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I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd like to get a different perspective on certain topics.

I would be glad to get some recommendations on books dealing with knot invariants, in particular the Arf invariant and the Alexander polynomial (I've noticed there are several approaches to define the former).

Thanks in advance!

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Have a look at Quantum Groups by Kassel: it contains, among others, modern algebraic structures (like tensor categories with additional data) related to knots, braidings and links. This is the content of Part III, where isotopy invariants of knots and links in the 3-dim. Euclidean space are introduced.

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I really like Quantum Invariants by Tomotada Ohtsuki. Unlike Kassel (mentioned in another answer), whose book is largely self-contained, Ohtsuki reads more like an encyclopedia of ideas in the intersection of representation theory, knot theory, and diagram algebra. The bibliography is essential, and fortunately, quite substantial, for when you want to dig deeper.

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