Abstract

To model knotted springy wires we minimize in prescribed knot classes a total energy consisting of the classic Euler-Bernoulli bending energy and an additive repulsive potential. The ultimate goal is to characterize the shape of such minimizing knots for various knot classes. For that we send a prefactor of the repulsive potential to zero and analyze the limiting configurations -- so-called elastic knots. For all torus knot classes T(2,b) we established the doubly-covered circle as the unique elastic knot, which confirms mechanical and numerical experiments. There are, however, instances when numerical gradient flows seem to get stuck in different configurations exhibiting some symmetry. To provide analytic support for these rare observations we use the symmetric criticality principle to find symmetric elastic knots exhibiting these symmetries. In this talk we give a survey on the analytic results, show some of the numerical simulations obtained by Bartels, Riege and Reiter, and address many open questions.

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Abstract

The unknotting number of a knot K is the minimal number of crossing changes required to turn K into the unknot, taken over all diagrams of K. There is no algorithm known to compute it, and even finding the unknotting number of a fixed diagram becomes computationally unfeasible when the crossing number is large. Reinforcement learning is a machine learning paradigm where an agent learns to perform various actions by interacting with an environment in order to maximise a reward. I will discuss how reinforcement learning can help with the study of the unknotting number. This is joint work in progress with Sam Blackwell, Alex Davies, Thomas Edlich, and Marc Lackenby.

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Neural networks are a class of parameterized functions that have proven remarkably successful at making predictions about unseen data from finite labeled data sets. They do so even in settings when classical results suggest that they ought to be overfitting (aka memorizing) the data. I will begin by describing the structure of neural networks and how they learn. I will then advertise one of the theoretical questions animating the field: how does the relationship between the number of parameters and the size of the data set impact the dynamics of how they learn? Along the way I will emphasize the many ways in which topology, geometry, and combinatorics play a role in the field.

Abstract

We present a range of topological questions associated with DNA self-assembly and three dimensional structures. The questions vary from topological graph theory related to DNA strand routing a three-dimensional mesh, to questions in knot theory related to structural embeddings in 3D, to algebraic descriptions related to Jones monoids associated with DNA origami.

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We'll review the universal construction of topological theories in low dimensions and describe examples and their applications in topology, representation theory and number theory.

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I will describe several ways in which tools from reinforcement learning and natural language processing can help with problems in knot theory and low-dimensional topology, including hard challenges such as the smooth Poincare conjecture in dimension 4 and its variants. I will try to make the talk self-contained; in particular, no prior knowledge of machine learning is required, and a large part of the talk will give a gentle introduction to various algorithms and architectures.

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Knots occur in all kinds of systems, including graphs. We're used to finding knots in loops, but what about knotted theta curves? Or knotted graphs? In this talk, we discuss a general framework for constructing random embeddings of arbitrary graphs. This is a conditional probability problem, but setting up the conditions correctly requires some ideas from algebraic topology. We'll view the problem as embedding a simplicial 1-complex and see that the embedding data is (in a precise sense) dual to the simplicial homology of the 1-complex. This will enable us to make some new exact calculations for random knots.

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Abstract

We develop a new random model for links based on meanders. Random meander diagrams correspond to matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of links. We prove that unlinks appear with vanishing probability, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained on every step. Then we give expected twist number of a diagram, and bound expected hyperbolic and simplicial volume of links.

Abstract

Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020 Proc. R. Soc. A 476, 20200124. ((doi:10.1098/rspa.2020.0124)), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time.

Abstract

My collaborators (Henrik Schumacher, Philipp Reiter) and I used self-repulsive energies to construct a useful Riemannian Metric on the above manifold. In the lightning talk, I will introduce a little infinite dimensional Riemannian geometry, sketch some difficulties in the construction and concepts on how to overcome them.

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In 2014 Jones established a correspondence between links and elements of the Thompson group. This correspondence arose as a consequence of Jones' construction of unitary representations of this group. I provide an analog of this program for annular links.

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Given any element g in oriented Thompson group, we construct a canonical grid diagram associated to it. We show that this grid construction is equivalent to Jones' representation theoretic construction of link L_g. Moreover, using this grid diagram we study the number of L_g's components and its link group. We also give a lower bound of max Thurston Bennequin number in terms of oriented Thompson index.

