INTERNATIONAL MATH CHAMP:Princeton High School Senior Nick Hagedorn, third from right in front row, celebrates his victory in the math competition at the 22nd Annual Taiwan International Science Fair in Taipei. (Photo courtesy of Princeton Public Schools)

By Donald Gilpin

Not a stranger to the worlds of science, math, and high-level competition, Princeton High School (PHS) senior Nick Hagedorn took his knot theory project to the 22nd Annual International Science Fair in Taiwan (TISF) last month and brought back a first place award in mathematics.

During the five-day event, Hagedorn competed with about 630 students from 27 different countries who were presenting their research projects to judges and peers at the National Taiwan Science Education Center in Taipei. The TISF is sponsored by Taiwan’s Ministry of Education.

“Going to Taiwan was a spectacular experience,” Hagedorn wrote in an email. “I loved trying new foods at the night markets. The fair also organized cultural tours across Taipei City and the surrounding Taiwan area.”

He described the experience of making friends across cultural and linguistic barriers. “In the absence of a common language, we bonded over each other’s dances and songs,” he noted. “I can now proudly say I have friends who don’t speak any English.”

He also pointed out that a visit to the National Palace Museum was a highlight of the week, and he modestly added, “The competition also went very well.”

Hagedorn discussed his project, as quoted in a Princeton Public Schools press release. “Imagine you tangle up a piece of string into a knot and then glue the ends of the string together,” he said. “My research finds mathematical equations that can relate different properties of that tangled mess.”

He added, “Knowing about these properties helps to
answer the fundamental question of the mathematical field of knot theory: given two knots, is there some way to twist one of the knots — without cutting the knot or passing any part of it through itself — so that it looks like the other knot?”

Admitting that nonscientists might not be impressed by the practical impact of his research on their lives, he continued, “It does help make progress towards some overarching goals of knot theory. A big purpose of knot theory is being able to distinguish distinct knots. Knowing how to do this has a number of applications. For instance: molecules created with the same atoms, but shaped into different knots, have different and often unique properties.”

He went on, “In cybersecurity, certain post quantum cryptography algorithms rely on the difficulty of classifying knots. And in biology, DNA reproduction requires topoisomerase to unknot the two DNA strands formed by replication, transcription, and recombination; some chemotherapy drugs work by halting this untangling process, which requires knowing what transformations untie certain knots.”

Hagedorn, who was a 2022 New Jersey Governor’s STEM Scholar, became interested in knot theory after reading The Knot Book by Williams College math professor Colin Adams.  “I reached out to Professor Adams, and he introduced me to the study of multi-crossing knot projections,” Hagedorn reported on his website.

Winning the grand prize at the Mercer Science and Engineering Fair last March qualified Hagedorn to compete at the 2023 Regeneron International Science Fair in Dallas last May. He won four awards in the Regeneron Dallas competition: the Second Place Grand Award in Mathematics, the First Place American Mathematical Society Karl Menger Special Award, the First Place Mu Alpha Theta Special Award, and the Taiwan International Science Fair Special Award, which sent him to Taiwan.

As he looks toward graduation from PHS in June, Hagedorn is non-committal about the future of his knot research, his ongoing scientific explorations, and his college plans. “I’m currently taking a math course at Princeton University, which I’m greatly enjoying,” he wrote. “I am really excited for college, though I haven’t decided which school I will be attending. Ask me again in May!”