Speakers

Group equivariant non-expansive operators (GENEOs) are mathematical tools for approximating data observers when data are represented by real-valued or vector-valued functions (https://rdcu.be/bP6HV, https://www.nature.com/articles/s41598-025-90152-7). These operators focus on the assumption that data interpretation depends on the observers' geometric properties. In this talk, we will show how GENEOs can be used to locate pockets in proteins.

The scientific enterprise enriches the debate about models. In particular, in the field of structural biology, the deep-learning neural network system called AlphaFold has been applied for many purposes. It allows us to predict a protein’s structure with high accuracy. I will present the system's output, namely the proteins' models, in light of the discussion of structure representation and argue for a specific kind of representational relation holding between the predicted model structure and its target-system. By doing so, I will criticise the artifactual approach advanced by Knuuttila (2021) and present the features that characterise the predicted structures of AlphaFold as representational models.

In 1970 John Conway created a notion of a "tangle". Tangles can be seen as a building blocks of knots and are a basis of knot tabulation. Using tangles Conway managed to tabulate all knots up to 11 crossings, and links up to 10 crossings. Moreover, he found a bijection relation between rational numbers and a subclass of tangles called "rational tangles", thus fully classifying this subset of algebraic tangles. Over 50 years later we extended Conway work. We introduced a binary tree notation, which helped us to define a unique "canonical representation" for algebraic tangles. This let us classify algebraic tangles up to 14 crossings. Moreover we studied symmetry groups of algebraic tangles. During the talk I'll introduce you to the theory of tangles and present our results.

We discuss representation of knots and links by piecewise linear embeddings in the 3-space. All tame links admit such presentations, that are naturally suited for computer implementation and manipulation. We present the computation of linking number of polygonal links and describe methods to provide other knot invariants from polygonal representations.

This talk discusses logical and topological interpretations that are at the basis of the structure of continued fractions, their arborescent generalizations, arborescent links and knots, their invariant polynomials, the Temperley Lieb algebra and the interpretations of these invariants in terms of state sums and link homology. Consider the calculus of indications as generated by the mark <> and satisfying the arithmetic rules < > < > = delta <> and <<>> = = E, an identity operator , where the space before the comma is an empty word. Add to this arithmetic a new entity X so that <X>=X and XX = E. X is a “nilpotent” self-referential element in the form. We will show that this crossing arithmetic is at the bottom of a deep topological well. The iconic relationship with topology is that if <Z> denotes the ninety degree turn of the mirror image of Z (when Z is a tangle), then <X> = X when X is a crossing. Iconic logic becomes parity calculus for counting components and detecting component changes in tangles and tangle closures. The talk will illustrate how such parity calculus applies to arborescent weaving patterns and their associated invariants.

Knotted topologies in biopolymers and proteins are a fascinating yet challenging phenomenon with growing relevance in structural biology and chemistry. These complex molecular shapes are increasingly recognized for their potential roles in biological function, but traditional mathematical tools often fall short in detecting and classifying them due to knot theory limitations. Recent advances in machine learning, particularly in sequence-based models like LSTMs and attention mechanisms, offer new possibilities for analyzing non-trivial topologies. Using only 3D structural data, ML models can learn to recognize non-trivial topologies, like knots, theta-curves or lassos, showing the possibilities of making the process easier and faster than classical algorithmic methods.

This talk will discuss the notion of link homotopy, which was first defined by John Milnor in his seminal paper "Link Groups" in 1954, and has been steadily studied since then. Defined for classical links, it is often colloquially described as "linking modulo knotting", as it tries to capture the linking interactions between different components of a link while leaving out the knottedness of each individual component. Milnor himself defined the classical invariants for link homotopy, the link group and the "higher-order linking number" numerical invariants. Since then, many generalizations, new invariants and different viewpoints have been introduced. This talk will explore some of them, along with potential future results.

An open problem in knot theory is whether there exists a universal knot invariant capable of uniquely identifying the topology of any given embedding of S^1 in 3-space. This talk will cover a recent advance using particular features of a given embedding and machine learning techniques to identify knots of up to 10 crossings with up to >95% accuracy.

In this talk we discuss how new advances in knot theory lead to novel metrics of structure of biopolymers that are continuous functions in the space of configurations. We apply our methods to proteins and show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data and is complementary to existing approaches. In particular, our methods enable the definition of the Topological Landscape of Proteins and its quantitative and qualitative analysis as a continuum. The methods are general and apply to quantify structural complexity and compare structures in different contexts, such as the SARS-CoV-2 Spike protein and tau protein. We employ the writhe and Local Topological Entropy (LTE), to investigate the evolutionary dynamics of proteins. Notably, we find the evolutionary behavior of thierodoxin to be well correlated with LTE of the native state. Furthermore, our results reveal a strong correlation between these topological parameters and established dynamical measures, such as the Dynamical Flexibility Index (DFI), indicating for the first time that the local topology/geometry of the native state reflects the dynamical landscape of a protein. These results suggest that these topological metrics could serve as valuable reaction coordinates, bridging the gap between protein structure and dynamics.

Twist-fold processes are ubiquitous in nature and are often governed by energy relaxation mechanisms. In this talk we show how the Reidemeister twist move associated with an inflexional instability of a physical knot leads to an energy change of the system, sometimes with dramatic effects. Three different cases are considered: when magnetic energy is released, when elastic energy is stored, and when tension energy is re-distributed through the topological re-organisation of a soap film.

Khovanov homology is a link invariant introduced by Mikhail Khovanov as a categorification of Jones polynomial. It detects the unknot and provides geometric and topological information about knots (for example, it gives a lower bound for the 4-genus and detects fiberedness among positive links). Computing Jones polynomial (and therefore Khovanov homology) is an NP-hard problem. However, Morton and Short proved that, when fixing the number of strands of a braid, computing the Jones polynomial of its closure can be done in polynomial time with respect to the length of the braid. In this talk we present a conjecture on an equivalent statement for Khovanov homology, and introduce some results supporting it. On the way, we will work with (a simplified version of) Khovanov spectra, a link invariant introduced by Robert Lipshitz and Sucharit Sarkar as a refinement of Khovanov homology.

We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots – closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots. In the rigid case, evaluating a bonded knot in the basis of this module yields an bonded knot invariant closely related to the APS bracket and the Simplified RNA polynomial.

Viral RNAs fold into more compact structures than equivalent random RNA having the same length and composition. Evolutionarily, their compactness has been connected to an evolutionary pressure to fit the RNA within the virion; structurally, it appears to be caused by a higher degree of branching in viral RNAs. To study these connections, it is generally assumed that on a large enough scale the folds of long RNAs can be mapped onto trees, and their physical size connected to some properties of the trees, from their diameter (Maximum Ladder Distance -MLD) to topological indices for branching. Comparing RNA folds of different lengths directly however, remains problematic. This is a general problem in chemistry: to compare the topological problems of two branched molecules one needs to somehow map them to trees, and then connect some topological indices to their physical properties. Topological indices however are strongly dependent on the tree size, which in turn depends both on the molecular weight and the mapping procedure. To alleviate this, we introduced a novel normalization of topological indices using estimates of their probability density functions. We determine two optimal normalized topological indices and construct a phase space that enables a robust discrimination between different architectures of branched macromolecules. In this talk I will discuss this approach in the specific setting of long RNAs.