### Grassmannian structures on polygons and space curves

#### Jason Cantarella (University of Georgia)

Joint work with Thomas Needham.This talk is an overview of some geometric structures on the space of closed curves and their utility in studying random knotsand knot energies. It has been known since the 1990's that the space of closed, framed space polygons with n edges and total length 2 can be identified with the Grassmannian of 2-planes in complex n-space. We’ll discuss how this gives a natural measure (and symplectic structure) on the space of closed framed space polygons, and use it to compute the expected value of a simple knot energy on polygons (radius of gyration).

The second half of the talk will discuss the corresponding structures in the smooth category. The collection of smooth closed framed space curves forms an infinite-dimensional manifold that is diffeomorphic to the Grassmannian of 2-planes in the loop space of the complex plane. This endows the moduli space of framed loops with a natural Riemannian metric and compatible symplectic structure. The metric admits explicit geodesics, and this property can be used to give a shape recognition algorithm for, e.g., proteins and oriented trajectories. These structures also give a geometric framework for studying energy functionals on curves, many of which have convenient representations in our complex coordinate system.

### Prediction of the optimal set of contacts to fold the smallest knotted protein (Poster)

#### Paweł Dabrowski-Tumanski (University of Warsaw)

Knotted protein chains represent a new motif in protein folds. They have been linked to various diseases, and recent extensive analysis of the Protein Data Bank shows that they constitute 1.5% of all deposited protein structures. Despite thorough theoretical and experimental investigations, the role of knots in proteins still remains elusive. Nonetheless, it is believed that knots play an important role in mechanical and thermal stability of proteins. Here, we perform a comprehensive analysis of native, shadow-specific and non-native interactions which describe free energy landscape of the smallest knotted protein (PDB id 2efv). We show that the addition of shadow-specific contacts in the loop region greatly enhances folding kinetics, while the addition of shadow-specific contacts along the C-terminal region (H3 or H4) results in a new folding route with slower kinetics. By means of direct coupling analysis (DCA) we predict non-native contacts which also can accelerate kinetics. Next, we show that the length of the C-terminal knot tail is responsible for the shape of the free energy barrier, while the influence of the elongation of the N-terminus is not significant. Finally, we develop a concept of a minimal contact map sufficient for 2efv protein to fold and analyze properties of this protein using this map.### Folded Ribbon Knots in the Plane

#### Elizabeth Denne (Washington and Lee)

Knots and links are modeled as folded ribbons lying in the plane. The ribbonlength of a knot is the the length of a knot divided by the width of the ribbon around it. This problem is related to the ribbonlength of immersed ribbons in the plane, as well as to the ropelength of thick knots. In this talk we'll discuss the construction of folded ribbon knots, and give examples of folded ribbon knots and their ribbonlength. We'll also discuss the topology of folded ribbon knots, and the problem of minimizing ribbonlength for a given knot type - it turns out there are several good candidates for this notion. This is joint work with undergraduate students from Smith College and Washington & Lee University.### Sedimentation of macroscopic rigid knots and its relation to gel electrophoretic mobility of DNA knots

#### Giovanni Dietler (EPFL)

I will present experiment comparing sedimentation of rigid knots to the mobility of DNA knots. It is a general question of the extent to which the hydrodynamic behaviour of microscopic freely fluctuating objects can be reproduced by macrosopic rigid objects. In particular, the sedimentation speeds of knotted DNA molecules undergoing gel electrophoresis is compared to the sedimentation speeds of rigid stereolithographic models of ideal knots in both water and silicon oil. The sedimentation speeds grow roughly linearly with the average crossing number of the ideal knot configurations, and the correlation is stronger within classes of knots. This is consistent with previous observations with DNA knots in gel electrophoresis.### Relative Frequencies of Alternating and Non-alternating Prime Knots in Random Knot Spaces

