Curvature based geometric functionals have received great attention during the past years. Many 'energy' functionals defined on curves and surfaces have been investigated that aim at relating analytical properties to the geometry and topology of the respective objects. So-called knot energies play a central rôle in geometric knot theory. There are several links to physical knot theory which comprises studying knots as physical objects, having a given length and thickness, and, more generally, any kind of knotted structure appearing in the sciences. For instance, self-avoidance which is the central concept for modeling the above-mentioned geometric functionals has an impact in this context. Self-repelling forces are observed in the behaviour of protein foldings and the motion of knotted DNA structures. Knot energies are also considered in topological fluid dynamics. The objective of this workshop is to present recent results from theory and applications and discuss future developments.

Acknowledgement: The workshop is supported by VARIOGEO (ERC Advanced Investigator Grant ERC-2010-AdG_20100224, Grant Agreement Number 267087).

Speakers

Dorothy Buck

Imperial College, United Kingdom

Elizabeth Denne

Washington & Lee University, USA

Thomas El Khatib

Technische Universität Berlin, Germany

Aya Ishizeki

Saitama University, Japan

Martin Meurer

RWTH Aachen, Germany

Takeyuki Nagasawa

Saitama University, Japan

Chiara Oberti

University of Milano-Bicocca, Italy

Sylwester Przybył

Poznań University of Technology, Poland

Renzo Ricca

Università di Milano, Italy

Sebastian Scholtes

RWTH Aachen, Germany

John Sullivan

TU Berlin, Germany

Marta Szumańska

University of Warsaw, Poland

Alexander Volkmann

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Germany

We'll discuss recent work on knotted and linked DNA molecules. Using several case studies as examples, we'll consider the topological techniques used to model the processes that knot and link DNA. We'll explore the biological ramifications of DNA knotting and linking, and how the results of these topological models can inform experimentalists.

In this talk we review some recent progress and present new results on the relaxation of magnetic knots to braids and to their groundstate energy minima. By referring to tubular knot embeddings in ideal magnetohydrodynamics, we illustrate how magnetic knots are generically driven by the Lorentz force to inflexion-free spiral knots and braids [1]. Then, by using classical results by Arnold, Moffatt and Freedman & He, we show that topological crossing number information provides a lower bound on the minimum magnetic energy of knots [2].
By combining recent analytical results on magnetic energy relaxation and data on numerical knot tightening, we establish new relationships between ropelength and groundstate energy minima of knots and links, showing a remarkable similarity between the two spectra. New relationships between ropelength and minimum crossing number are presented and compared with current results [3].
This work provides useful applications for the study of astrophysical flows, and helps to establish a mathematical foundation for the classification of physical knots and links based on a one-to-one correspondence between energy and topology.
[1] Ricca, R.L. (2013) New energy and helicity lower bounds for knotted and braided magnetic fields. Geophys. Astrophys. Fluid Dyn. 107, 385-402.[2] Ricca, R.L. (2008) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293-300.[3] Ricca, R.L. & Maggioni, F. (2014) On the groundstate energy spectrum of magnetic knots and links. J. Phys. A: Math. & Theor., submitted.

Magnetic fields confined into thin tubes in the shape of torus knots and unknots are considered and the relationships between their length, bending and torsional energy and their geometrical and topological complexity are analyzed. The winding number is taken as natural index of knot complexity. Numerical information on self-linking are compared with theoretical results by E. Fuller (2003) and the invariance of the self-linking number under dilatation is discussed.
This research is part of a PhD Project carried out under the guidance of R.L. Ricca.

Shapes of the simple and twisted clasps will be described. Presented analysis will be based on numerical simulations performed with the use of an appropriately modified Finite Element Method. The calculations have been made at a sufficiently large resolution where the length of a single segment of the simulated rope of unit radius is in the order of up to 10^(-4), thus the whole representation of the simulated clasp contains about 10^5 points. The final conformations of various clasps will be analyzed from the physical point of view.

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy~$E_{\mathrm{bend}}=\int\kappa^2$ and add a small multiple of ropelength~${\mathcal R}=\textnormal{length}/\textnormal{thickness}$ in order to penalize selfintersection. Our main objective is to characterize {\it elastic knots}, i.e., all limit configurations of energy minimizers of the total energy $E_{\vartheta}:=E_{\mathrm{bend}}+\vartheta{\mathcal R}$ as $\vartheta$ tends to zero. For every odd $b>1$ and the respective class of $(2,b)$-torus knots (containing the trefoil) we obtain a complete picture showing that the respective elastic $(2,b)$-torus knot is the twice covered circle.

We consider the Möbius energy defined for a closed curve in $ \mathbb{R}^n $: \[ \mathcal{M} ( \mbox{\boldmath $ f $} ) = \iint_{ ( \mathbb{R} / \mathcal{L} \mathbb{Z} )^2 } \left(\frac 1 { \| \mbox{\boldmath $ f $} ( s_1 ) - \mbox{\boldmath $ f$} ( s_2 ) \|_{\mathbb{R}^n }^2 } - \frac 1 { \mathscr{D} ( \mbox{\boldmath $ f $} ( s_1 ) , \mbox{\boldmath $f $} ( s_2 ) )^2 } \right) d s_1 d s_2 .\] Here $ \mathcal{L} $ is the length of closed curve, $ s_i $'s are arc-length parameters, and $ \mathscr{D} $ is the distance along the curve. In this talk we show that the energy can be decomposed into three parts: \[\mathcal{M} ( \mbox{\boldmath $ f $} ) = \mathcal{M}_1 ( \mbox{\boldmath $ f $} ) + \mathcal{M}_2 ( \mbox{\boldmath $ f $} ) + 4 .\] The first one is an analogue of Gagliardo semi-norm of $ \mbox{\boldmath $ f $}^\prime $ in the fractional Sobolev space $ H^{1/2 } $. This implies the natural domain of $ \mathcal{M} $ is $ H^{ 3/2 } \cap H^{1,\infty } $, which was shown by Blatt. The integrand of second one has the determinant structure, which shows a cancellation of integrand. The energy $ \mathcal{M} $ is invariant under the Möbius transformations. We discuss the Möbius invariance of each $ \mathcal{M}_i $'s.

