This one-day conference brings together both Dutch and international topologists and knot theorists with interests in applications of these areas to molecular and material design. The conference will showcase foundations and new theoretical developments in topology and knot theory with actual or potential relevance for MMD. In addition to the speaker presentations, there will opportunities for informal knowledge exchange between speakers and participants and between theorists and experimentalists.
Several of the invited speakers have experience applying their theoretical tools in the areas of molecular and materials design. We invite participants to bring structural questions about molecular and material design to share during the round table discussions. We aim to pair laboratory questions with theoretical tools to advance research in both areas.
Location: Auditorium of the Matrix ONE building, UvA Science Park
Registration is free, but we kindly ask all participants to register as soon as possible so we can give accurate information to the caterers. Please register by following this link.
This conference is organized by Jo Ellis-Monaghan and Wout Moltmaker, and is supported through the KdVI and MMD TechHub.
Abstracts:
Roland van der Veen - The Theta invariant of knot(oid)s
After briefly recalling the topological meaning of the Fox matrix and its relation to the Alexander polynomial of knots we will show how by inverting this matrix one gets a much stronger invariant that we call Theta. Theta is a two variable polynomial that seems to have many attractive symmetries and relations to topology waiting to be explored. Unlike most quantum invariants one can actually compute Theta for knot diagrams of over a hundred crossings. As an illustration of its strength, Theta distinguishes two (previously intractable) pairs from the table of unresolved 5 crossing prime knotoid pairs (Table 6.1 in Wout's thesis).
A handout for this talk will be available at http://www.rolandvdv.nl/Talks/Wout25/.
This is joint work with Dror Bar-Natan.
Sofia Lambropoulou - From Planar to Toroidal to Doubly Periodic Knottings
We present a topological approach for studying knotted structures in the thickened torus and their invariants, via the transition from the planar objects and the annular ones and via their planar mixed representatives. We then consider the universal cover of the torus for studying doubly periodic entanglements. Our main diagrammatic categories of focus will be those of pseudo knots and knotoids.
(Joint works with Ioannis Diamantis, U Maastricht, and Sonia Mahmoudi, Tohoku U)
Neslihan Gügümcü - Algebraic invariants of linkoids
In this talk, we discuss knotoids and linkoids, and a number of invariants for them. Specifically, we present a new invariant for linkoids introduced by myself and Runa Pflume, so called the fundamental pointed quandle of an n-linkoid. We show that the fundamental pointed quandle serves as a a stronger invariant than the fundamental quandle of linkoids. We also cover some results on topological analysis of proteins via knotoids and discuss possible applications of linkoids.
Senja Barthel - Studying crystal nets for materials discovery
A molecule can be described by its molecular graph. Similarly, the bond structure of a crystal can be described by a spatial infinite periodic graph. The study of this so called net, in particular its isomorphic type (called net topology), symmetry, and tilings of the surrounding space that it can be obtained from it, allows for insight into structure-property relationships of crystalline materials. I will give some examples of material discovery projects that are based on the study of crystalline bond networks.
Iain Moffatt - Why are graph polynomials knot theory weight systems?
In knot theory “weight systems” are linear functionals on chord diagrams. Their significance arises from the Vassiliev-Kontsevich Theorem which gives that every Vassiliev knot invariant determines and is determined by a weight system. There is a method for constructing weight systems from Lie algebras and many important and familiar knot invariants arise in this way. For example, the Jones polynomial is determined by a weight system coming from sl(2).
Turning to combinatorics, a “graph polynomial” is a polynomial-valued invariant of graphs. Over the last few years there has been interest in constructing weight systems from graph polynomials. For example, Chmutov showed that Gross, Mansour, and Tucker's ”partial dual genus polynomial” gives a weight system, and Kodaneva and Netrusova showed that Arratia, Bollob\'as and Sorkin's “interlace polynomial” is a weight system. In both cases, the proofs rely on direct combinatorial analysis.
In this talk I will explain how it follows from a quick application of some results of Bar Natan that various common graph polynomials are Lie algebra weight systems. In particular, I will resolve a question of Chmutov by showing that the partial dual genus polynomial is a Lie algebra weight system.
I will keep the exposition gentle and won't assume prior knowledge of graph polynomials, knot theory, weight systems or Lie algebras.
Jasper Stokman - Quantum invariants and q-deformed CFT
The Reshetikhin-Turaev invariants of ribbon-links involve colorings of link diagram components by objects from a ribbon category. The ribbon structure of the category is used to assign morphisms to elementary events in the link diagram, like crossings. In case of the ribbon category of finite dimensional quantum group representations, one obtains the colored Jones polynomials as quantum invariants.
In this talk I will present a recent program aimed to extend these quantum topological techniques to the context of q-deformed conformal field theory. This requires colorings by highest weight representations of quantum groups ,as well as by dynamical braidings. This is joint work with Hadewijch De Clercq and Nicolai Reshetikhin.
Louis H. Kauffman - Knot Logic and Arborescent Links
We discuss an algebraic method for finding the number of components of an arborescent link and for any of its smoothings. We prove, using this method that the rational link with reduced fraction P/Q has one component if and only if P is odd and two components when P is even, a result that is surely known to those who know it. This method leads to algorithms for computing invariants of arborescent links. That our algebra yields component counts leads to questions about bracket states and Khovanov Homology. We discuss relationships of our formalism with points of view about logic and multiple valued logic, and problems in knot theory that are at the interface between combinatorics and topology.
Wout Moltmaker - Invariants of Planar Knotoids and Linkoids
Knotoids and linkoids are natural generalizations of knot and link diagams, in which we allow for open-ended components as well as the usual closed knotted curves. This generalization is distinct from e.g. braids since the open ends are allowed to lie anywhere in the diagrams, also in interior regions. Knotoids and linkoids can be applied to many settings involving open-ended tangled structures, and have notably been applied to the study of protein topology. However, knotoids have proven difficult to classify even for low crossing numbers, and in particular knotoids drawn on the plane are relatively poorly understood. In this talk I will give a summary of some of the work I did on knotoids and linkoids, particularly planar ones, during my PhD. I will briefly introduce knotoids and describe how they are applied to study proteins and other systems of tangled open curves. Then I will discuss several invariants of (planar) knotoids and linkoids, defining new invariants and giving new results for existing ones. These include polynomial invariants, as well as quandle- and quantum invariants.
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