#### Claus Ernst (Western Kentucky University)

We are interested in the following question: Are there knots that are intrinsically easier to form in most random knot spaces without obvious topological biases? For example, in the space of random equilateral polygons with a fixed length, it is known that composite knots have higher frequencies than the prime knots when the polygons are long. Here we concentrate on the average behavior of the knot spectrums for knots up to 16 crossings. Using random polygons of various lengths under different confinement conditions as the random knot spaces, we demonstrate that the relative spectrums of the knots when divided into groups by their crossing numbers, remain surprisingly robust as these knot spaces vary. For a given knot space and a given crossing number $c$ , let ${P}_{c}$ be the probability that a uniformly chosen random knot from the space is of crossing number $c$ , ${P}_{c}\left(A\right)$ be the probability that a uniformly chosen random knot from the space is an alternating prime knot of crossing number $c$ and ${P}_{c}\left(N\right)$ be the probability that a uniformly chosen random knot from the space is a non-alternating prime knot of crossing number $c$ . It had been reported earlier that the order among the ${P}_{c}$ 's changes as the polygon lengths and the confinement conditions change. However, we find surprisingly that the order among the ratios ${P}_{c}\left(A\right)/{P}_{c}$ remain the same. Furthermore, our results indicate that tighter confinement condition favors alternating prime knots, i.e. ${P}_{c}\left(A\right)/{P}_{c}$ increases as the confinement radius decreases. Interestingly, we also observe that ${P}_{c}\left(N\right)$ behaves very similarly to ${P}_{c-1}\left(A\right)$ .### A decomposition of the Möbius energy and consequences (Poster)

#### Aya Ishiseki (Saitama University)

### Hamiltonian Knots in Cubic Lattices

#### Gyo Taek Jin (KAIST)

In this work, we studied knots in cubic lattice boxes. We showed that every knot can be presented as a Hamiltonian cycle in a cubic lattice box of size n x n x n for some odd integer n. We also showed that the number n is closely related with the arc index and the minimal crossing number.This is a joint work with Choonbae Jeon, Hyuntae Kim, Gyoung Hoon Ko, Sang Min Ko, Chang Yong Kim, and Jeong Min Shin.

### A new skein invariant for classical links from the Yokonuma-Hecke algebras (Poster)

#### Konstantinos Karvounis (U Zürich)

We study the Juyumaya-Lambropoulou invariants ${\Theta}_{d,D}$ for classical knots and links constructed from the Yokonuma-Hecke algebras, and in particular their relationship to the Homflypt polynomial. We first prove that the Markov trace $tr{d}_{D}$ on the Yokonuma-Hecke algebras can be computed on classical knots and links by five rules which do not involve the framing generators. Using this we show that the invariants ${\Theta}_{d,D}$ on classical knots are topologically equivalent to the Homflypt polynomial. We then re-define the invariants ${\Theta}_{d,D}$ via skein relations. Using this we show that the invariants ${\Theta}_{d,D}$ are not topologically equivalent to the Homflypt polynomial on classical links by providing computational data for six pairs of Homflypt-equivalent links which are distinguished by ${\Theta}_{d,D}$ and a diagrammatic proof for one of these pairs. Finally, we give a new 3-variable skein link invariant $\Theta $ which specializes to the invariants ${\Theta}_{d,D}$ and to the Homflypt polynomial.This is joint work with M. Chlouveraki, J. Juyumaya and Sofia Lambropoulou.

### Geometric curvature energies in calculus of variations (Poster)

#### Slawomir Kolasinski (Albert-Einstein Institute, Golm)

### How to tie your shoe laces and other open problems in knot theory

#### John Maddocks (EPFL)

I will describe some 30 year old work on understanding the role friction plays in tying (real) knots, and then describe some possible connexions to ideal knot shapes.Ropes in Equilibrium, J H Maddocks and J B Keller, SIAM J Applied Math vol 47, 1987

### Course Grained 3-Space Simulations For Knotting in Polymeric Systems

#### Kenneth C. Millett (University of California, Santa Barbara)

Polymeric systems are modeled by collections of mathematical curves that are entangled due to the effects of knotting or linking, both local and global. This mutual interference is implicated in large-scale effects making their characterization and quantification an objective of substantial interest. In this review, I will bring together several mathematical streams:(1) classical knot theory and its application to the course grained, off lattice, study of polymeric systems closed chains (Description of the topological entanglements of DNA catenanes and knots by a powerful method involving strand passage and recombination, J. Mol. Biol. (197) 1987, 585-603);

