Knot Theory and Low-dimensional topology

The key topics of this mini-symposium will be devoted to links and braids, invariants of knots and links and related topics such as three-dimensional manifolds, quantum invariants, combinatorial invariants, link homology, braids, skein modules and knot algebras. Focus will also be put on applications in areas of science such as long chain molecules and molecular biology, molecular chemistry, textiles and meta-materials. From knotoids and linkoids to the real world of physical entanglements, the mathematical tools of knot theory become essential to describe natural phenomena around us.

Tara Brendle The Burau representation (1:30 pm)

We will survey known results about the faithfulness question for the Burau representation of the braid group.  This will include a brief discussion of recent work joint with Vasudha Bharathram and Joan Birman.

Kasturi Barkataki Linkoids, Parallel Algorithms, and Machine Learning the Jones Polynomial of Open Curves (2:00 pm)

The computation of topological complexity for entangled closed and open curves becomes rapidly intractable as the complexity of their corresponding link or linkoid diagrams increases, limiting its applicability to large-scale datasets arising in physical and biological filamentary systems. This challenge motivates the development of both new theoretical frameworks and scalable computational approaches for analysing entanglement. We build on the theory of linkoids, multi-component generalisations of knotoids, and describe how invariants such as, the Jones polynomial, can be extended to open curves via a linkoid spectrum, providing a closure-independent formulation of topological complexity. This perspective further enables a parallel algorithm for the exact computation of the Jones polynomial, in which a diagram is decomposed into simpler constituent open pieces (linkoids) that can be evaluated independently and recombined through a gluing procedure. While this approach significantly improves computational efficiency, large datasets and highly complex systems remain challenging. To address this, we introduce a machine learning framework to predict coefficients of the Jones polynomial directly from geometric representations of open curves. By learning from data, the model provides fast approximations of topological complexity and extends to predicting related invariants and entanglement measures. This enables the classification of open curves into similarity classes and suggests a scalable, data-driven approach to studying complex entanglement, as well as a new perspective on understanding knots through their local structure.

Lucrezia Beatrice Lorenzi  Bonded links and related algebraic structures (2:30 pm)

In a recent work, Diamandis, Kauffman, and Lambropoulou introduced bonded links: a modern structure in the realm of knot theory, consisting of classical links endowed with embedded arcs with endpoints on the link, called bonds. Analogously to the classical Alexander theorem and the standard closure of braids, which relate classical links to the braid group (on n strands, B_n), in the suitable topological context their algebraic behaviour is described by the monoid of bonded braids (on n strands, BBM_n). The existence of a map BB_n \to B_n, which sends bonds to Artin pure braid generators A_ij, sheds further light on the nature of bonds. Based on a joint project with Sofia Lambropoulou.

Sofia Lambropoulou TBA (3:30 pm)

Alessia Cattabriga "Plat-like" representations of links in 3-manifolds (4:00 pm)

In this talk I will present two different representation for links in 3-manifolds via surface braid groups: use uses (orientable)  Heegaard splitiing and the other one-sided (non-orientable) Heegaard splittings and describe similarities and differences among them.

Marithania Silvero Khovanov homology, closed braids and complexity (4:30 pm on-line)

Khovanov homology is a link invariant refining the celebrated Jones polynomial. It provides geometric and topological information about knots (such as detecting the unknot and bounding the 4-genus). Although its definition is conceptually simple, computing Jones polynomial (and therefore Khovanov homology) is NP-hard. However, Morton and Short proved that, for a fixed number of braid strands, computing the Jones polynomial of its closure takes polynomial time with respect to the braid length.