One Knot at a Time: PIMS CRG PDF Wenzhao Chen, on Knot Theory and Classification.

By Ruth A. Situma. PIMS Program and Communications Manager

Traveling across continents is not new for Wenzhao Chen. After completing a Ph.D. at the University of Michigan, Wenzhao moved to Europe for a two-year postdoc position at the Max Planck Institute for Mathematics in Bonn. He is back in North America, Vancouver more precisely, to start a second PDF with the PIMS Collaborative Research Group on Low Dimension Topology, at the University of British Columbia. He arrived last August, slightly before UBC’s Term 1 winter session started. “Staff at the UBC Mathematics Department were very efficient. My fellow topologists and postdocs here are very friendly as well, so the process of settling down, blending in, and resuming work with focus happened really smoothly” he says. I connected with Wenzhao for a quick check-in as he settles into his second semester. Our interview has been edited for clarity.

On a trip to Rome, Wenzhao Chen at Piazza Navona.

How did you get into your current field of research and your eventual connection to your current PDF sponsor, Professors Liam Watson?

I did my Ph.D. at Michigan State University under the supervision of Prof. Matt Hedden. He really is the one who guided me to low-dimensional topology. I became interested in the 4-dimensional aspect of knot theory. For example, a typical question would ask which knot in the 3-dimensional sphere bounds disks in the 4-dimensional ball, where the 3-dimensional sphere is regarded as the boundary of the 4-dimensional ball. The primary tool I use to investigate these types of questions came from Heegaard Floer homology, which is defined using pseudo-holomorphic curve theory; it is a package that contains invariants for knots, 3-manifolds, and 4-manifolds, etc.

Figure 1. An example of a satellite knot.

Understanding the knot Floer homology of a class of knots called satellite knots is particularly helpful when considering knot theory from a 4-dimensional perspective. However, previous methods of computing these knot invariants are rather involved. A few years ago, Prof. Liam Watson (who is now my current PDF sponsor), together with his collaborators Prof. Jonathan Hanselman and Prof. Jacob Rasmussen, introduced the immersed-curve technique in bordered Heegaard Floer homology, which significantly improves the way we perceive and compute Heegaard Floer homology of 3-manifolds. Recently, joint with Prof. Jonathan Hanselman, I have been working on extending their immersed-curve technique to compute the knot Floer homology of satellite knots.

Figure 2: The immersed-curve diagram for computing the knot Floer homology of the Mazur satellite of the trefoil knot. The diagram on the left shows the curves on the torus. The diagram on the right shows the lifts of the curves in a covering space of the torus (after simplification); the rank of the knot Floer homology is equal to the number of intersection points.

Roughly, a satellite knot is obtained by looping a knot (called pattern knot)in the solid torus in the shape of another knot (called companion knot). Our method features associating to each of the pattern knot and the companion knot a curve on a torus with two base points, from which one can read o the knot Floer invariants of the satellite knot. Studying the behavior of these invariants then becomes a game of manipulating and understanding curves.

We have been moving between in-person and online learning and teaching. What has stood out for you in your classes?

I'm teaching Linear System, which is a linear algebra course for engineer students. The course started online and is scheduled to move back to being in-person later. Before the semester started, I once worried about everyone will just be sitting behind the screen silently and I will not be able to access whether students are following. As it turned out, my students are very comfortable with asking me questions and they are using the chat feature smartly. So, it is going smoothly so far, and I look forward to interacting with these lively students in a real classroom when the situation permits.

Research can be intense. What do you do when you need to “unknot” your mind?

I think as long as one can enjoy both work and life, then it is a good balance. Most mathematicians enjoy research since the discoveries that came out of research are already rewarding, so I don't have anything special to add. What is currently helping me a lot, is staying connected with friends and colleagues. It is enjoyable and I can often learn something from hearing my friends sharing their experiences from work or life.

There is probably another thing one can do at UBC, which is walking around the beautiful campus when the brain is tired. The main mall greenway, the UBC rose garden, and the beaches nearby are all good places to refresh the mind. On sunny days, one can see the snow-capped mountains when looking out the window in PIMS.

Quiet green spaces and blue sky, on the far side of Main Mall at UBC.

Wenzhao Chen obtained his Ph.D. at Michigan State University in 2019, where he studied Heegaard Floer homology and low dimensional topology under the supervision of Dr. Matt Hedden. He was a postdoc in the Max Planck Institue for Mathematics in Bonn from 2019 to 2021. Currently, He is a PIMS Postdoctoral Fellow at the University of British Columbia. He is working with Dr. Liam Watson in low-dimensional topology.

Wenzhao will be speaking at the PIMS Emergent Research Seminar Series, on February 9, 2022, at 9:30 AM Pacific. Details on his talk, Knot Floer homology of satellite knots, can be found here