Tangled in Knot Theory
Institute for Computational and Experimental Research in Mathematics (ICERM)
May 22, 2023  May 25, 2023
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Monday, May 22, 2023

8:50  9:00 am EDTWelcome11th Floor Lecture Hall

9:00  9:45 am EDTElastic knots11th Floor Lecture Hall
 Speaker
 Heiko von der Mosel, RWTH Aachen University
 Session Chair
 Elizabeth Denne, Washington & Lee University
Abstract
To model knotted springy wires we minimize in prescribed knot classes a total energy consisting of the classic EulerBernoulli 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  socalled elastic knots. For all torus knot classes T(2,b) we established the doublycovered 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.

10:00  10:20 am EDTCoffee Break11th Floor Collaborative Space

10:20  11:05 am EDTThe unknotting number and reinforcement learning11th Floor Lecture Hall
 Speaker
 András Juhász, University of Oxford
 Session Chair
 Elizabeth Denne, Washington & Lee University
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.

11:20 am  12:05 pm EDTThe topology, geometry, and combinatorics of feedforward neural networks11th Floor Lecture Hall
 Speaker
 Julia Grigsby, Boston College
 Session Chair
 Elizabeth Denne, Washington & Lee University
Abstract
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.

12:15  1:45 pm EDTLunch/Free Time

1:45  2:30 pm EDTDNA self assembly and DNA knots11th Floor Lecture Hall
 Speaker
 Nataša Jonoska, University of South Florida
 Session Chair
 Adam Lowrance, Vassar College
Abstract
We present a range of topological questions associated with DNA selfassembly and three dimensional structures. The questions vary from topological graph theory related to DNA strand routing a threedimensional mesh, to questions in knot theory related to structural embeddings in 3D, to algebraic descriptions related to Jones monoids associated with DNA origami.

2:45  3:30 pm EDTUniversal construction and its applications11th Floor Lecture Hall
 Speaker
 Mikhail Khovanov, Columbia University
 Session Chair
 Adam Lowrance, Vassar College
Abstract
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.

3:45  4:05 pm EDTCoffee Break11th Floor Collaborative Space

4:05  4:50 pm EDTMachine learning and hard problems in topology11th Floor Lecture Hall
 Speaker
 Sergei Gukov, Caltech
 Session Chair
 Adam Lowrance, Vassar College
Abstract
I will describe several ways in which tools from reinforcement learning and natural language processing can help with problems in knot theory and lowdimensional topology, including hard challenges such as the smooth Poincare conjecture in dimension 4 and its variants. I will try to make the talk selfcontained; 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.

5:00  6:30 pm EDTReception11th Floor Collaborative Space
Tuesday, May 23, 2023

9:00  9:45 am EDTRandom graph embeddings11th Floor Lecture Hall
 Speaker
 Jason Cantarella, University of Georgia
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
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 1complex and see that the embedding data is (in a precise sense) dual to the simplicial homology of the 1complex. This will enable us to make some new exact calculations for random knots.

10:00  10:20 am EDTCoffee Break11th Floor Collaborative Space

10:20  10:25 am EDTRandom meander model for linksLightning Talks  11th Floor Lecture Hall
 Virtual Speaker
 Anastasiia Tsvietkova, RutgersNewark/IAS
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
We develop a new random model for links based on meanders. Random meander diagrams correspond to matching pairs of parentheses, a wellstudied 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 nonisotopic knots obtained on every step. Then we give expected twist number of a diagram, and bound expected hyperbolic and simplicial volume of links.

10:25  10:30 am EDTThe Jones polynomial of collections of open curves in 3spaceLightning Talks  11th Floor Lecture Hall
 Speaker
 Kasturi Barkataki, Arizona State University
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
Measuring the entanglement complexity of collections of open curves in 3space 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 3space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a welldefined single variable polynomial that is a topological invariant, which, for linktype 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 3space. For collections of open curves in 3space, 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 3space for the first time.

10:30  10:35 am EDTOn the manifold of embedded curves with fixed lengthLightning Talks  11th Floor Lecture Hall
 Speaker
 Elias Döhrer, TU Chemnitz
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
My collaborators (Henrik Schumacher, Philipp Reiter) and I used selfrepulsive 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.

10:35  10:40 am EDTConstructing Annular Links from Thompson's Group TLightning Talks  11th Floor Lecture Hall
 Speaker
 Louisa Liles, University of Virginia
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
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.

10:40  10:45 am EDTThompson link and grid diagramLightning Talks  11th Floor Lecture Hall
 Speaker
 Yangxiao Luo, University of Virginia
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
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.

