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Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups. Suppose $V$ is a dualizable object in a braided monoidal category $(C, \otimes)$. This means, among other things, that $V^{\otimes n}$ acquires an action of the braid group $B_n$, and ...


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One of the earliest appearances of the ingredients for a skein relation can be found in Romilly Allen's 1904 book on Celtic Knotting. There he explains that designers of Celtic knot patterns first start with a regular weave and then systematically replace some of the crossings by "horizontal" and "vertical" smoothings. These days, it is easy to find working ...



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