(2) knotting of open chains used to characterize aspects of, for example, macromolecules such as protein structures (Linear random knots and their scaling, Macromolecules, (38) 2005, no. 2, 601-606; Conservation of complex knotting and slipknotting patterns in proteins, Proceedings of the National Academy of Science, USA, vol 109, no 26, 2012, E1715-E1723; KnotProt: a database of proteins with knots and slipknots, Nucleac Acids Research, (2014) 1, doi: 10.1093/nar/gku1059);

(3) methods of sampling collections of open and closed ideal chains and, more recently, with specified thickness (results of Laura Plunkett and Kyle Chapman) and several facets of the analysis of the resulting data.

### Magnetic torus knots and unknots: geometric properties, energy and helicity

#### Chiara Oberti (University of Milano-Bicocca)

This talk is organized in two parts. In the first part, by using a standard parametrization of torus knots/unknots, we prove the existence and determine the location of inflection points, and we prescribe the condition for removing the singularity associated with torsion at the inflection point. We show that to first approximation total length grows linearly with the number of coils, and it is proportional to the minimum crossing number of the knot type. By using the concept of tangent indicatrix, we relate the development of inflectional configurations to the growth in writhing number, and we show how the passage through inflectional state is responsible for the jump of the intrinsic twist of framed torus knots/unknots.In the second part three different approaches to calculate the helicity associated with a magnetic torus knot in ideal MHD are compared to show that the helicity of toroidal knots/unknots is dominated by writhe contribution. We provide lower and upper bounds on magnetic energy, and we establish estimates of magnetic energy in terms of geometric properties. Since torus knots are also a good mathematical model for studying braided magnetic fields, these results provide useful information for many possible applications, from solar physics and astrophysics to fusion physics.

This work is part of my PhD thesis, under the supervision of R.L. Ricca.

### Regularization of energies of knots and surfaces

#### Jun O'hara (Tokyo Metropolitan University)

I will talk on geometric quantities for knots, surfaces, and compact bodies which we can obtain by applying either Hadamard regularization or analytic continuation to the divergent integral of energy.(joint work with Gil Solanes (Universitat Autonoma de Barcelona))

### Generation of supercoils in nicked and gapped DNA drives, DNA unknotting and postreplicative decatenation (Poster)

#### Dusan Racko (University of Lausanne)

Due to helical structure of DNA the process of DNA replication is topologically complex so that freshly replicated DNA molecules are entangled with each other and are frequently knotted. For proper functioning of DNA it is necessary to remove all these entanglements. This is done by DNA topoisomerases that pass DNA segments through each other using cutting and pasting mechanism. However, it has been a riddle how DNA topoisomerases select their places of action, since in highly crowded DNA in living cells random passages between contacting segments would only increase the extent of entanglement. Using molecular dynamics simulations we observed that in actively supercoiled DNA molecules the entanglements resulting from DNA knotting or DNA catenation spontaneously approach sites of nicks and gaps in the DNA. Type I topoisomerases that preferentially act at sites of nick and gaps are thus naturally provided with DNA-DNA juxtapositions where cutting and pasting operation results in an error-free DNA unknotting or DNA decatenation.### What knots lurk inside other knots?

#### Eric Rawdon (U St. Thomas)

For a fixed knot configuration, subknots are the knot types seen in the open subarcs of the configuration. For nice knot configurations (like ones minimized with respect to some knot energy), the subknots are typically simpler knot types than the host knot type. We compare and contrast the set of subknots coming from KnotPlot configurations, tight knot configurations, and random configurations. This is joint work with Ken Millett and Andrzej Stasiak.### HOMFLYPT polynomial for vortex knots and cascade process

#### Renzo Ricca (U. Milano-Bicocca)

In this paper we present and discuss the derivation of the skein relations of the HOMFLYPT polynomial for vortex knots in terms of helicity contributions [1]. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT becomes then function of knot topology and vortex circulation, and it can be interpreted as a new, powerful invariant of topological fluid mechanics.By analyzing the branching process associated with the polynomial skein relations, we also show that the process of vortex cascade through continuous reduction of topological complexity by stepwise unlinking – observed experimentally in the production of vortex knots in water, and tested by various numerical simulations – can be reproduced by the recursive application of the skein relations to minimal knot diagrams [2]. This observation suggests a possible interpretation of the skein reduction process in terms of minimal energy pathway for the natural decay of topologically complex vortex systems.

