Linear fibres are fundamental building elements in the biological world: following the same principles from the molecular to the visible level, nucleic acids, amino acids, proteins or polysaccharides associate into polymers and bundles which are organized to form the macroscopic world around us. Fascinatingly, and unbeknownst to many, these nano- and microscopic fibres can become entangled, form knots and links, and be weaved much like macroscopic ropes and threads. Cell- and micro-biology provide many examples. Knots and links have been found embedded in the structure of several proteins; although their function here is far from understood, the topological state of the polypeptide chain represents a novel degree of freedom to employ in the engineering of artificial enzymes for pharmaceutical and industrial applications. The mitochondrial DNA of kinetoplastida, a group of flagellated protists, is formed by thousands of mini-circles linked to one another to form a large interconnected network, which is a primary pharmaceutical target in the treatment of some important tropical diseases. The stratum corneum, which forms the protective layer of mammalian skin, is a woven network of fibres. In other cases, the absence of knots and other entanglements is itself the manifestation of an underlying organization. That is the case of DNA in the cell nucleus: this can be naively seen as a very thin 2 metres-long rope “stuffed” into a box with linear dimensions of the order of 10 micrometres. Clearly, such a situation should generate many knots and entanglements, which nonetheless must be prevented by cells in order to function and replicate. A similar situation is to be found outside the realm of living matter. In recent years, in fact, great interest in the design of next generation materials has emerged: thanks to advancements in computational as well as in experimental techniques, physicists, chemists and engineers have been able to demonstrate the impact of knots, links, networks and other global entanglements in systems ranging from knotted molecules to artificial polymers and gels. Knotting and entanglement of defect lines in the orientational fields of liquid crystals and flow fields of simple fluids were also identified.

In all the cases presented above, the types of entanglement can be characterised in terms of the kind of knots, links, or textures. These properties are independent of the specific geometrical details of the system and can be formally defined within the context of their abstract “topological” state, i.e. the state that is conserved upon smooth deformations. It appears thus evident that topology underlies two fundamental challenges of current research. i) Understanding the architecture, function, and evolution of complex, self-assembling biological structures ranging from proteins to chromosomes and supra-cellular networks; and ii) designing novel soft materials with tunable physical and functional properties.
Unfortunately, the deeply interdisciplinary nature of the problem as well as the presence of economical and geographical barriers has led to the formation of a series of scattered, mostly disconnected research groups all over Europe. It is therefore the aim of this Action to overcome these barriers, creating an interdisciplinary network of researchers encompassing the fields of physics, chemistry, molecular biology, mathematics and engineering. Bringing together this complementary expertise will result in a substantial boost to the capacity of the European scientific community to investigate the fundamental connection between the topological state of a system and its behaviour, properties, and function.