The first and second variational formulas of the Möbius energy was calculated by several mathematicians. Direct calculation products a lot of terms which are not integrable even in the sense of Cauchy's principal value. By combining several terms appropriately, the integrability recovers, however, it is a quite hard job. Using the decomposition which was given in the previous talk, we can calculate the variational formulas relatively easily. One can find their explicit expressions, and can show the following estimates. Let $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ be integrands of the energy, the first variation, and the second variation of $ \mathcal{M}_i $. Assume $ \mathcal{M} (\mbox{\boldmath $ f $} ) < \infty $. (For $ \mathcal{M} $ and $ \mathcal{M}_i $, see the abstract of the previous talk.) If the curve and test functions are in $ H^{ 3/2 } \cap H^{1,\infty} $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ L^1 $. If the curve and test functions are in $ C^{1,1} $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ L^\infty $. If the curve and test functions are in $ C^2 $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ C^0 $.

In this talk we derive some sharp geometric inequalities for compact surfaces with boundaries. Making a new observation, we are able to link these inequalities with the L1-tangent-point energy of closed curves in euclidean space. This enables us to answer a question raised by Strzelecki, Szuma\'nska and von der Mosel. Afterwards, we will present an alternative proof due to Simon Blatt.

We investigate the relationship between a curvature energy $\mathcal{F}$, for example the M\"obius energy or thickness, and discrete versions $\mathcal{F}_{n}$ of this energy. These discrete energies $\mathcal{F}_{n}$ are defined on equilateral polygons with $n$ vertices, while the energy $\mathcal{F}$ is initially defined on all curves, but finite energy usually assures that the curve has some higher regularity, i.e. the arc length parametrisation typically belongs to some (fractional) Sobolev space. It will turn out that the energy $\mathcal{F}$ is the $\Gamma$-limit of the corresponding discrete energies $\mathcal{F}_{n}$ for $n\to\infty$. This directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to a minimizer of the smooth energy in the same knot class, where, depending on which energy $\mathcal{F}$ represents in particular, we additionally might have to guarantee that the limit of the polygons belongs to the same knot class. Moreover, we show that the unique absolute minimizer of the discrete energies is the regular $n$-gon.

We describe the possible critical curves for the ropelength problem in the absence of struts, that is, curves minimizing length subject only to a limit on curvature. These includes certain concatenations of circular arcs and straight segments, as well as a two-parameter family we call the supercoiled helices

Knots and links are modeled as ribbons immersed in the plane. This is a 2-dimensional analogue of thick knots and the ropelength problem. This talk will introduce the idea of a flat ribbon link and give examples of 'tight' ribbon length knots. It turns out there are some surprising technicalities involved - moving from 3 to 2 dimensions does not necessarily simplify the mathematics.

Using optimal control theory, the Markov-Dubins problem of finding a shortest path between two points in the plane with given initial and final tangent direction and an upper bound on curvature has been thoroughly solved. Minimizers for that problem have a simple form consisting only of straight line segments and circular arcs. Naturally, such curves also occur as strongly critical curves for the ropelength problem, if we neglect critical self-distance. Criticality was shown before by analyzing possible kink tension functions, but a more direct way to show criticality of minimizers would be to prove that they are thickness regular, which means that there is a variation vector field in space around the curve in the direction of which the right-derivative of the Thi∞-functional is positive. But it turns out that this is not always possible.

In 1999 J.C. L\'{e}ger proved that a one-dimensional set with finite total Menger curvature is 1-rectifiable. We are trying to generalize this result to higher dimensional sets and curvature energies, where it is not clear how to define those curvature energies. In this talk we give a characterization of integral Menger-type curvatures and a motivation why those may be suitable for this purpose.

During the talk I will show an application of $t$-energies defined as \[I_t(\mu) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{1}{|x-y|^t} d\mu(x) d\mu(y)\] which allows to state whether certain measures supported on spiral curves are locally in $H^{-1}(\mathbb{R}^2)$. The measures we consider correspond to vortex sheets i.e. describe vorticity of irregular flows. Spiral vortex sheets and their evolution were first observed and analysed by physicists in 1930s, but it is still not known whether they are solutions to the 2d Euler equation. The question whether the spirals are elements of $H^{-1}$ was motivated by the fact, that the existence of the vortex solution of the Euler equation was proved [1] under the assumption that the initial vorticity is a compactly supported Radon measure belonging to $H^{-1}$. The theorem I will present applies to a broader class of measures than measures supported on spiral curves -- namely to all compactly supported Radon measures with prescribed (in a certain way) relation between measure of a ball centred at the origin and its radius. If time permits I will also show how to use $t$-energies to get a new proof of the fact that the Morrey space of measures embeds compactly in $H^{-1}$ (the embedding theorem was first proved in [2]).J.-M. Delort, Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 553-586, (1991).M. C. Lopes Filho, H.J. Nussenzveig Lopes, S. Schochet, A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution. Trans. Amer. Math. Soc. 359 4125-4142, (2007).