[1] Liu, X. & Ricca, R.L. (2015) On the derivation of HOMFLYPT polynomial invariant for fluid knots. J Fluid Mech. 773, 34-48.

[2] Ricca, R.L. (2015) Vortex knot cascade in polynomial skein relations. In Numerical Analysis and Applied Mathematics ICNAAM 2015 AIP Conf. Proc., to appear.

### Torsion of Khovanov homology

#### Radmila Sazdanovic (NCSU Mathematics)

Categorification has made a huge impact on many areas of pure and applied mathematics by revealing deeper fundamental structures and unexpected relations. Recent examples include Khovanov and Floer link homology theories that have led to amazing progress in knot theory and low dimensional topology. In this talk we will focus on torsion in Khovanov homology and related graph homology theories. This is joint work with A. Lowrance and J. Przytycki.### Discrete Curvature Energies

#### Sebastian Scholtes (RWTH Aachen University)

We give an overview of available methods for discrete curvature energies. Discrete energies are not only designed to make swift numerical computations and thus opening up the field to computational methods but, more importantly, they provide an independent, geometrically pleasing and consistent discrete model that features a similar behavior as the original model. The focus of the talk will be on knot energies such as Möbius energy, integral Menger curvature or thickness. Time allowing, we also discuss the bending energy regarding curves subject to clamped boundary conditions.### Numerical Optimization of Geometric Energies

#### Henrik Schumacher (Universität Hamburg)

For many applications in science and engineering, one needs trustworthy approximations of the minimizers of a given geometric optimization problem.Since the feasible set of such an optimization problem is often infinite dimensional, a common approach for the approximation of their solutions is the Ritz-Galerkin method: Computing minimizers of the energy on a finite dimensional subspace, a so-called discrete ansatz space.

Classical results, such as Cea's lemma, guarantee that the minimizers in the discrete ansatz space are arbitrarily close to the minimizers of the infinite dimensional problem if the discrete ansatz space is chosen sufficiently large---and provided that the energy is convex. Alas, many interesting geometric energies are nonconvex such that these standard results cannot be applied to prove the convergence of discrete solutions.

Moreover, it may be inefficient or even impossible to calculate the energy exactly, to use subsets of the feasible set as discrete ansatz space, or to determine discrete minimizers exactly.

In this talk, we outline a way to overcome these hardships and to reduce the convergence analysis for Ritz-Galerkin schemes to approximation theory. The developed tools will be illustrated by an application to minimal surfaces. Finally, we will realize that a priori knowledge of the regularity of minimizers is crucial for obtaining reasonable convergence rates.

### Equilibrium shapes of closed developable elastic strips

#### Eugene Starostin (UCL)

Joint work with Gert van der Heijden (UCL)We study equilibrium configurations of isometrically deformed closed elastic strips of constant width and constant geodesic curvature of the middle line. The description of the model is reduced to an invariant one-dimensional variational problem and the equations are derived in the Euler-Poincare form. Closed configurations are obtained by numerical solution of boundary value problems for a system of ODEs. We consider various classes of symmetric solutions, including knotted. Particular attention is paid to variation of shapes depending on the aspect (width-to-length) ratio. Examples of folded flat configurations for extreme values of the aspect ratio are discussed.

