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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?

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

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  • $\begingroup$ Why isn’t this major news? Why hasn’t this result been published in journals? $\endgroup$ – Keshav Srinivasan Sep 12 at 20:26
  • $\begingroup$ I believe it is currently being written up. I saw him give a talk about this result at a recent conference. $\endgroup$ – Josh Howie Sep 12 at 20:43
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    $\begingroup$ It does say on that page that the paper would appear on arXiv in June 2018, so what happened? $\endgroup$ – liuyao Sep 12 at 20:47
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    $\begingroup$ 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. $\endgroup$ – Mike Miller Sep 12 at 21:50
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    $\begingroup$ 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 Sep 13 at 3:18

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