Seungwon Kim and team solve a 40-year-old problem in topology

July 6, 2022

The mathematics alumnus helped show that certain 2D objects remain distinct in 4D space.

Two genus-1 Seifert surfaces Z 0 (left) and Z 1 (right) for the same knot K that are not isotopic.
Two Seifert surfaces for the same knot that are not isotopic. (Image courtesy of Kyle Hayden, Seungwon Kim, Maggie Miller, JungHwan Park, and Isaac Sundberg)

An international group of mathematicians have finally answered a 40-year-old question in the field of topology — whether any two Seifert surfaces of a knot can be deformed to look alike in four-dimensional space.   

Graduate Center alumnus Seungwon Kim (Ph.D. ’17, Mathematics) was one of five researchers who arrived at the answer in a proof published on the ArXiv preprint server in May. Their work is covered in a recent issue of Quanta Magazine.

“Every knot has many Seifert surfaces, which are the orientable surfaces whose boundary is the knot, much like soap film after dipping the knot into soapy water,” said Kim’s Ph.D. adviser Professor Ilya Kofman, (GC/College of Staten Island, Mathematics).

“The open question was whether any two Seifert surfaces become equivalent when they are pushed off into the fourth dimension. This paper proved that the answer is ‘no.’”

The team solved the problem using new mathematical ideas that were developed without any thought of the fourth dimension, Kofman said. “So, not only is their result exciting, their methods revealed important new tools that may now be used to solve other open problems,” he said.

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The group is now in a unique position to address some of the most important problems in four-dimensional topology, said Kofman, who noted that Kim works in both three and four-dimensional spaces. “Such mathematical dexterity is very impressive,” said the professor. “This work captures a lot of what's exciting in research mathematics.”

Kim is a senior researcher at the Center for Quantum Structures in Modules and Spaces at Seoul National University, where he works on projects in geometric topology, low-dimensional topology and knot theory.

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