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?