I would like to know a bit about applications of the Alexander Theorem from Knot and Braid Theory. I would be very interested in learning about possible applications for the description of everyday objects and phenomena and, above all, about eventual usages for computing, formal languages, language processing and the like.
Thanks in advance.
Generally speaking, trying to fish around for practical applications of a specific mathematical theorem or theory --- such as Alexander's Theorem from knots and braid theory, or surgery theory from your previous question --- does not usually work out well. It's sort of like asking which diseases can be cured using a hammer.
That's not to say that mathematical theories do not have practical applications. In fact it happens quite commonly that a scientific question leads to development of a new mathematical theory, or refinement of an existing mathematical theory, for purposes of applying that theory to the question. Newton's invention of calculus to solve questions about gravity is usually the prime example. Chomsky's development of a hierarchy of linguistic structures, and Turing's development of a theory of computing, which are perhaps closer to your interests, are further good examples.
Mathematics does work extremely well as a tool for solving specific problems. But the problem solving process is not about flipping through a card catalog of theorems to find a match between theorem and problem. Instead, the process is about knowing how to develop the mathematics that you need to address the problem at hand.
