Knots and Physics.*World Scientific, Singapore. - Sossinsky, A. (2002)
*Knots - Mathematics with a Twist.*Harvard U. Press, Cambridge USA.

### Knots and Applications 1: An Elementary Introduction (in Italian)

February 9, 2017.

First lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 2:00:32.

## Knots and Applications 1: An Elementary Introduction (in Italian)

February 9, 2017.

First lecture of a mini-course (in Italian) organized by the Accademia dei Lincei and Scuola Normale Superiore
and held at the Scuola Normale Superiore in Pisa.

Video on YouTube and on
SNS video library.

Information on SNS portal.

Duration: 2:00:32.

Summary

Knots and links have accompanied human life since its very origin, from rudimentary manufactures till the most elaborate ornaments, with merely practical, aesthetic or symbolic functions. The recent discoveries of knotted DNAs and highly entangled polymer structures in chemical-physics demonstrate that knots and links form at all scales, with a purpose or function yet to decipher. The mathematics of knots, born with Tait to complement the vortex atom theory of Lord Kelvin, is now undergoing an extraordinary development due to the continuous and spontaneous ramifications in various sub-fields of pure mathematics, from topology to algebra, but also due to the unexpected, innumerable applications is having in many disparate fields of science and technology, from the study of complex systems to molecular genomics, from cryptography to neural networks and big data analysis. Lecture 1 (of 3). (i) An intuitive approach: knots in art and science: from arts and crafts to DNA biology; knots in mathematics: theorems, numerics and platonic figures. (ii) A little bit of history: origin of knot theory: Kelvin’s string theory and Tait’s tabulation. (iii) The key concept: topological equivalence and invariants: playing with continuity. (iv) Let’s get serious: elements of knot theory: classification issues and basic definitions; Gauss linking number: from definition to application; minimal crossing number and minimal diagram; Tait’s tabulation re-visited: a source of inspiration; Reidemeister’s moves to encode topological equivalence.

- Adams, C.C. (1994)
*The Knot Book.*W.H. Freeman & Co., New York. - Ashley, C. (1944)
*The Ashley Book of Knots.*Doubleday, New York.

### Vortex knots cascade by HOMFLYPT polynomial

September 28, 2016.

Lecture given at the International Centre for Theoretical Physics in Trieste.

Video on YouTube and at
ICTP Events.

Information at ICTP Workshops.

Duration: 0:41:02.

## Vortex knots cascade by HOMFLYPT polynomial

September 28, 2016.

Lecture given at the International Centre for Theoretical Physics in Trieste.

Video on YouTube and at
ICTP Event.

Information at ICTP Event.

Duration: 0:41:02.

Summary

Since Moffatt’s original work of 1969, it is well-known that the kinetic helicity admits topological interpretation in terms of linking numbers. For a single vortex filament helicity can be expressed in terms of the Călugăreanu-White self linking number, and geometric decomposition in terms of writhe and twist. By applying knot theoretical techniques Liu and Ricca derived well-known knot polynomials – most notably the HOMFLYPT polynomial– from the helicity of fluid systems, hence showing that these are new invariants of ideal fluid mechanics. In the case of HOMFLYPT the two polynomial variables are shown to be related to the writhe and twist of the vortex knot. Due to reconnection or recombination of neighboring strands vortex knots are found to undergo an almost generic cascade process, that tend to reduce topological complexity by stepwise unlinking. Here, by using the adapted HOMFLYPT polynomial for fluid knots, we prove that under the assumption that topological complexity decreases by stepwise unlinking by anti-parallel reconnection this cascade process follows a path detected by a unique, monotonically decreasing sequence of numerical values. This result holds true for any sequence of standardly embedded torus knots T(2, 2n + 1) and torus links T(2, 2n). By this result we demonstrate that the computation of this adapted HOMFLYPT polynomial provides a powerful tool to quantify topological complexity of various physical systems.

- Liu, X. & Ricca, R.L. (2012)
The Jones’ polynomial for fluid knots from helicity.
*J. Phys. A: Math. and Theor.***45**, 205501. - Liu, X. & Ricca, R.L. (2015)
On the derivation of HOMFLYPT polynomial invariant for fluid knots.
*J. Fluid Mech.***773**, 34-48. - Liu, X. & Ricca, R.L. (2016)
Knots cascade detected by a monotonically decreasing sequence of values.
*Sci. Rep.***6**, 24118.

### Knot Polynomials as a New Tool in Turbulence Research

November 27, 2013.

Lecture given at the Lomonosov State University in Moscow.

Video on YouTube and at
Math-Net.ru.

Information at Math-Net.ru.

Duration: 0:37:18.

## Knot Polynomials as a New Tool in Turbulence Research

November 27, 2013.

Lecture given at the Lomonosov State University in Moscow.

Video on YouTube and at
Math-Net.ru.

Information at Math-Net.ru.

Duration: 0:37:18.

Summary

In recent decades there has been overwhelming evidence that vorticity tends to get concentrated to form coherent structures, such as vortex filaments and tubes (the "sinews of turbulence"), in both classical and quantum fluids. In the case of vortex tangles, structural complexity methods have proven to be useful to investigate and establish new relations between energy, helicity and complexity. Indeed, this approach can be pursued further by introducing knot polynomials. By using a suitable transformation in terms of helicity, it has been recently shown that the standard Jones polynomial of knot theory can be interpreted as a new invariant of topological fluid mechanics. By briefly reviewing this work, we show how to compute this invariant for some simple, but non-trivial topologies. Then, by considering the standard decomposition of the helicity of a vortex filament in terms of writhe and twist, we focus on geometric aspects of vortex reconnection, a key feature in real vortex dynamics, and present a recent result on the conservation of writhe under reconnection. For dissipative systems, this means that any deviation from helicity conservation is entirely due to twist, inserted or deleted locally at the reconnection site. This result has important implications for helicity and energy considerations. We conclude the talk with some speculations on future work, by showing how a combination of these results may lead to the development of a novel approach to investigate fundamental aspects of turbulence research.

- Liu, X. & Ricca, R.L. (2012)
The Jones’ polynomial for fluid knots from helicity.
*J. Phys. A: Math. and Theor.***45**, 205501. - Laing, C.E., Ricca, R.L. & Sumners, De W. L. (2014)
Conservation of writhe helicity under anti-parallel reconnection.
*Sci. Rep.***5**, 9224. - Ricca, R.L. (2014)
Structural complexity of vortex flows by diagram analysis and knot polynomials.
In
*How Nature Works*(ed. I. Zelinka et al.), pp. 81-100. Emergence, Complexity and Computation**5**. Springer-Verlag.

### Topological Fluid Mechanics – From Gauss’ linking Number to Modern Topological Dynamics

March 28, 2013.

Lecture given at the École Normale Supérieure in Paris.

Video on YouTube and at ENS Savoirs.

Information at ENS Savoirs.

Duration: 1:01:24.

## Topological Fluid Mechanics – From Gauss’ linking Number to Modern Topological Dynamics

March 28, 2013.

Lecture given at the École Normale Supérieure in Paris.

Video on YouTube and at ENS Savoirs.

Information at ENS Savoirs.

Duration: 1:01:24.

Summary

Contrary to common perception topological fluid mechanics is an old subject, rooted in Helmholtz’s (1858) conservation laws and Kelvin’s (1870) vortex atom theory. In this talk we would like to address some of the most recent developments in the field, in the light of the original ideas rooted in the works of Kelvin and Tait. We review some recent results on vortex knots and on the energy groundstate of magnetic knots and links based on the author’s own research. These results may find useful applications in various physical contexts, providing further grounds to establish a mathematical foundation of topological field theory based on a one-to-one correspondence between energy and topology.

- Ricca, R.L. (1998) Applications
of knot theory in fluid mechanics.
In
*Knot Theory*(ed. V.F.R. Jones et al.), pp. 321-346. Banach Center Publs.**42**, Polish Academy of Sciences, Warsaw. - Ricca, R.L. Ed. (2001)
*An Introduction to the Geometry and Topology of Fluid Flows*. NATO ASI Series II,**47**. Kluwer. Dordrecht, The Netherlands. - Ricca, R.L. Ed. (2009)
*Lectures on Topological Fluid Mechanics*. Springer-CIME Lecture Notes in Mathematics**1973**. Springer-Verlag. Heidelberg, Germany. - Ricca, R.L. (2013)
New energy lower bounds for knotted and braided magnetic fields.
*Geophys. Astrophys. Fluid Dyn.*,**107**, 385-402.

### On the Energy Spectrum of Magnetic Knots and Links

December 6, 2012.

Lecture given at the Isaac Newton Institute for Mathematical Sciences in Cambridge.