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The framed curvature flow is a generalization of the curve shortening flow and the vortex filament equation, where the magnitude of the velocity vector is determined by the curvature, and its direction is given by an associated time-dependent moving frame. The flow can be defined in such a way that it sweeps out trajectory surfaces of constant mean or Gaussian curvature.

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"The Most Important Missing Infrastructure Project in Knot Theory" -Dr. Dror Bar-Natan There are a number of great knot and link tables available to researchers; be that mathematicians, biologists, physicists, and many other domains. However, with only knot and link tables we are in the position of a chemist with a table of fatty acids but no periodic table. Our group at University of Iowa are striving to build that periodic table of knot "elements", the two string tangles. Where a 2-string tangle is an embedding of 2 disjoint string-segments into the interior of a 3-ball.

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Given a Legendrian Knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery. Using the naturality result we can provided new families of Legendrian non--simple knot.

Abstract

In this short talk, we introduce the generalization of tangent-point energies to higher dimensions. In particular, we study surfaces embedded into the space. We conjecture that there are two distinct symmetric surfaces that are ciritcal points of the energy. The idea is that sufaces cannot have certain symmetries simultaneously. We sketch the proof of this conjecture.

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Links in a thickened torus are natural objects of study as quotients of periodic structures in 3-space on one hand and the simplest realizations of virtual links on the other. This talk will provide a brief overview of an algorithm that can generate a knot table for such links up to a given number of crossings, and address the challenges of this task in terms of computational power.

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An electromagnetic field consists of two time-dependent vector fields E_t and B_t on R^3 that together satisfy Maxwell's equations. I will briefly explain why for every knot type K there is an electromagnetic field such that at every moment in time t there is a periodic orbit of E_t and B_t in the shape of K. Similarly, for every embedded torus T there is an electromagnetic field whose electric or magnetic part has an invariant torus isotopic to T for all time.

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In the talk we discuss the change that writhes namely odd writhe and n-writhes undergoes when we alter a virtual knot diagram by a local move. The study allows us to relate odd writhe with a numerical invariant defined using the particular local move. In addition, we find that n-writhes on the contrary do not behave very nice under the move and blow boundlessly. As one of its impact, the coefficients of the affine index polynomial grow without bounds for the local move under study.

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An important ingredient of Sumners and Whittington's proof that almost all sufficiently long ring polymers are knotted relies on finding a suitable tangle that guarantees certain global properties. I will survey some tangle insertion techniques that have been used to study linking statistics, which includes the setting of links confined in lattice tubes that I investigated with Jeremy Eng, Rob Scharein, and Chris Soteros.

Abstract

In this talk I will be describing a generalization of Jones polynomial and Khovanov homology for the Legendrian knots in $(\mathbb{R}^3,\xi_{st})$ utpo Legendrian knot type. We will see that the Legendrian Jones polynomial $P_K(A,r)$ reduces to the Jones polynomial for the corresponding topological knot after substituting $r=1$, where $K$ is a Legendrian knot. We will also see that one of the classical invariant of Legendrian knot type, known as the Thurston-Benniquin number appears as grade shift in the Legendrian Khovanov homology.

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Abstract

Self-avoiding energies were originally constructed to simplify knots and links in R^3. The driving idea was to design barrier functions for the feasible set, the set of curves of prescribed isotopy class. Such functions must blow up whenever a path of curves tries to escape the feasible set. This singular behavior makes it challenging to perform numerical optimization for self-avoiding energies, in particular, when ""close"" to the boundary of the feasible set. Motivated by the Riemannian metric in the Poincaré model of hyperbolic space, my collaborators and I constructed Riemannian metrics that provide nice preconditioning for the Möbius energy (with Philipp Reiter) and for tangent-point energies (with Keenan Crane). These metrics work extremely well in numerical experiments. In particular, the elementary gradient schemes that we employ require rarely any line search in the form of collision detection. That is, a step of finite, reasonable size in the knot space (almost) never escapes from the feasible set. That lead us to the conjecture that these metrics (or some mild modifications) must be geodesically complete. In this talk I will present ongoing work with Elias Döhrer and Philipp Reiter. After explaining the notion of geodesic completeness and its applications, I will introduce a certain class of Riemannian metrics on the space of knots. These are closely related to tangent-point energies whose energy spaces are Hilbert spaces. Finally, I will sketch a proof for geodesic completeness.