10:45  10:50 am EDTTrajectory Surfaces of Framed Curvature FlowLightning Talks  11th Floor Lecture Hall
 Speaker
 Jiri Minarcik, Czech Technical University in Prague
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
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 timedependent moving frame. The flow can be defined in such a way that it sweeps out trajectory surfaces of constant mean or Gaussian curvature.

10:50  10:55 am EDTThe Tanglenomicon: Tabulation of two string tanglesLightning Talks  11th Floor Lecture Hall
 Speaker
 Joseph Starr, University of Iowa
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
"The Most Important Missing Infrastructure Project in Knot Theory" Dr. Dror BarNatan 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 2string tangle is an embedding of 2 disjoint stringsegments into the interior of a 3ball.

10:55  11:00 am EDTNaturality of LOSS invariant under contact +n surgery and Legendrian nonsimple knotLightning Talks  11th Floor Lecture Hall
 Speaker
 Shunyu Wan, University of virginia
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
Given a Legendrian Knot L in a contact 3 manifold, one can associate a socalled 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 nonsimple knot.

11:00  11:05 am EDTSymmetric critical surfaces for tangentpoint energiesLightning Talks  11th Floor Lecture Hall
 Speaker
 Axel Wings, RWTH Aachen University
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
In this short talk, we introduce the generalization of tangentpoint 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.

11:05  11:10 am EDTClassifying links in a thickened torusLightning Talks  11th Floor Lecture Hall
 Speaker
 Max Zahoransky von Worlik, Technische Universität Berlin
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
Links in a thickened torus are natural objects of study as quotients of periodic structures in 3space 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.

11:10  11:15 am EDTStable knots in electromagnetic fieldsLightning Talks  11th Floor Lecture Hall
 Virtual Speaker
 Benjamin Bode, Consejo Superior de Investigaciones Científicas
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
An electromagnetic field consists of two timedependent 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.

11:15  11:20 am EDTVariations in writhes for Virtual KnotsLightning Talks  11th Floor Lecture Hall
 Virtual Speaker
 AMRENDRA GILL, HarishChandra Research Institute Prayagaraj, India
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
In the talk we discuss the change that writhes namely odd writhe and nwrithes 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 nwrithes 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.

11:20  11:25 am EDTTangle insertions and linking probabilityLightning Talks  11th Floor Lecture Hall
 Virtual Speaker
 Puttipong Pongtanapaisan, University of Saskatchewan
 Session Chair
 Allison Moore, Virginia Commonwealth University
Abstract
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.

11:25  11:30 am EDTOn a generalization of Jones polynomial and its categorification for Legendrian knotsLightning Talks  11th Floor Lecture Hall
 Virtual Speaker
 Monika Yadav, Indian Institute of Science Education and Research, Bhopal, India
 Session Chair
 Allison Moore, Virginia Commonwealth University
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 ThurstonBenniquin number appears as grade shift in the Legendrian Khovanov homology.

12:10  12:15 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:15  1:45 pm EDTLunch: NetworkingWorking Lunch  11th Floor Collaborative Space

1:45  2:30 pm EDTGeodesically complete Riemannian metrics on the space of knots11th Floor Lecture Hall
 Speaker
 Henrik Schumacher, University of Technology  Chemnitz
 Session Chair
 Sonia Mahmoudi, Drexel University
Abstract
Selfavoiding 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 selfavoiding 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 tangentpoint 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 tangentpoint energies whose energy spaces are Hilbert spaces. Finally, I will sketch a proof for geodesic completeness.

2:45  3:30 pm EDTApplications of Knot Theory to Characterizing Entanglements for Polymers in Tubes or Nanochannels11th Floor Lecture Hall
 Virtual Speaker
 Chris Soteros, University of Saskatchewan
 Session Chair
 Sonia Mahmoudi, Drexel University
Abstract
Motivated in part by recent experimental data for knotting of DNA in nanochannels and nanopores, 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, transfermatrix methods and Monte Carlo computer simulation. For the smallest tube that admits knotting, we prove longstanding 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.

3:45  4:05 pm EDTCoffee Break11th Floor Collaborative Space

4:05  4:50 pm EDTTwisted topological tangles or: the knot theory of knitting11th Floor Lecture Hall
 Speaker
 Sabetta Matsumoto, Georgia Institute of Technology
 Session Chair
 Sonia Mahmoudi, Drexel University
Abstract
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 twoperiodic weft knitted textiles using a construction we call the swatch [1]. Using this construction, we can prove that all twoperiodic 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 DMR1847172, 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 sarahmarie 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.
Wednesday, May 24, 2023

9:00  9:45 am EDTPartial Quadratics, their Pushwards, and Signature Invariants for Tangles11th Floor Lecture Hall
 Speaker
 Dror BarNatan, University of Toronto
 Session Chair
 Van Pham, University of South Florida
Abstract
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

10:00  10:20 am EDTCoffee Break11th Floor Collaborative Space

10:20  11:05 am EDTThe shapes of (tangled) thickened curves.11th Floor Lecture Hall
 Speaker
 Rhoslyn Coles, Technical University Berlin
 Session Chair
 Van Pham, University of South Florida
Abstract
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.

