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The category of tangles

Tangled Web

I want to get back to discussing tangles. So far we’ve been thinking about tangles entirely topologically. But as it turns out, tangles are also fundamentally algebraic objects. The algebraic gadget we need to understand tangles is that of a free ribbon category. Indeed, Shum’s theorem states that framed, oriented tangles form the morphisms of a free ribbon category on a single generator.

To begin to understand this deep statement we must start with the definition of a category. A category is a set of objects $latex A,B,C,ldots$ along with a class (for technical reasons a class, not a set) of morphisms $latex f,g,h,ldots$. Each morphism has a source object and a target object so that we can think of a morphism as an arrow $latex Bleftarrow A$. There is a composition operation of morphisms $latex gf$ which is defined only if the source of $latex g$ is the target…

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