Video on YouTube and at INIMS.

Information at INIMS.

Duration: 0:17:37.

## On the Energy Spectrum of Magnetic Knots and Links

December 6, 2012.

Lecture given at the Isaac Newton Institute for Mathematical Sciences in Cambridge.

Video on YouTube and at INIMS.

Information at INIMS.

Duration: 0:17:37.

Summary

The groundstate energy spectrum of the first 250 zero-framed prime knots and links is studied by using an exact analytical expression derived by the constrained relaxation of standard magnetic flux tubes in ideal magneto-hydrodynamics and data obtained by the RIDGERUNNER tightening algorithm. The magnetic energy is normalized with respect to the reference energy of the tight torus and is plotted against increasing values of ropelength. A remarkable generic behavior characterizes the spectrum of both knots and links. A comparative study of the bending energy reveals that curvature information provides a rather good indicator of magnetic energy levels.

- Ricca, R.L. (2008) Topology
bounds energy of knots and links.
*Proc. R. Soc. A***464**, 293-300. - Maggioni, F. & Ricca, R.L. (2009)
On the groundstate energy of tight knots.
*Proc. R. Soc. A***465**, 2761–2783. - Ricca, R.L. (2009) New
developments in topological fluid mechanics.
*Nuovo Cimento C***32**, 185-192

### Impulse of Vortex Knots from Diagram Projections

July 24, 2012.

Lecture given at the Isaac Newton Institute for Mathematical Sciences in Cambridge.

Video on YouTube
and at INIMS.

Information at INIMS.

Duration: 0:22:05.

## Impulse of Vortex Knots from Diagram Projections

July 24, 2012.

Lecture given at the Isaac Newton Institute for Mathematical Sciences in Cambridge.

Video on YouTube
and at INIMS.

Information at INIMS.

Duration: 0:22:05.

Summary

By using methods based on the analysis of standard plane projections of complex tangles, we can extract geometric information and use it to determine the impulsive forces associated with vortex knots and links. This method relies on the interpretation of linear and angular momentum of ideal vortex filaments in terms of projected areas. An immediate application of this method allows one to make predictive estimates of the evolution and dynamical features of vortex knots and links. This will be illustrated by a number of examples, some related to well-known results from laboratory and numerical experiments, where vortex ring collisions and vortex linking have been studied, and some others, such as the production of a trefoil vortex knot, proposed as "thought" experiments.

- Ricca, R.L. (2008) Momenta
of a vortex tangle by structural complexity analysis.
*Physica D***237**, 2223-2227. - Ricca, R.L. (2009) Structural
complexity and dynamical systems.
In
*Lectures on Topological Fluid Mechanics*(ed. R.L. Ricca), pp. 169-188. Springer-CIME Lecture Notes in Mathematics**1973**. Springer-Verlag. - Ricca, R.L. (2009) New
developments in topological fluid mechanics.
*Nuovo Cimento C***32**, 185-192.

### Recent Progress in Topological Fluid Dynamics: from Helicity to Jones Polynomials

June 27, 2012.

Lecture given at the Kavli Institute for Theoretical Physics in Santa Barbara (UCSB).

Video on YouTube and at KITP.

Information at KITP.

Duration: 1:00:47.

## Recent Progress in Topological Fluid Dynamics: from Helicity to Jones Polynomials

June 27, 2012.

Lecture given at the Kavli Institute for Theoretical Physics in Santa Barbara, University of California.

Video on YouTube and at KITP.

Information at KITP.

Duration: 1:00:47.

Summary

In this talk I review some of the work that I have done on aspects of topological fluid mechanics. I start with a historical review of the key contributions by Gauss, Lord Kelvin, Tait and Maxwell to end up with a broad view of the current state of the art in the subject. I then consider magnetic knots and links in ideal magnetohydrodynamics, magnetic energy and helicity, and the energy relaxation problem. I show how inflectional magnetic knots relax to form minimal braids and how these eventually relax further to tight knots and links. The groundstate energy spectrum of the first prime knots and links is presented and discussed in relation to ropelength data. Some simple results on vortex torus knots are presented for LIA as well as Biot-Savart evolution, and an energy-complexity relation is discussed for superfluid vortex tangles. I finally present some work on the derivation of the Jones polynomial for vortex knots in Euler equations, by showing some explicit calculations of this invariant for simple standard knot and link types. An outline of future work concludes the talk.