11:20 am  12:05 pm EDTThe Braid Indices of the Reverse Parallel Links of Alternating Knots11th Floor Lecture Hall
 Speaker
 Yuanan Diao, UNC Charlotte
 Session Chair
 Van Pham, University of South Florida
Abstract
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.

12:15  1:45 pm EDTLunch/Free Time

1:45  2:30 pm EDTKhovanov homology and Wall's stabilization problem11th Floor Lecture Hall
 Speaker
 Kyle Hayden, Columbia University
 Session Chair
 Jason Cantarella, University of Georgia
Abstract
In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. Walltype stabilization problems aim to quantify this instability. I will discuss an approach to these problems that uses Khovanovtype 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 Khovanovtype tools for knots and surfaces to calculate Floertheoretic invariants for associated 3 and 4manifolds.

2:45  3:30 pm EDTA tale of two tangles11th Floor Lecture Hall
 Speaker
 Allison Moore, Virginia Commonwealth University
 Session Chair
 Jason Cantarella, University of Georgia
Abstract
A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a twostring 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 BarNatan's tangle invariant in terms of immersed curves on the fourpunctured 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.

3:45  4:05 pm EDTCoffee Break11th Floor Collaborative Space

4:05  4:50 pm EDTA threepronged approach to using machine learning in knot theory11th Floor Lecture Hall
 Speaker
 Mark Hughes, Brigham Young University
 Session Chair
 Jason Cantarella, University of Georgia
Abstract
With recent breakthroughs in AI like ChatGPT and DALLE 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 lowdimensional topology. These applications will range from approaches that have already been successfully implemented to more speculative worksinprogress.
Thursday, May 25, 2023

9:00  9:45 am EDTDesigned selfassembly of molecular knots, links and topological gels11th Floor Lecture Hall
 Speaker
 Cristian Micheletti, International School for Advanced Studies (SISSA)
 Session Chair
 Danielle O'Donnol, Marymount University
Abstract
Supramolecular constructs with complex topologies are of great interest across softmatter 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 selfassembly. Specifically, by controlling the geometry of the selfassembled 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, 931935 (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 handson demonstration of the designed selfassembly of ""macroscopic"" trefoil knots see the video at this link: www.youtube.com/watch?v=XKsuMlp2PLc

10:00  10:20 am EDTCoffee Break11th Floor Collaborative Space

10:20  11:05 am EDTDynamics of DNA knots with excess link11th Floor Lecture Hall
 Speaker
 Abby Pekoske Fulton, Worcester Polytechnic Institute
 Session Chair
 Danielle O'Donnol, Marymount University
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.

11:20 am  12:05 pm EDTComputational Complexity of Finite type Invariants11th Floor Lecture Hall
 Speaker
 Nancy Scherich, Elon Univeristy
 Session Chair
 Danielle O'Donnol, Marymount University
Abstract
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.

12:15  1:45 pm EDTLunch: Open Problems SessionWorking Lunch  11th Floor Collaborative Space

1:45  2:30 pm EDTStable Khovanov homology of torus links and volume11th Floor Lecture Hall
 Speaker
 Christine Ruey Shan Lee, Texas State University
 Session Chair
 Clayton Shonkwiler, Colorado State University
Abstract
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 ChampanekarKofman 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 JonesWenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by ChampanekarKofman, extending the results by Stosic and Rozansky. Since the JonesWenzl 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.

2:45  3:30 pm EDTA Bit More on Chirality of Crooked Curves11th Floor Lecture Hall
 Speaker
 Rob Kusner, University of Massachusetts, Amherst
 Session Chair
 Clayton Shonkwiler, Colorado State University
Abstract
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 tracefree, so if a curve has a righthanded twist about some axis, there is also a perpendicular axis about which the twist is lefthanded. 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 higherdimensional — 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/longgoing project with Giovanni Dietler, Wöden Kusner, Eric Rawdon, and Piotr Szymczak — a bit was conducted at ICERM].

3:45  4:05 pm EDTCoffee Break11th Floor Collaborative Space

4:05  4:50 pm EDTOn alternating knot types and their diagrams11th Floor Lecture Hall
 Speaker
 Claus Ernst, Western Kentucky University
 Session Chair
 Clayton Shonkwiler, Colorado State University
Abstract
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 2bridge knots are curly. Finally, we report on a new result by Y. DIao on the ropelength of alternating knots.
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