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.

Kasturi Barkataki Linkoids, Parallel Algorithms, and Machine Learning the Jones Polynomial of Open Curves

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.

Marithania Silvero Khovanov homology, closed braids and complexity

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.