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Motivated in part by recent experimental data for knotting of DNA in nano-channels and nano-pores, we have been studying the entanglement complexity of polymers confined to lattice tubes. To model ring polymers in solution, we consider systems of polygons confined to tubular subsets of the simple cubic lattice and study their entanglement complexity using a combination of knot theory, transfer-matrix methods and Monte Carlo computer simulation. For the smallest tube that admits knotting, we prove long-standing conjectures about the knot and link statistics as the system size grows. Monte Carlo simulations are used to explore the conjectures for larger tube sizes. I will review these results including our most recent results for the special case of a system of two polygons that span the tube.

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Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure. At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity. The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. We have developed a formalization of the topology of two-periodic weft knitted textiles using a construction we call the swatch [1]. Using this construction, we can prove that all two-periodic weft knits form ribbon links [2]. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics. The authors were partially supported by National Science Foundation grant DMR-1847172, by the Research Corporation for Science Advancement and by the International Center for Sustainability with Chiral Knotted Meta Matter (SKCM²). We would like to thank sarah-marie belcastro, Jen Hom, Jim McCann, Agniva Roy, Saul Schleimer and Henry Segerman for many fruitful conversations. [1] S. Markande and S. Matsumoto, in: Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Culture, (Tesselations Publishing, 2020), pp. 103–112. [2] M. Kuzbary, S. Markande, S. Matsumoto and S. Pritchard, 2022.

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Following a general discussion of the computation of zombians of unfinished columbaria (with examples), I will tell you about my recent joint work with Jessica Liu on what we feel is the "textbook" extension of knot signatures to tangles, which for unknown reasons, is not in any of the textbooks that we know. http://drorbn.net/icerm23

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I will present results of computer experiments on thickened–tubified–curves. A thickened curve is a curve that may be thickened to an embedded tube of unit radius, allowing us to choose a scale at which we may measure the curve’s length, a scale invariant length. A closed (tangled) thickened curve’s scale invariant length is lower bounded by the knot invariant ropelength but such curves need not be tight, they can be loosely tied. This work investigates the shapes of loosely tied thickened curves by performing deformation experiments without changing the curves’ scale invariant length. The main inspiration of this research is the role of tangling in filament–like biomaterials in solution, such as proteins existing in the aqueous environment of living cells. From this physically motivated perspective the experiment optimises the curve trajectory towards thickened curve shapes with a high degree of thermodynamic stability in solution. How can entanglement hinder or coordinate the energetically driven shape change of a thickened curve? This is joint work with Prof. Myf Evans of the University of Potsdam.

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The braid indices of most links remain unknown as there is no known universal method that can be used to determine the braid index of an arbitrary knot. This is also the case for alternating knots. In this paper, we show that if K is an alternating knot, then the braid index of any reverse parallel link of K can be precisely determined. The formula can be easily obtained from any reduced alternating diagram D of K. As a consequence of this result, we can now prove that the ropelength of any alternating knot is at least proportional to its minimum crossing number.

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In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. Wall-type stabilization problems aim to quantify this instability. I will discuss an approach to these problems that uses Khovanov-type homology theories and relies on blending theoretical calculations with intensive machine computation. Time permitting, I will discuss how one might push these techniques further and use these Khovanov-type tools for knots and surfaces to calculate Floer-theoretic invariants for associated 3- and 4-manifolds.

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A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. We will discuss two instances in which tangle decompositions can be used to address classic problems in knot theory. In the first instance, we will use a very simple tangle decomposition to prove an equivariant version of the cosmetic surgery conjecture. The proof strategy relies on a reinterpretation of Khovanov homology and Bar-Natan's tangle invariant in terms of immersed curves on the four-punctured sphere. The second instance involves a complicated tangle, the role of which is to generalize the statement 'unknotting number one knots are prime' to spatial theta graphs. This spans joint work with two different sets of authors.

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With recent breakthroughs in AI like Chat-GPT and DALL-E 2 making headlines, it's natural to ask what role machine learning will play in mathematics. In this talk I will outline three broad modes through which machine learning can be applied to the field of knot theory and low-dimensional topology. These applications will range from approaches that have already been successfully implemented to more speculative works-in-progress.