- Ricca, R.L. (1998) Applications
of knot theory in fluid mechanics.
In
*Knot Theory*(ed. V.F.R. Jones et al.), pp. 321-346. Banach Center Publs.**42**, Polish Academy of Sciences, Warsaw. - Ricca, R.L. (2013)
New energy lower bounds for knotted and braided magnetic fields.
*Geophys. Astrophys. Fluid Dyn.***107**, 385-402. - Liu, X. & Ricca, R.L. (2012)
The Jones polynomial as a new invariant of topological fluid dynamics.
*Phys. A: Math. & Theor.*,**45**, 205501.

### On Călugăreanu's Theorem

June 27, 2012.

Lecture given at the Kavli Institute for Theoretical Physics in Santa Barbara (UCSB).

Video on YouTube and at KITP.

Information on the KITP.

Duration: 0:09:16.

## On Călugăreanu's Theorem

June 27, 2012.

Lecture given at the Kavli Institute for Theoretical Physics in Santa Barbara, University of California.

Video on YouTube and at KITP.

Information on the KITP.

Duration: 0:09:16.

Summary

The Călugăreanu invariant (after the Romanian mathematician G. Călugăreanu) is the limiting form of the Gauss linking number and measures the self-linkage of a knot. It is one of the most fundamental invariants in the topology of knots and links and has received important applications in several contexts of contemporary science. The recent rigourous derivation of the Călugăreanu invariant from first principles of fluid mechanics and the discovery of its intimate relationship with the helicity of knotted vortex filaments and magnetic flux tubes, has led to a deeper understanding of its role in many aspects of topological fluid mechanics. In this open discussion I examine how the mathematical study of Călugăreanu (1961) on the continuous deformation of a knot may entail the study of inflectional configurations and how this is associated with changes in the bending and torsional energy, when kinks and twists appear in elastic strings or magnetic flux tubes. The Călugăreanu invariant plays a central part in this approach, which is crucial for our progress in understanding mechanisms of energy re-structuring and release.

- Ricca, R.L. & Moffatt, H.K. (1992) The helicity of a knotted vortex filament.
In
*Topological Aspects of the Dynamics of Fluids and Plasmas*(ed. H.K. Moffatt et al.), pp. 225-236. Kluwer. - Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Calugareanu invariant.
*Proc. R. Soc. A***439**, 411-429. - Ricca, R.L. (1995) The energy spectrum of a twisted flexible string under elastic relaxation.
*J. Phys. A: Math. & Gen.***28**, 2335-2352. - Maggioni, F. & Ricca, R.L. (2006) Writhing
and coiling of closed filaments.
*Proc. R. Soc. A***462**, 3151-3166.

### Open Problems in Knot Theory and Applications – Forward Look and Plenary Discussion

July 7, 2011.

Open discussion on future challenges in knot theory and applications held at the Scuola Normale Superiore
in Pisa.

Video on YouTube.

Information at De Giorgi Mathematical Research Center.

Duration: 0:23:44.

## Open Problems in Knot Theory and Applications – Forward Look and Plenary Discussion

July 7, 2011.

Open discussion on future challenges in knot theory and applications held at the Scuola Normale Superiore
in Pisa.

Video on YouTube.

Information at De Giorgi Mathematical Research Center.

Duration: 0:23:44.

Summary

Renzo Ricca with Ken Millett, Keith Moffatt, Carlo Petronio, De Witt Sumners and many others in an open discussion on future challenges in knot theory and applications, at a gathering held at the Scuola Normale Superiore in Pisa in 2011. At the end, Keith Moffatt reads a poem by James Clerk Maxwell on Peter Guthrie Tait.

- Bajer, K., Kimura, Y. & Moffatt, H.K., Eds. (2013)
*Topological Fluid Dynamics: Theory and Applications.*Procedia IUTAM**7**, Elsevier. - Kauffman, L.H., Lambropoulou, S., Jablan, S. & Przytycki, J.H., Eds. (2012)
*Introductory Lectures on Knot Theory.*Series on Knots and Everything**46**.World Scientific. - Millett, K.C. & Sumners, DeW. L., Eds. (1994)
*Random Knotting and Linking.*Series on Knots and Everything**7**. World Scientific. - Stasiak, A., Katritch, V. & Kauffman, L.H., Eds. (1998)
*Ideal Knots.*Series on Knots and Everything**19**. World Scientific.