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I'm wondering if there's compiled literature on well-known algorithms and their bounds for computing various knot invariants (I'm writing a Master's thesis on the subject).

I can find individual results and papers on, say, the Jones polynomial (eg polynomial quantum algorithms in the general case or faster ones in specific cases), and reading through the source code of mathematical packages like the Knot Theory one in SageMath also yields helpful references, but I was wondering if there are survey papers in this respect, compendiums, seminars, really anything that will yield direct and explicit references in this area (and the more up-to-date the better).

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    $\begingroup$ I don't know if this is what you had in mind, but the following paper by Clément Maria gives an algorithm and some complexity bounds for computing link invariants from ribbon categories in general (and from quantum groups in particular) : arxiv.org/abs/1910.00477. $\endgroup$
    – Adrien
    Commented Mar 20, 2023 at 19:59
  • $\begingroup$ @Adrien I actually already know of this paper (came across it in this presentation along with other useful references I've been hunting down) but do appreciate your linking to it, it's exactly the sort of thing I'm looking for and in particular I'm going to use it in my section on the Jones and HOMFLY polynomials. It'd be fantastic if I could find more presentations and papers like these ones on other knot invariants. $\endgroup$
    – Nobilis
    Commented Mar 20, 2023 at 20:49

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There's a survey paper by Marc Lackenby on computing 3-manifold invariants. He discusses many cases of how these algorithms apply to knot theory.

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