In Honor of Professor Mikio Furuta

This article is an English translation of a piece I wrote in March 2025 on the occasion of the retirement of Professor Mikio Furuta, who was my advisor in both the master’s and doctoral programs in graduate school.

I entered the master’s program of the Graduate School of Mathematical Sciences at the University of Tokyo in April 2016, and under the supervision of Professor Mikio Furuta, I received my master’s degree in March 2019. I then continued on to the doctoral program in the same graduate school in April of that year, and received my doctorate in March 2022. Since April 2022, I have been engaged in mathematical research as a Special Postdoctoral Researcher at RIKEN, and from April 2025 I am scheduled to continue my research there as a Research Scientist.

In February 2025, I was surprised to hear directly from Professor Furuta, in a message to his former students, that he had decided to retire earlier than planned. At the time, I had a new paper that I had begun writing in January. The paper aimed to substantially extend the results of my master’s thesis, and two years earlier I had presented the underlying idea in a lab seminar in front of Professor Furuta and received encouraging comments from him. I thought that, rather than trying to thank him for all his guidance with clumsy words, it would convey my gratitude better if I completed this paper in a form I could truly be satisfied with and showed him how much I had grown. So I resolved to finish the paper before his retirement.

In this article, I would like to look back on how, under Professor Furuta’s guidance in my master’s program, I encountered my research theme, found the seeds of my research, and wrote my master’s thesis, and to describe how that experience continues to shape my work today.

Research in the Master’s Program

From March 2007, when I graduated from the Department of Mathematics in the Faculty of Science at the University of Tokyo, until I entered the Graduate School of Mathematical Sciences at the same university, I worked for nine years as a software engineer in private companies. In 2015, through a variety of circumstances, I decided to study mathematics again. After an intense crash course of study, I took the graduate entrance examination that year, passed, and entered the master’s program in 2016. I had decided to go on to graduate school solely because I wanted to try mathematics again after having once failed at it, so I had no concrete idea at all of what I wanted to work on in the master’s program. Before entering, guided only by a vague desire to do topology, I met with several professors, and among them I felt an extraordinary aura from Professor Furuta and hoped to study under him.

Once I had actually entered and saw the senior students giving seminars, I immediately realized, “I’ve come to a place far beyond my level.” Looking at my seminar presentations at the time, Professor Furuta must surely have thought, “How did someone at this level manage to pass the entrance exam?” Right after I entered, rigorous weekly seminar training began, and a whole year passed while I remained in a state of understanding nothing. Even by around the summer of my second year, I still had no research topic, and realizing that “at this rate I won’t be able to write a master’s thesis,” I decided to extend my enrollment by one year.

Near the end of my second year, Professor Furuta told me about a knot homology theory called Khovanov homology, and through a student of Professor Yukio Kameya at Keio University, who had written a master’s thesis on Khovanov homology, I was able to learn an overview of the theory. As a former software engineer, I had been tinkering, as a hobby (or perhaps as a form of escapism), with a program to compute the homology groups of simplicial complexes as a break from studying, and I realized that with a small extension it might also compute Khovanov homology. I immediately wrote the program and carried out various computational experiments. In the course of doing so, I observed that the homology class called the Lee class, determined from a knot diagram, is a generator of the homology group over Q, but not over Z. Moreover, for various knots, powers of 2 always seemed to appear in its vector components.

I presented this in seminar and said, somewhat uncertainly, “I feel that the appearance of powers of 2 may mean something...”. Then Furuta’s eyes lit up, and he advised me, “You should investigate that thoroughly.” The most obvious idea was that the exponent of the power of 2 — which I denote by d_2​ and call the 2-divisibility — might be a knot invariant, but it was easy to see that the value changes even if one merely deforms the trivial knot ◯ into an ∞-shaped diagram. When I said, rather disappointedly, “It seems it probably won’t be an invariant,” Furuta immediately replied, “No — if that change happens to coincide with the change of some known quantity, then the difference between those two quantities becomes an invariant.”

I thought, “I see...,” and decided to sit down and investigate how the factor 2 appears under the Reidemeister moves. After a lengthy case-by-case analysis, I was able to prove that, although there is a sign ambiguity, the factor 2 always appears exactly to the extent of the change in a certain quantity (r − w) / 2​ determined from the knot diagram. Thus, if one tentatively sets

s = d_2 − (r − w) / 2

then one obtains a knot invariant. (I did not tell my advisor that this s was named after the initial of “Sano.”)

The next question was whether this s was genuinely new, or whether it coincided with some known invariant. Once again, I computed values for about fifteen knots with small crossing numbers and lined them up, and then asked Kouki Sato, a postdoctoral researcher who had been attending our weekly seminar as well (now Assistant Professor at Meijo University), “Does this sequence look familiar to you...?” After a little while, he told me, “It looks like 2s + 1 may coincide with Rasmussen’s s-invariant.”

