13
$\begingroup$

In this 1998 journal paper, all the prime knots with 16 or fewer crossings are found (some of which were found earlier by others). There are over 1.7 million such knots. But the prime knots with 17 crossings have not yet been tabulated. Here is what this book says:

This is probably hard and requires new ideas.

But this book was written in 2004, so things may have changed since then. There have certainly been a lot of developments in knot theory over the past 15 years.

So my question is, what is the state of research on finding all prime knots with 17 crossings? Are we relatively close to doing so, and have partial results been discovered?

Improve this question
$\endgroup$
4
  • 4
    $\begingroup$ Is there a reason that 17 is qualitatively different than 16? This question suggests that planarity of the Dowker code starts to become rare. mathoverflow.net/questions/19745/… $\endgroup$
    – MyNinthAccount
    Commented Sep 11, 2019 at 21:08
  • 1
    $\begingroup$ The problem of tabulating knots via minimal-crossing diagrams, at this point, is less about mathematics and more about accounting. How do you keep track of such vast quantities of data? Provided you can store the data, we have the algorithms to compare and deal with duplicates. Given that 19-crossing knots have over 300-million distinct types, the quantity of redundant diagrams must be astronomical. $\endgroup$
    – Ryan Budney
    Commented Jun 11, 2022 at 0:48
  • $\begingroup$ Dear Ryan - Yes, organising the data (of the 19 crossing census) is a huge task. However, the “dealing with duplicates” is also highly non-trivial. Ben has said that with each increase in crossing number he finds new difficult examples that require mathematical work. $\endgroup$
    – Sam Nead
    Commented Jun 11, 2022 at 7:49
  • $\begingroup$ @SamNead: I agree, but it's mathematics that's all set up. Finite index subgroup enumeration and interfacing Regina with GAP is something that is fairly direct -- Regina used to interface with GAP, in some of its first iterations. I had been planning to extend that line of reasoning to build the corresponding covering spaces (triangulated) but I did not finish that project. Curious to see if Ben did that as well. Scanning the paper it looks like the diagram moves must be some of the most recent coding additions. $\endgroup$
    – Ryan Budney
    Commented Jun 11, 2022 at 8:17

1 Answer 1

Reset to default
17
$\begingroup$

Ben Burton has found that there are 352,152,252 prime non-trivial knots with up to 19 crossings. See here for the tables.

2022-06-11 update: The details of this enumeration have now been published in a conference proceedings. It describes the variety of algorithmic techniques that were required, and the pairs of knots that were most difficult to distinguish.

Improve this answer
$\endgroup$
5
  • 1
    $\begingroup$ Why isn’t this major news? Why hasn’t this result been published in journals? $\endgroup$
    – Keshav Srinivasan
    Commented Sep 12, 2019 at 20:26
  • 1
    $\begingroup$ I believe it is currently being written up. I saw him give a talk about this result at a recent conference. $\endgroup$
    – Josh Howie
    Commented Sep 12, 2019 at 20:43
  • 2
    $\begingroup$ It does say on that page that the paper would appear on arXiv in June 2018, so what happened? $\endgroup$
    – liuyao
    Commented Sep 12, 2019 at 20:47
  • 3
    $\begingroup$ Things sometimes take longer than expected, as I'm sure every mathematician has experienced first-hand. If the precise details matter to you, you can of course e-mail Ben. $\endgroup$
    – mme
    Commented Sep 12, 2019 at 21:50
  • 4
    $\begingroup$ If you think a one year lag on a paper is unusual, you must be young. I've got a pre-print that I've been getting ready for publication for the past 11 years... $\endgroup$
    – Ryan Budney
    Commented Sep 13, 2019 at 3:18

You must log in to answer this question.

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