#### by Pierre de la Harpe

Vaughan Jones was born in New Zealand on December 31, 1952. His lifelong interest in math and science began very early. As he liked to remember from his first years at school, he discovered by himself that

a hundred plus a hundred is two hundred. These were gigantic numbers. I knew that one plus one is two and I deduced that a hundred plus a hundred is two hundred. That was great. I was the only one in the classroom to have thoughts of this kind. (Translated from [e17].)

He did his undergraduate studies at the University of Auckland in the
years 1970–73, and attended several masters-level courses there that
he made good use of later. He heard of von Neumann algebras in a
course by
Michael Lennon,
who is still remembered in Auckland as an
outstanding teacher. A course by
Paul Hafner
taught him Galois theory,
which may have inspired him later when he thought of a subfactor __\( N
\subset M \)__ as an analogue of a field extension __\( L / K \)__
[10].
And his first encounter with knot theory may well
have been a course by
David Gauld,
during which he and another student
calculated the Alexander polynomials of the granny and square knots.

He was always interested both in mathematics and physics. Paul Hafner remembers that

he was clutching Jauch’s book1 on quantum mechanics, pointing out that this was a major attraction. I do not know how he had come across it, maybe Mike Lennon, or possibly people from the Physics Department?

Just after his MSc, he contributed a paper in the 1974 New Zealand Mathematics Colloquium: “Dimension theory of orthomodular lattices”; the abstract is reproduced on page 148 of a historical article by Garry Tee [e14].

We quote Paul Hafner again:

I could tell him that the Swiss Government supported international academic collaboration by way of federal scholarships (“Bundesstipendien”), with one available to New Zealanders. Traditionally, these scholarships were ‘owned’ by the Department of Germanic Languages and Literatures in Auckland. At an occasion where I met the Swiss Ambassador to New Zealand, maybe a year prior, the Ambassador expressed his displeasure that there did not appear to be any scientists among the applicants — now here was my chance to disrupt by pointing Vaughan in the right direction. He applied and was successful.

This is how he arrived in Geneva to work with Josef-Maria Jauch, whose main interests were the mathematical foundations of quantum mechanics.

In 1974, his first three months in Switzerland were in Fribourg, for a
language course to learn French. “It was the coolest three months in
my life,” (*les mois les plus sympa de ma vie*) as he has
said. The course was a real success: Vaughan became almost bilingual.
A good lesson for all students and colleagues who believe that English
is enough to settle down anywhere in the academic world, forgetting
that thinking together with other people and in another language is
both possible and worthwhile. It is during this period in Fribourg
that he met his future wife Wendy. Later, in Geneva, they got married
in 1979
(and they celebrated their forty years in 2019).
Vaughan also visited Jauch once, and working in his group looked
promising and pleasant. But Jauch died suddenly the next week, and
Vaughan
had to modify his plans. He started to work with
Constantin Piron,
a
colleague and former student of Jauch, and also with
Jean-Pierre Eckmann.
His first published article recorded in MathSciNet is from
this period (see
[1]).

As he continued to be interested in both mathematics and physics, he attended lectures by André Haefliger, probably one on foliations and one on de Rham theory. Flavio da Silveira, another PhD student of Haefliger, remembers that Vaughan often asked interesting questions and, in the corridor before or after the lectures, was often to be seen in the center of a group of students actively discussing the course. At one point, Vaughan asked André if he could possibly become a teaching assistant in the mathematics department. The answer was “Yes, I need somebody for my course, and you can start tomorrow.” This was in February of 1976, if I remember correctly.

It happened that Mahnaze Ardjomande, the wife of da Silveira, also an assistant in Geneva as a PhD student of Narasimhan, had decided to work with her two sisters on a project to found a scientific museum for children in Teheran (this was before the 1979 Iranian Revolution). The da Silveira’s left Geneva for Teheran without much notice, and thus there were now two vacant assistant positions. Vaughan took the one offered by Haefliger. As I was working in Lausanne and visiting Geneva weekly, Claude Weber asked me if I knew somebody who could take the other position; I was interested myself, and started the next week. (Administrative details would follow.)

This is how Vaughan Jones became the PhD student of André Haefliger.
Haefliger has a very strong and important place in mathematics, and he
had a strong influence on Jones, but he is certainly not a specialist
of von Neumann algebras; this shows
the originality of the student and the open mind of the advisor.
Together with the unofficial and decisive codirection of
Alain Connes,
the outcome was a very strong thesis
[2].
The main
result is a classification of the actions of finite groups on __\( \mathbb{R} \)__, the
hyperfinite factor of type II__\( _1 \)__, in terms of three invariants. As
Jones writes modestly, “the method of proof that I use is a
development of that of
[e3]”,
where Connes classifies the
actions of finite cyclic groups on __\( \mathbb{R} \)__. Subfactors of __\( \mathbb{R} \)__ appear
already here, as fixed-point algebras of appropriate finite group
actions
([2], end of Section 2.1).

I note in passing that Jones spent 6 years altogether in Geneva as a PhD student; this shows that even talented people like him may need time to finish their work, despite what politicians sometimes seem to suggest.

In Geneva, as in all places, Vaughan Jones was a marvelous companion. His informal style of working encouraged exchanges of all kinds, and he was always ready to share his ideas. He was not only a first-class mathematician, he was also an excellent baritone and for many years sang in the University choir, and enjoyed practicing chamber music with his advisor André Haefliger and the choir’s director Chen Liang-Sheng. He shared his love of sports with his many friends, and always conducted himself with modesty, humor, and respect for everybody. On one occasion, however, he comported himself with glorious panache: during his thesis defense, he dressed in a superb smoking, like a king addressing his courtiers — with the jury members on the front bench: André Haefliger, Alain Connes, Michel Kervaire, and me.

