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Can we find the minimal crossing number for a given knot or prove that a given diagram or code for a knot is the one with the minimal number of crossings?

For the 8-knot it always has at least four crossings. Can that be generalized for any knot or any diagram or knot code, that the diagram of this particular knot implies that the knot has a minimal number of crossings?

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Knot tables traditionally give (prime) knots according to their minimal crossing number. The way this works is that it's relatively easy to enumerate all knot diagrams for a given crossing number, so supposing you have a table for all knot types up to $n$ crossings, you can add in the knot types for $n+1$ crossings by enumerating the diagrams and throwing out all the ones that are equivalent to one in the table so far (this can take a fair amount of work).

Sometimes you can tell immediately whether a knot diagram has minimal crossing number. For example, diagrams of alternating knots (or more generally adequate knots) that are reduced have minimal crossing number (this is the first Tait conjecture, proved in 1997 using the Jones polynomial by Kauffman, Murasugi, and Thistlethwaite). Reduced means there are no crossings along which you can twist half the diagram to remove -- or in other words, there is no crossing such that there are fewer than four distinct regions of the plane around it.

Your example of the figure-eight knot is an alternating knot, and any four-crossing diagram of it you might have in mind is reduced, so four is the minimal number of crossings in any diagram of the knot.

As it happens, knot tables are organized such that for a given minimal crossing number, the prime knots are enumerated in the following order:

  • alternating torus knots
  • the remaining alternating knots
  • non-alternating torus knots
  • the remaining non-alternating knots

For eight-crossing knots, $8_1$ through $8_{18}$ are alternating knots (none torus knots), $8_{19}$ is a non-alternating torus knot, and $8_{20}$ and $8_{21}$ are non-alternating knots. This is using the Alexander-Rolfsen-Briggs notation -- as far as I know the subscript has no meaning other than the index of the knot in the table.

By the way, this is a useful resource for knot data: https://knotinfo.math.indiana.edu/

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  • $\begingroup$ Thanks but I was looking for a way to calculate the minimal crossing number given a projection that might be very different from its equivalent. I read that b(D) ≤ c(D) ≤ b(D)(b(D)− 2) for a minimal diagram D where b(D) is the bridge number and c(D) is the crossing number but can that also be true for non-minimal diagrams? If it only holds for minimal diagrams then it's possible to check if the projection is the minimal. $\endgroup$ Jun 28, 2021 at 5:54
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    $\begingroup$ @NiklasR. Determining the minimal crossing number of a knot is hard in general -- what I was trying to communicate is that to calculate it you need to either find which knot in the table it is equivalent to or to use specific properties of your specific knot (like whether it is alternating). It would be a much more significant undertaking to write a survey on crossing number. We don't even know basic things like whether the minimal crossing number of a connect sum of knots is the sum of the minimal crossing numbers, except in certain specific cases. $\endgroup$ Jun 28, 2021 at 6:19
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    $\begingroup$ That formula involving bridge number and crossing number does not seem to be correct. The figure-eight knot has crossing number 4 and bridge number 2, but it's not true that $2 \leq 4 \leq 2(2-2)$. I also wouldn't expect an inequality like that to be correct because there are bridge-number-2 knots with arbitrarily high minimal crossing number. $\endgroup$ Jun 28, 2021 at 6:21
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    $\begingroup$ @NiklasR. It would have been nice if the first text had cited the source of that result... I believe I found it: Theorem 2.9, "On the bridge number of knot diagrams with minimal crossings". What they're saying is that if you know the number of crossings is minimal, then the number of bridges in that particular diagram will satisfy the inequality. For the figure-eight knot, the number of bridges in the alternating 4-crossing diagram is 4, so it checks out. The last section mentions the inequality holds for some non-minimal diagrams, too. $\endgroup$ Jun 28, 2021 at 16:32
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    $\begingroup$ Thanks for sharing the slides -- I had missed this development! I'm not sure what you're seeing that's dubious, but in any case here's the corresponding paper: arxiv.org/abs/1908.04073 (And by the way, if you fix k, the problem is in NP. It becomes hard if k is part of the problem. Sort of the intuition is that the number of Reidemeister moves you might need to try to simplify a diagram grows way too fast. But concretely, the authors reduced a pre-existing NP-hard problem to the crossing number problem.) $\endgroup$ Jun 28, 2021 at 16:43

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