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Supramolecular constructs with complex topologies are of great interest across soft-matter physics, biology and chemistry, and hold much promise as metamaterials with unusual mechanical properties. A particularly challenging problem is how to rationally design, and subsequently realize, these structures and the precise interlockings of their multiple molecular strands. Here we report on the combined use of theory and simulations to obtain complex supramolecular constructs via programmed self-assembly. Specifically, by controlling the geometry of the self-assembled monomers we show that the assembly process can be directed towards ""privileged"", addressable topologies of molecular knots, and extended linked structures, such as Olympic gels and catenanes. We conclude presenting an overview of the unique static and dynamical properties of linear catenanes. The talk will cover results based on the following publications [1] E. Orlandini and C. Micheletti, J. Phys. Condensed Matter, 34, 013002 (2022) [2] M. Marenda, E. Orlandini and C. Micheletti, Nature Communications, 9, 3051 (2018) [3] G. Polles, E. Orlandini and C. Micheletti, ACS Macro Letters, 5, 931-935 (2016) [4] G. Polles, D. Marenduzzo, E. Orlandini and C. Micheletti, Nature Communications, 6, 6423 (2015) [5] M. Becchi, R. Capelli, C. Perego, G.M. Pavan and C. Micheletti, Soft Matter, (2022) [6] For an actual hands-on demonstration of the designed self-assembly of ""macroscopic"" trefoil knots see the video at this link: www.youtube.com/watch?v=XKsuMlp2PLc

Abstract

The goal of this work is to describe the dynamics of DNA knots with excess link in an ionized fluid. To do so, we employ two models: 1. the Generalized Immersed Boundary (GIB) method, which is a deterministic method that accounts for the fluid, structure interaction of an immersed DNA molecule in an ionized fluid; 2. the Stochastic Generalized Boundary (SGIB) Method, which is an extension of the GIB method that also takes into account the random thermal fluctuations within the fluid. Using the GIB and SGIB methods, we numerically explore the energy landscape of a closed DNA segment with excess twist in a trefoil knot configuration. We first analyze the symmetry of stable knotted equilibrium configurations, approximate saddle configurations, and examine the elastic energy throughout the deterministic process. We then use the SGIB method to model DNA knot dynamics as a continuous time Markov chain. We classify boundaries within the energy landscape using the Procrustes distance. Finally, we obtain a steady state distribution for the Markov process given a fixed linking number and compare this to energy estimates obtained from the GIB method.

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This is an introductory talk about finite type invariants for knots with a focus on computation complexity. I will give some new language to help the knot theory community to talk about our current understanding of knot invariants. I will conclude the talk with a new and very efficient algorithm to compute finite type invariants.

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Let T(n, k) denote the (n, k)-torus link. It is well known that the Jones polynomial and the Khovanov homology of torus links stabilize as k approaches infinity by the work of Champanekar-Kofman and Stosic. In particular, Rozanksy showed that the stable Khovanov homology of torus links exists as the direct limit of the Khovanov homology of T(n, k)-torus links, and the stable Khovanov homology recovers the categorification of the Jones-Wenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by Champanekar-Kofman, extending the results by Stosic and Rozansky. Since the Jones-Wenzl projector can be used to define the colored Jones polynomial, we will discuss potential relationship between the stable invariant to the hyperbolic volume of a knot in the spirit of the volume conjecture.

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Chiral rigid filaments rotate when placed in a wind. We exploit this phenomenon to construct a tensor measure of chirality for rectifiable space curves. Our tensor is trace-free, so if a curve has a right-handed twist about some axis, there is also a perpendicular axis about which the twist is left-handed. This measure places minimal smoothness requirements on the curve, hence it can be used to quantify chirality for biomolecules or polymers, and it can be readily generalized to study rather rough — or even higher-dimensional — geometric objects in space. We also speculate on what to expect when a wind of scattered particles is replaced by wave scattering. [This is part of an ongoing/long-going project with Giovanni Dietler, Wöden Kusner, Eric Rawdon, and Piotr Szymczak — a bit was conducted at ICERM].

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For a given an alternating knot type K any two minimal diagrams K are related by flypes. Using flypes one can create a list of all minimal diagrams of K. What knot types have the largest or smallest number of diagrams? Many alternating knot types are curly, that means they admit a diagram without inflection points that has fewer maxima than their braid index. We show that almost all 2-bridge knots are curly. Finally, we report on a new result by Y. DIao on the rope-length of alternating knots.

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