By Allison Henrich

My friend Aaron Wootton and I were chatting at a conference recently, and he told me a great story about a time he tried to get something published that was rejected over and over only to later turn into something far better. It got me thinking… I have a story like that! Though my story is more embarrassing, I thought it would be fun to share it with you.

It all started in my first year as a professor, with a project I was working on with Lou Kauffman. We were trying to extend a complicated knot invariant to strange types of knots we really like called virtual knots. In the process, we read an inspiring paper by Ivan Dynnikov. Well, we read the first half of it—the second half seemed to be a really long, super technical proof of his amazing main result. The point is that we realized that Dynnikov’s work had implications for the unknotting complexity problem, a problem we were fascinated by. (To learn more about this problem, I highly recommend listening to Laura Taalman’s interview on the podcast My Favorite Theorem!)

So, we shifted gears and got to work writing a paper which would eventually be titled “Unknotting Unknots.” In this paper, we gave an upper bound on the number of local moves you need to do to an unknotted knot diagram to turn it into a simple circle in the plane. This wasn’t the first upper bound to be found, nor was it the best, but it was a different approach that lent itself to improvements over time. Anyway, we thought it was interesting, and so we dedicated the better part of a year researching and writing about it.

Once we were satisfied with our work, we submitted our paper to a respected topology journal. Three days later—without even sending it out for review—the editor rejected it. He pointed out the obvious shortcomings of the paper and sent us on our way. I was mortified that an editor would find our submission so bad that they would reject it so quickly and definitively. If the project had been mine alone, I probably would have taken the hint and abandoned the paper right then. But Lou and I regrouped, did some computations, consulted some other knot theory experts about potential improvements, and revised the paper to honestly address the issues with the work upfront and discuss ideas for future research that could improve the results.

Round two. We submitted our paper again, this time to a good but lower ranked journal. This is where I have to tell you something that will seem superfluous. When Lou and I were working on this project, we constantly referred back to the Dynnikov paper. This was back in the day when people mostly had printed, hard copies of articles in hand when they worked on research. When I had gone to print the 45-page Dynnikov article, the printer had run out of paper after the first 18 pages. No problem! The rest of the paper was mostly technical proofs anyhow. All the good stuff we needed was in that first 18 pages.

Fast forward to the day I received our referee report from the second journal. The referee had no trouble recommending rejection once again, but this rejection was confusing. They had written, among other criticisms, “The main result of this paper was known to Dynnikov.” I called Lou, panicked and devastated. He and I digested the review. “It was known to Dynnikov! How does the referee know? Is our referee Dynnikov himself? Or a mind reader?” We frantically pulled up the Dynnikov paper from arXiv. And there it was. Right on page 38: one of the main results from our paper. Nooooo! I can still hear Lou’s voice. It had an unmistakable smile in it. “Bad mathematicians. Bad!”

Then, in his next breath, Lou said, “Let’s submit it to the Monthly!” The American Mathematical Monthly, that is. It’s only one of the most widely read math journals in the world. “Let’s turn this into an expository article about the unknotting complexity problem.” I was floored that he could get over the disappointment and embarrassment and pivot so quickly. I wanted to curl up into a ball and throw away that paper so it could never, ever see the light of day again. Instead, I reluctantly decided to go along with his preposterous plan. I’ve reflected a lot on that moment over the years. Lou’s response has taught me so much about how to be resilient and believe in the value of my work.

So, what happened? We did make a few revisions to “Unknotting Unknots,” and we did submit it to the Monthly. Not only was it accepted, but it was the lead article for the issue it appeared in. Later, we won the Halmos-Ford Award for expository excellence for “Unknotting Unknots.” At a party for the award winners at MathFest, the editor of the Monthly at the time, Scott Chapman, told me that it was one of the most well-written papers he had published. Yes, he was talking about that paper I had wanted to throw in the trash. Unbelievable.

Now, Aaron Wootton and I are organizing a storytelling event at the 2024 Joint Mathematics Meetings where others can share their stories of rejection and triumph. We hope that you will come join us. The event, called “Inspiring Stories: How an Academic Rejection Led to Something Amazing” will be held on Friday, January 5th at 3-4:30 in Room 307 of the Moscone Center. And if you have a story yourself and will be attending the JMM, we’d love to hear it! We’re accepting speaking requests for this session via this form from now through November 15th. We hope to see you soon!