3
$\begingroup$

I am looking for a bit of orientation with regards to computational topology resources, as I am personally totally ignorant on the subject. I have lots of different links in $S^3$ (hundreds of millions) lying around in some boxes my garage and I would like to start sorting through them and getting rid of the ones that are not unlinks. What software should I load these links into to do this (efficiently)?

I know there are some invariants I can compute that will detect unlinks but this is probably a bit computationally expensive to compute on every link. For example, it will probably cut down on the computation a lot if I first just compute some pairwise linking numbers of the components and throw out any links where these do not vanish. After doing that maybe I should check that the individual components at least seem to be unknotted (maybe by computing the Alexander polynomial?). What sorts of invariants would you compute to try and sort through the whole garage in a reasonable amount of time?

Improve this question
$\endgroup$
5
  • $\begingroup$ It would probably help if you provided (1) how these links are described - e.g., do you have diagrams? triangulations of the complement? - and (2) roughly how complex these descriptions are on average - e.g., average crossing number of the diagrams. $\endgroup$
    – dvitek
    Commented Jul 27, 2020 at 1:28
  • $\begingroup$ SnapPy is actually pretty good at detecting unlinks. Just compute the fundamental group, and it can usually see that it is a free group. $\endgroup$
    – Ian Agol
    Commented Jul 27, 2020 at 3:28
  • $\begingroup$ @dvitek Sure thing - yeah I have diagrams of all of these links and crossing numbers are in the 10-20 range. $\endgroup$
    – user101010
    Commented Jul 27, 2020 at 14:13
  • $\begingroup$ @user101010 In that case computation of knot polynomials is also pretty feasible. Naive algorithms should take time $\mathrm{poly}(n)\cdot 2^n$, whereas more sophisticated algorithms can look like $\mathrm{poly}(n)\cdot 2^{\sqrt{n}}$. See the Knot Atlas page on computations of the Jones polynomial for how to implement this for polynomials satisfying skein relations. I think Cotton Seed has some code floating around under the name knotkit that might implement good algorithms, but you'd have to check. (Caveat: I don't have experience with the relevant parts of SnapPy.) $\endgroup$
    – dvitek
    Commented Jul 28, 2020 at 6:47
  • $\begingroup$ @user101010 If you need to ensure you've detected all of the unlinks and no more, I'm not sure if SnapPy can guarantee that. (Other people in this thread would know more.) For polynomials this is somewhat harder - once you start hitting 15-16 crossings Thistlethwaite has examples of nontrivial links whose Jones polynomials are those of the appropriate unlinks. However, Khovanov homology detects unlinks (Batson-Seed) and you only have to work over $\mathbb{F}_2$, not integrally. This will be slower than other approaches but may be your only bet for unlink detection with no false positives. $\endgroup$
    – dvitek
    Commented Jul 28, 2020 at 6:49

2 Answers 2

Reset to default
3
$\begingroup$

It will of course depend on where your examples are coming from. But here are some lightweight approaches.

  1. Randomize the triangulation of the link complement a few times and then simplify. Do you get a standard triangulation of the unlink complement? (Regina and Snappy will both do something like this.)

  2. Compute the Alexander polynomial. (Snappy running under sage will do this.)

  3. Compute a presentation of the fundamental group. (Regina and Snappy will both do this.)

  4. Think about using hyperbolic geometry or normal surface theory. But these will be slower...

Improve this answer
$\endgroup$
2
$\begingroup$

I have never used this particular piece of software, but I have heard good things about the Book Knot Simplifier, which is capable of doing much of what you are describing that you'd want it to -- though importantly, if I understand their documentation correctly, it does not solve the unknotting problem as much as it tests out various simplifications of the knots involved. However, the input and editing of links seems so user-friendly that this seems like it would be a perfect place to start cleaning out your garage.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .