# School of Mathematics

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## Knots at Leeds

### Disciplines and synergies

Structures and propertiesIn many physical applications, such as textiles, molecular biology, polymer physics and chemistry, the behaviour and the performance characteristics of structurally complex objects depend on the physical properties of the constituent elements, and the structure in which those elements are arranged. The topology of that structure often affects its behaviour. For example, polymeric fluids consist of chain-like molecules arranged randomly which become entangled under flow; in supramolecular chemistry, molecules or assemblies with a complex knotted structure can be controlled, leading to molecular switch and electronics applications; in textiles, physical properties depend on how the constituting fibrous elements are knotted or entangled. Understanding the consequences of the knotted, tangled, or braided topology in each case provides insight into the other areas of applications. For example, random textiles have similarities with polymer solutions and biological molecules. Across this wide range of applications lies the fundamental question of how robust a knot is to changes in the immediate environment, and such a question of recognition and classification has a universal, mathematical flavour: given two abstract representations of a knot or link, can we determine whether these represent the same knot?

Recognising knots

One mathematical strategy is to associate a computable number to each representation, and show that the same underlying knot leads to the same number. This idea continues to drive developments in pure knot theory, representation theory and analysis, and particularly in crossover areas of homological algebra studied at Leeds. It is a fundamental problem to define and calculate such invariants, whether simple diagram-colouring techniques, polynomial invariants, or knot homology. Knot invariants have immediate applications in topological quantum computation, for example, which aims to identify quantum states of matter, or describing behaviour such as polymer and DNA physics. Recognising knots is also a fundamental task in qualitative spatial reasoning. Naturally classification becomes more challenging as knots get more complicated, but an increase in complexity also inspires theory appropriate to dynamic, framed, or constrained knots.

Complexity of knots

How complicated a knot is may be described in a variety of ways. A notion which spans several different areas is that of complexity. Understanding topological complexity in static knots has wide-ranging applications, from liquid crystals, to molecular structure of DNA, to magnetic fields in plasma. One common approach is to ascribe energy to knots, such as rope-length energy, O'Hara's electrostatic energy, and Kirchhoff's elastic energy, and such measures provide useful bounds on topological complexity. The topological complexity, or entropy, of dynamically changing knots and braids can produce chaotic advection in devices designed to mix fluids in applications including food science, DNA hybridisation and polymers.

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