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## Knots and entropy project (joint with Rob Sturman)

How might we prevent this with better cable design?

This is a project about applying maths to solve real world problems.
Real world problems of a certain broad type. (Not necessarily headphone cable tangles, but perhaps something analogous; or something related, like how to achieve good washing results by a good design for a washing machine, or how to minimize the risk of umbilical chord tangling in the womb* ...)

So, the first question is, what kind of challenge are we going to try to apply maths to? -- For definiteness let's think about the cable design problem.
In which case the _aim_ is to design a better (less tangly) cable.
(The project need not actually achieve this aim, but it should try to take a helpful step in the right direction.)

This challenge is going to start with the problem of understanding _why_ the cable tangles.
And perhaps even before that, the problem of establishing exactly what it means to say that the cable tangles!
(It might be a good idea for a project student or team working in this area to begin by getting some quantitative information on real cable tangling -- that is, by doing some controlled experiments with real cables. The exact details of the experiment are not so important here. The idea is just to get a feel for what really happens when a cable tangles.)

The next question is: how do we apply maths to help us with this challenge?!
This is the problem of ``mathematical modelling''.
A good place to look for clues is in the way maths already models knots and tangles -- in the Theory of Knots and Tangles (which already exists! See e.g. the book Introduction to Knot Theory by R H Crowell and R H Fox).
Then we can see if this existing maths can be adjusted to suit our cable modelling problem.

Next we will need to think about what is happening physically (i.e. mechanically) with our cable? ...But let's leave that until we've thought about how to model the cable `statically'.

Project themes:

• What are mathematical knots and tangles? That is, what is the formal definition of a mathematical knot?
• In what ways are they similar or different to real (static) knots?
• How can they be worked with mathematically?
• How can their "pictorial" representation be used in rigorous calculations?
• How do they connect with their applications?
• What are the dynamics, the mechanics, of real knots? How can _these_ be modelled?
• What is entropy and how can it help us?
• ...

Keywords: knots, tangles, energy and entropy

References:

• Introduction to Knot Theory by R H Crowell and R H Fox here!
• Geometric Topology in Dimensions 2 and 3 by E E Moise