Rasmussen’s s-invariant is an integer-valued knot invariant derived from Khovanov homology, and is also deeply related to the Lee class, so it seemed natural that it would appear in this context. Although I was a little disappointed that I had not obtained a new “Sano invariant,” I thought that, as a master’s thesis, the result “giving another expression for the Rasmussen invariant” was respectable enough. So I reset

s_2 = 2d_2 + w − r + 1

and made it my goal to show that s_2 ​= s. However, s, defined using the filtration structure of the homology group over Q, and s_2​, defined using the 2-divisibility over Z, did not seem at all easy to compare directly.

When I reported this again in the next seminar, Furuta advised me that it might be good to investigate how far one could develop arguments parallel to those for s on the side of s_2​. At the time, I thought, “If they are going to coincide anyway, then whatever holds for s will also hold for s_2, so maybe this is not very meaningful...” However, by then the summer of my third year in the master’s program was already approaching, so I decided to take this up as a concrete research problem.

After that, as I carried out my own investigation and research, I came to understand where the factor 2 comes from. It is the difference between the two roots of the quadratic polynomial X² − 1 used to define the Lee class. If one replaces this polynomial by X² − 3X and considers the 3-divisibility d_3​, one obtains a corresponding invariant s_3; and if one introduces an indeterminate H and uses X² − HX, one can consider an invariant s_H​ defined by H-divisibility. In this way, one obtains a family of invariants {s_c}, indexed by pairs (R, c) consisting of a commutative ring R and a noninvertible element c. This made it possible to ask whether these invariants coincide with s, and also how far one can develop arguments parallel to those for s independently of it.

The results I was ultimately able to prove in my master’s thesis were as follows:

1. Each s_c​ has properties analogous to those of s (such as concordance invariance and bounds on the slice genus), and

2. In particular, when (R,c) = (Q[H],H), the invariant s_H​ coincides with Rasmussen's s-invariant.

For this research, I was able to receive the Dean’s Award at the graduation ceremony. Later, in joint work with Kouki Sato in 2022, we also found that there exist knots for which s_2​ and s do not coincide. Had I kept aiming only to prove s_2 = s, I would have obtained a far smaller result.

Subsequent Work

The content of my master’s thesis was published in 2020, in the first year of my doctoral program, as “Divisibility of Lee’s class and its relation with Rasmussen’s invariant” in the journal Journal of Knot Theory and Its Ramifications. After that, almost all of my solo research has been a development of the work I began in the master’s program.

1. “Fixing the functoriality of Khovanov homology: a simple approach” (2020) — By examining more precisely the sign changes associated with Reidemeister moves of the Lee class, I showed that one can fix the sign ambiguity in the functoriality of Khovanov homology.

2. “A Bar-Natan homotopy type” (2021) — By imitating the construction of the Khovanov homotopy type, which is a spatial refinement of Khovanov homology, I constructed a spatial refinement of Bar-Natan homology, a variant of Khovanov homology. My original goal was to obtain a spatial refinement of the s-invariant using this, but I have not yet succeeded.

3. “A family of slice-torus invariants from the divisibility of Lee classes” (2022) — Joint work with Kouki Sato. We defined reduced versions of the invariants constructed in my master’s thesis and showed that they have the “good property” known as the slice-torus property.

4. “Involutive Khovanov homology and equivariant knots” (2024) — For knots with a special symmetry called strong invertibility, I extended Khovanov homology and the Rasmussen invariant, and as an application reproved that a certain family of strongly invertible knots each admit an exotic pair of slice disks.

5. “A diagrammatic approach to the Rasmussen invariant via tangles and cobordisms” (2025) — described in detail below.

In the fifth paper, within Bar-Natan’s framework extending Khovanov homology to tangles, I reinterpreted the Lee class as the chain homotopy class of a certain “dotted cobordism,” and reinterpreted the indeterminate H as a “2-dotted sphere,” thereby redefining its H-divisibility d_H. I then defined a knot invariant by exactly the same formal expression as in my master’s thesis,

s_H = 2d_H + w − r + 1.

When this quantity is considered over Q, it once again recovers the Rasmussen invariant. This formulation using cobordisms makes it possible to consider divisibility even in a setting where homology itself has no meaning, and allows one to partially estimate or determine s_H​ from tangle diagrams cut out from a knot diagram. In other words, it gives a new interpretation and a new computational method for the Rasmussen invariant, which previously could not even be defined unless the knot was given as a whole.

The long proof of invariance that I wrote in my master’s thesis has now been rewritten in a far more refined form. However, the ideas that Professor Furuta and Sato nurtured together with me at that time still remain alive at the root of my research.

In Closing

I feel that the way I approach research now is entirely based on what Professor Furuta taught me in the master’s program. In particular, my way of facing mathematics — when one finds an interesting phenomenon, investigate it thoroughly; do not cover things over with superficial understanding; even if an initial conjecture turns out to be wrong, do not let it go too quickly; be honest about the joy of exploration and about the joy that comes when deep understanding is attained — these are things that I feel were instilled in me directly through the seminars. I also constantly feel that the good relationships we have continued to maintain with our lab mates even after graduating are thanks to the environment Professor Furuta created: one in which we could pursue mathematics seriously while discussing things with one another as equals.

I would like once again to express my deep gratitude for his guidance over these six years, and I hope that from now on as well, he will continue to enjoy mathematics freely together with us.

March 2025, Taketo Sano.