Forming notes and deforming knots with Puttipong Pongtanapaisan, PIMS PDF at the University of Saskatchewan.
By Ruth A. Situma, PIMS Programs and Communications Manager
Puttipong has spent the last six months balancing his research and settling into the “PDF-Life” in Saskatoon— he arrived in July 2021. As a PIMS postdoctoral scholar at the University of Saskatchewan, Puttipong’s preference is to think about mathematics while sitting at a coffee shop. What could be better than studying knot theory while eating a knotted bagel and sipping coffee? Forming musical notes, or riffing on his guitar! Unfortunately, his coffee shop routine was cut short at the start of the pandemic — just when he was completing his Ph.D. at the University of Iowa — but quickly resumed when he landed in Canada, and Saskatoon started opening up.
With the increase in Omicron transmissions, the only other way to safely study knot theory may be at home, with a playlist of “Coffee shop white noise” but sans knotted bagels — an acceptable compromise for the time being! “My PDF research has more direct applications to biology and involves more computer simulation. I have used the software KnotPlot as a graduate student to perform basic tasks, but my PDF research really taught me to appreciate the software’s capabilities” says Puttipong. Our interview has been edited for clarity.
You studied knot theory with your supervisor, Prof. Tomova. How did you decide to pair with your PDF supervisor, Prof. Soteros?
So, very simply, knot theory is a branch of topology that deals with the study of knotted loops. In graduate school, I was exposed to various tools that can be utilized to analyze knots, but I was particularly inspired by Professor Maggy Tomova’s geometric and combinatorial techniques. Roughly, a prominent theme that appears in Professor Tomova’s and my research is the following: instead of investigating the knot itself, we focus on how 2D shapes situated around the knot interact with one another. On the other hand, Professor Chris Soteros’ research focuses on linking statistics in lattice tubes, but we were able to translate questions that are of interest to Professor Soteros into topological problems. For example, the exponential growth rate of a link type turns out to be closely related to local moves on link diagrams, and, determining the size of the smallest tube that can contain the knot is equivalent to studying how level 2D planes intersect the knot.
To elaborate, given an embedding of a knot, one can consider the vertical level plane that intersects the knot the most and record that intersection number. Now, if we deform the knot (without cutting the knot or letting it cross through itself) and perform the same task again, we may get a lower number. The lowest possible number recorded overall embeddings of a knot type is the trunk of the knot type. This quantity was defined and studied by Makoto Ozawa. Later, Arsuaga et al. showed that if a knot has a large trunk, then we need a large tube to confine it. Therefore, problems about confinements of knots and links in lattice tubes can be phrased in terms of 2D shapes lying around the knot. For instance, if a knot does not contain certain 2D shapes called essential surfaces, then the trunk is completely determined by the number of local maxima of the knot (minimized overall embeddings).
Are you teaching online or in person? What has the experience been like?
The most recent course that I taught was in 2020. Since it was during the start of the pandemic, I taught in person while being live-streamed so that some students can also participate remotely. It felt difficult at times trying to focus on teaching and at the same time making sure that the technology was working properly. The students and I got more accustomed to the format as the semester went on, and I was really thankful that the students were supportive when we would encounter technical difficulties. I am excited to start teaching again in January 2022; we will be online for the first few days, but I’m looking forward to teaching in person after that.
What do you do to balance your research and life and what has been your best discovery since arriving in Saskatoon?
My research usually involves drawing and looking at pictures of shapes, which I find a bit more relaxing than staring at complicated equations. I think doing research in knot theory is a really fun hobby. Some mathematicians probably agree with me since there are games created from knot invariants. For example, Allison Henrich and her coauthors wrote about the Region Unknotting Game.
In addition to mathematics, I enjoy playing guitar/piano and listening to Jazz. (I started taking guitar and piano lessons when I was 15 and did a double major in math and music as an undergraduate student). I grew up listening to jazz tunes composed by the ninth King of Thailand (King Bhumibol Adulyadej) and I love playing these tunes arranged for guitar by Hucky Eichelmann. I am also a fan of stand-up comedy and live acoustic music. Back in Iowa City, many famous comedians and musicians would stop at Iowa City as a part of their tours, so I miss how easy it was to go see my favourite artists.
The South Saskatoon River has beautiful views. On the days that are not extremely cold, I like to walk the trail along the river.
Puttipong Pongtanapaisan obtained his Ph.D. at the University of Iowa, where he studied knot theory and low dimensional topology under the supervision of Dr. Maggy Tomova. He is currently a PIMS Postdoctoral Fellow at the University of Saskatchewan. He is working with Dr. Chris Soteros to explore knotted objects in lattice tubes by analyzing the arrangement of local maxima and minima of knots and links.
Puttipong will be speaking at the PIMS Emergent Research Seminar Series, on January 12, 2022, at 9:30 AM Pacific. Details on his talk, Knotted Objects Confined to Tubes in the Simple Cubic Lattice, can be found here.