### Generation of supercoils in nicked and gapped DNA drives DNA unknotting and postreplicative decatenation

#### Andrzej Stasiak (University of Lausanne)

Joint work with Dusan Racko, Fabrizio Benedetti, Julien Dorier, Yannis BurnierCenter for Integrative Genomics, University of Lausanne, 1015-Lausanne, Switzerland

Due to the helical structure of DNA the process of DNA replication is topologically complex. Freshly replicated DNA molecules are catenated with each other and are frequently knotted. For proper functioning of DNA it is necessary to remove all of these entanglements. This is done by DNA topoisomerases that pass DNA segments through each other. However, it has been a riddle how DNA topoisomerases select the sites of their action. In highly crowded DNA in living cells random passages between contacting segments would only increase the extent of entanglement. Using molecular dynamics simulations we observed that in actively supercoiled DNA molecules the entanglements resulting from DNA knotting or catenation spontaneously approach sites of nicks and gaps in the DNA. Type I topoisomerases, that preferentially act at sites of nick and gaps, are thus naturally provided with DNA-DNA juxtapositions where a passage results in an error-free DNA unknotting or DNA decatenation.

### Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

#### Paweł Strzelecki (Warsaw University)

Joint work with S. Kolasinski and H. von der Mosel
We prove isotopy finiteness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials. That is, there are only finitely many isotopy types of (say, Lipschitz) submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy.

Moreover, we establish compactness (any sequence of submanifolds with uniformly bounded energy contains a subsequence converging to a limit submanifold with the same energy bound) and we show that all geometric curvature energies under consideration are lower semicontinuous with respect to Hausdorff-convergence of sets, which can be used to minimise each of these energies within prescribed isotopy classes.

### The Ribbonlength of Knot Diagrams

#### John M Sullivan (TU Berlin)

The ropelength problem asks to minimize the length of a knotted space curve such that a unit tube around the curve remains embedded. A two-dimenaional analog has a much more combinatorial flavor: we require a unit-width ribbon around a knot diagram to be immersed with consistent crossing information. Attempting to characterize critical points for ribbonlength leads us to new results about the medial axis of an immersed disk in the plane, including a certain topological stability for thin disks.This is joint work with Elizabeth Denne and Nancy Wrinkle.

### Pulling knotted proteins through the pores: topological traps and how to avoid them

#### Piotr Szymczak (Warsaw University)

If we tie a knot on a piece of a rope and then pull it through a narrow hole, the knot tightens, and can block the opening. We argue that a similar phenomenon can take place in microworld, during the transport of knotted proteins through the pores in cellular membranes. The radius of gyration of the tight knot is about 7-8 Angstrom (for a trefoil), whereas the radii of the narrowest constrictions of e.g. the mitochondrial pores are about 6 Angstrom, which means that the knot is a shade too large to squeeze through the pore opening. This leaves us with two possibilities: either the knot diffuses towards the end of the chain and slides away or gets tightened and jams the opening. Based on the results of the molecular dynamics simulations of the translocation process, we analyze the probability of pore jamming as a function of the magnitude of the pulling force. The results are shown to be consistent with a simple, two-pathway model, with one pathway leading to the knot tightening and the other - to the successful translocation. Next, we show how such topological traps might be prevented by using a pulling protocol of a repetitive, on-off character. During the off-force period some stored length is inserted into the knotted core, and the knot loosens, thus escaping the tightened configuration. Subsequently, during the next on-force period the protein makes another attempt at the translocation. Since the probability of getting trapped every time in n attempts rapidly decreases with n, repetitive trying always leads to a final success. Importantly, such a repetitive pulling is biologically relevant, since molecular import motors are ATP-hydrolysis driven and thus cyclic in character. Finally, we analyze the dependence of the translocation rate on the frequency and magnitude of the periodic force and show that there exists an optimum range of these parameters which leads to the most efficient translocation.### On elastic torus knots

#### Heiko von der Mosel (RWTH Aachen University)

Motivated by the elastic behavior of knotted loops of springy wire we minimize the classic bending energy ${E}_{bend}={\displaystyle \int}{\kappa}^{2}$ in given knot classes, where we add a small multiple of ropelength $\mathcal{R}=length/thickness$ in order to penalize selfintersection. Our main objective is to characterize*elastic knots*, i.e., all limit configurations of energy minimizers of the total energy ${E}_{\vartheta}:={E}_{bend}+\vartheta \mathcal{R}$ as $\vartheta $ tends to zero. For every odd $b$ with $\left|b\right|\ge 3$ 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.

This is joint work with H. Gerlach and Ph. Reiter.