Commutative monads, diagrams and knots

There is certain diverse class of diagram that is found in a variety of branches of mathematics and which all share this property: there is a common scheme for translating all of these diagrams into useful functional code. These diagrams include Bayesian networks, quantum computer circuits [1], trace diagrams for multilinear algebra [2], Feynman diagrams and even knot diagrams [3]. I will show how a common thread lying behind these diagrams is the presence of a commutative monad and I will show how we can use this fact to translate these diagrams directly into Haskell code making use of do-notation for monads. I will also show a number of examples of such translated code at work and use it to solve problems ranging from Bayesian inference to the topological problem of untangling tangled strings. Along the way I hope to give a little insight into the subjects mentioned above and illustrate how a functional programming language can be a valuable tool in mathematical research and experimentation.