**Leader: Antti Niemi, Uppsala University, Sweden**

**Co-leader: Simon Copar, University of Ljubljana, Slovenia**

**General overview. **The mathematical definitions of knots, links and networks are of a purely topological nature, in that different geometrical realizations of these entanglements constitute equivalent classes under a particular set of transformations, called ambient isotopies. The geometric details of the particular realization of these objects are irrelevant and the classes can be identified by using some topological invariants such as the Jones and HOMFLY polynomials. Hence, mathematically no knot, link or network can be defined on finite open strings, as these can be transformed continuously to a set of straight lines. This is clearly at odds with what is observed in physical systems where knots in linear strings – like shoelaces or earphones cables – are not only possible but even abundant. To overcome this impasse, physicists have started investigating possible ways to define physical knots and links. These include closure schemes to circularize linear chains, thus allowing the evaluation of standard polynomial invariants, as well as the definition of novel mathematical entities, e.g. virtual knots. Nevertheless, there are obstacles to the identification of the topological state when the physical structure of a knot is either loose or highly entangled, due to the uncertainty related to closure schemes and the computational time required to evaluate polynomial invariants. In the field of experimental biology, the progress in the detailed probing of chromosome arrangement is calling for new theoretical ways to characterize the inter-chain entanglement in dense solutions of linear polymers, so as to better understand and predict the resulting kinetic, rheological and mechanical properties, which all come into play in the cell cycle. The recently introduced concept of physical link holds much promise for becoming, together with the classic primitive path analysis, the method of choice for such quantitative profiling of inter-chain entanglement. Finally, a fundamental area that has barely been broached is the topological characterization of branched graphs, which naturally arise in the network of chemical bonds in cross-linked polymers. While research on the effect of knotted topology in polymers and other soft matter systems is well under way, the generalization of such concepts to branched topological networks is still in its infancy.

**General objectives of WG****1****. **A unification of the various aspects of topological entanglement in soft matter, chemistry and molecular biology will only be made possible by advances in the study of the underlying theoretical problems. These include the definition and characterization of physical knots, links, branched graphs and networks. Such an effort will require a combination of expertise in formal mathematical topology and the foundations provided by experience in physical and biological systems. The following points are particularly urgent for the development of the field.

- The systematic development, implementation and application of efficient methods to profile the physical knotting and linking of long polymers in various types of spatial confinement.
- The introduction of new, topologically inspired definitions and algorithms to characterize physical entanglement will be accompanied by the design of a suite of computational utilities and data structures to be used by the whole community, to facilitate the analysis and exchange of data.
- The great majority of the research in this field has been performed numerically due to the great difficulties of introducing non local objects such as knots and links in the framework of the field theory used to describe fluctuating filaments. The Action will develop novel methods to overcome this limitation.
- While the entanglements of finite loops (knots and links) have been explored from various perspectives within knot theory, the properties of weavings and networks (where nodes are shared by multiple filaments) are just beginning to be investigated. It is of fundamental scientific interest to develop tools to classify, quantify and examine entanglements of periodic weavings and nets.

**Specific tasks of WG****1****. **This WG will cover all aspects of the formal description of topological entanglements in physics, chemistry and biology. WG1 will focus on the following tasks:

- Extend the mathematical definitions and tools used to identify knots, links, and networks in order to accurately characterize their physical realizations. This will impact the productivity of all WGs.
- Develop novel mathematical techniques to explore possible knotting and entanglement of more complicated spatial structures, with a view to improved characterisation in physical systems.
- Investigate the possibility to develop novel field-theoretical models describing knotted and linked rings (in collaboration with WG2, WG4, WG5).
- Characterize the emergence of knots in the phase diagram of simple polymer models with an increasing level of sophistication, in order to provide well defined guidelines to future experiments in biopolymers such proteins and DNA (with WG2, WG3, WG4).
- Devise a standard file format to transfer data regarding topological objects such as knotted and linked molecules, networks, etc. between WGs. Produce a set of computational tools and web-tools to facilitate the analyses of other WGs and their collaboration.

**Deliverables**. STSMs within the WG and to other WGs. At least one article with 3 international members per each task. Standardized file format. Suite of analysis software.