On 16 February 2012, Colin Day passed away after a year long bout with stomach cancer. In his memory, I would like to discuss his thesis
A topological construction of Vassiliev style invariants of links.
Thesis (Ph.D.)–The University of North Carolina at Chapel Hill. 1993. 99 pp.
Available from Jim Stasheff’s homepage (new and improved scan; scroll down to Colin).
In his thesis, Colin Day extends Vassiliev’s construction of finite-type invariants from knots to links, and elucidates Vassiliev’s construction in the process. I don’t think that Vassiliev’s construction ever really got enough attention, and Colin’s thesis is surely the best entry-point to it, and deserves to be widely read.
The concept of a finite type invariant occured initially to two different people in two different contexts. They didn’t know each other’s work, and the world didn’t know of their work, until much later, and then it was mostly by way of Birman and Lin’s skein relation reformulation.
Goussarov, in 1986, considered combinatorial modifications on knots, as a tool to study exotic smooth structures on 4-manifolds. His approach, anticipating Habiro’s beautiful theory of claspers, was not published until after his death in 1999.
Vassiliev, on the other hand, was interested in singularity theory, a field which at first sight would seem to have little relationship to knots and links. He published his results in 1990. Hundreds and perhaps thousands of papers have been written about finite type invariants, but virtually all of them use Birman-Lin as opposed to Vassiliev’s approach. One reason for this is that Vassiliev’s approach is technically very difficult, and the most important contribution of Colin Day’s thesis might be to make Vassiliev’s approach more accessible.
To present Vassiliev’s approach with a minimum of compication and technical baggage, I’ll follow Joan Birman’s beautiful 1997 MSRI talk on the topic and make use of the later technical simplification of considering `holonomic knots’. A holonomic knot is a 2-jet extension of a smooth function , i.e. a map given by . Vassiliev proved that any knot is ambient isotopic to a holonomic knot, and Birman-Wrinkle showed that any two ambient isotopic holonomic knots are ambient isotopic in the space of holonomic knots. Thus, there is no loss of generality in taking knots to be holonomic. The corresponding statement is not known for links.
Inside the space sits the space consisting of the non-embeddings. The space is called the discriminant. The space of knots is . Because we’re working with holonomic knots, a knot is determined by a Maclaurin series
(assume this converges everywhere to ). You can truncate a Maclaurin series to approximate :
The space of truncated series to series with terms is denoted . By viewing the coefficients as a vector, is isomorphic to . Its discriminant is .
A (rational) knot invariant is a locally constant map (locally constant implies that it takes one value for each knot type, which is what `invariant’ means). Thus, a knot invariant is an element of the reduced cohomology group
But , so by Alexander duality
, where denotes closed homology. Vassiliev didn’t have holonomic knots at his disposal, so the original argument at this step was more involved.
Vassiliev defined a finite type invariant as a stable homology class as . For singularity theory reasons, we may restrict ourselves to double-point singularities in . This gives a filtration, which is used to define the Vassiliev spectral sequence. A type invariant is a stable class coming from the the stage of the filtration. To calculate, Vassiliev computes an actuality table, which is a table of calculated values which can be plugged into more general formulae. This reminds me of how Sumerians would multiply large numbers, 5000 years ago, by using pre-calculated tables of squares of numbers which they would then plug into the formula
Vassiliev’s paper is technically formidable, and many details are left to the reader.
Vassiliev’s papers came to the hands of Birman and Lin, who realized that all of the complexity can be sidestepped by defining a finite type invariant to be an invariant which satisfies the Vassiliev skein relation.
An invariant of type is an invariant which vanishes on all singular knots with double-points, however there exists a knot with double points for which it does not vanish. This definition is entirely combinatorial, and no algebraic topological baggage is required. A-priori it’s an unmotivated definition, but the motivation comes from the fact that finite-type invariants are seen to be invariants analogous to polynomials.
Michael Polyak explained the above assertion to me in a way which I liked a lot. His explanation if to be found here. In the years 1909-1929, Frechet worked on the notion of a differential (of the first and higher orders) in various spaces, using only an additive structure of the space. A byproduct of his research was the following theorem:
Theorem: (Frechet, 1909)
Given , and an -tuple , let . A continuous function is a polynomial of degree less than if and only if for any choice of .
The analogy is between and positive or negative crossings, correspondingly.
So the left hand side of Vassiliev’s skein relation (introducing a transverse double point) should be thought of as the derivative, and a type invariant is one whose st derivative vanishes. Let’s quickly verify that its th derivative is a constant. The th derivative of a finite type invariant is a knot with double points. A type invariant sees only the combinatorics of where these double-points lie relative to one another, because if we perform a crossing-change anywhere else along the knot, by the Vassiliev skein relation we have changed our type invariant by its value on a knot with double points, and by definition vanishes on all such knots.
Thus, Birman-Lin’s approach sidestepped the technical aspects of Vassiliev’s approach, and provided an alternative motivation for finite-type invariants, as analogues of polynomials. It also clearly and cleanly extends to links and to tangles. An excellent reference is Dror Bar-Natan’s On the Vassiliev knot invariants, which must still be considered the canonical introduction to finite-type invariants via the Birman-Lin approach. For textbooks, Chmutov-Duzhin-Mostovoy’s Introduction to Vassiliev knot invariants is very good indeed.
Colin Day’s thesis
The goal of Colin Day’s thesis was to extend Vassiliev’s construction to links, which he achieved. Note that we don’t know whether two ambient isotopic holonomic links are ambient isotopic in the space of holonomic links, so the full brunt of Vassiliev’s machine must be applied.
More than the generalization of Vassiliev’s results from knots to links, the thesis’s major contribution, to my mind, is its much more lucid exposition of Vassiliev’s results and methods, some calculations and examples, and proofs of various, often quite involved, arguments which Vassiliev had left to the reader. Unfortunately, Colin’s thesis was not widely read, because it was never published and wasn’t super-easy to access. But that has changed now, and I do hope that lots of you will read Colin Day’s thesis; Vassiliev’s original approach gives a different way of looking at finite-type invariants, and has really more-or-less been ignored up until now.
I conclude with some tributes to Colin Day, from comments on MathOverflow:
I’m shocked and saddened to learn that Colin has passed away! I knew him from several interactions, including a summer at Park City, where I first really got to know him. He was a wonderful guy, and I remember having a conversation with him and Is Singer, and Is saying how very much he had enjoyed meeting Colin. I’m sure that he impressed many people the same way. – Robert Bryant
I recall him as a lad attending the Georgia topology conference. He was full of enthusiasm and joy of life. -Scott Carter
The following photographs are from Colin Day’s widow, Helen Moore. The first shows Colin teaching a math class at the Cate School in 2010-2011. The second is from spring 1993, when Colin received his PhD and his mother and grandmother attended his graduation. The math building (Phillips Hall) is in the background. The third is from the 1990’s, but she does not know where it is set. It could be at Columbia/Barnard, where he was offered a Ritt instructorship in 1993. Or it could have been taken when he was living in New York after his PhD when he had returned to ballet.