Shortly after his thesis, Vaughan Jones established his remarkable
result concerning subfactors in a factor of type II__\( _1 \)__ and their
indices, which are positive numbers in the union of the discrete set
__\begin{equation}\label{eqon}
\{ 4 \cos^2 (\pi/n) \}_{n \ge 3} = \{ 1, 2, (3 + \sqrt{5})/2, 3,
\dots \}
\end{equation}__
and the continuum __\( \mathopen[ 4, \infty \mathclose] \)__. His first result
for indices was the existence of a first gap, between 1 and 2, a
real surprise since anything known so far related to dimensions and
factors of type II__\( _1 \)__ was continuous; very wisely, and following the
advice of Connes, Jones did not publish this preliminary result and
worked out the complete picture as it appears in his spectacular
article
[4],
and already in the previous announcement
[3].
The starting step in the proof is to associate to a
pair __\( N \subset M \)__ of a factor and a subfactor a projection __\( e_N \)__ from
__\( M \)__ onto __\( N \)__, and the pair __\( M \subset \langle M, e_N \rangle \)__; the
algebra __\( M_2 := \langle M, e_N \rangle \)__ generated by __\( M \)__ and __\( e_N \)__
makes sense because both __\( M \)__ and __\( e_N \)__ act on a Hilbert space which is
a natural completion of __\( M \)__; it is shown that __\( M_2 \)__ is a finite factor
and that the index of __\( M_1 := M \)__ in __\( M_2 \)__ is equal to the index of
__\( M_0 := N \)__ in __\( M \)__. The next step is to iterate this construction to
obtain a nested sequence of II__\( _1 \)__ factors
__\[
M_0 = N \subset M_1 = M \subset M_2 = \langle M_1, e_N \rangle \subset
M_3 = \langle M_2, e_2 \rangle \subset \cdots
\]__
and a sequence __\( e_1 = e_N, e_2, e_3, \dots \)__ of projections (__\( e_i^2 =
e_i \)__, and __\( e_i \)__ self-adjoint) which among other things satisfy
__\[
e_i e_{i \pm 1} e_i = \tau e_i, \hskip.5cm e_i e_j = e_j e_i
\quad\text{if} \hskip.2cm \vert i - j \vert \ge 2 ,
\]__
where __\( \tau \)__ is the reciprocal of the index of __\( N \)__ in __\( M \)__. The final
step is an analysis of the algebra generated by the __\( e_i \)__’s, which
gives the possible values of __\( \tau \)__.

I like one thing that he says in [11] about his work on the subject of [4]:

In this research for index of subfactors, I think I spent at least 80% of my time on that trying to prove something that I thought was true and turned out to be wrong.

This shows a way in which some of his results were indeed a surprise, even for him.

One day, after lecturing on subfactors in Geneva, he spent some time
in discussion
with
Didier Hatt-Arnold,
at this time a PhD student of Kervaire
and now a musician in Berlin. Didier pointed out that some of his
equations suggest a relation with braid groups, and he started to
teach Vaughan about braids; I added a reference to a survey by
Magnus
[e2].
The point is that
Alexander
had already shown in 1923
that any knot can be obtained as a closed braid and that, for each
positive __\( n \)__, braids on __\( n \)__ strands constitute a group __\( B_n \)__. The
following presentation
__\begin{equation}\label{eqtw}
B_n = \biggl\langle \sigma_1, \dots, \sigma_{n-1}\biggm|
\begin{aligned}
& \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}
\text{ for} \hskip.2cm i \in \{1, \dots, n-2 \} \\ &
\sigma_i \sigma_j = \sigma_j \sigma_i \text{ for} \hskip.2cm
i, j \in \{1, \dots, n-1 \} , \hskip.1cm \vert i - j \vert \ge 2
\end{aligned}
\biggr\rangle ,
\end{equation}__
appears in an article of 1925 by
Emil Artin.
Jones understood that
what is formally common between __\eqref{eqon}__ and __\eqref{eqtw}__ indicates an interesting
relation between the algebra __\( TL_n \)__ generated by __\( e_1, \dots,
e_{n-1} \)__, which is called the Temperley–Lieb algebra, and the
group __\( B_n \)__. More precisely, there is for each __\( t \in \mathbf C^* \)__ a
homomorphism __\( r_t \)__ from __\( B_n \)__ to the group of units of __\( TL_n \)__, given
by __\( r_t(\sigma_i) = \sqrt{t} ( te_i - (1 - e_i)) \)__. There is a natural
trace on __\( TL_n \)__ and the Jones polynomial __\( V_L (t) \)__ of an oriented link
__\( L \)__, which is the closure of a braid __\( b \in B_n \)__,
is defined by __\( V_L (t)
= (-(t+1)/ \sqrt{t} )^{n-1} tr \left( r_t(b) \right) \)__
[5].
This genius discovery owes a lot to a crucial conversation with
Joan Birman
at Columbia University; see also
[13].
There have
been many expositions of parts of Jones’ new ideas; let us quote one
by his unofficial thesis codirector
[e6]
and one from
seminars in Bern and Geneva
[e7].

The Jones polynomial marked a new period in knot theory, and more generally in low-dimensional topology. Some of the early consequences were the solution by Kauffman, Murasugi, Menasco, and Thistlethwaite of conjectures on knots formulated by Peter Tait in articles published between 1876 and 1885; see [e11], [e9], [e10], [e13], [e15]. The origin of these conjectures was Tait’s attempt to tabulate all knots; for this it is useful to establish that appropriate projections of a knot on a plane show the minimal possible number of crossings. More recent tabulations use, among other things, much of Tait’s strategy, plus the Jones polynomial to distinguish knots. “After over 25 years of laborious handwork, Tait, Kirkman, and Little had created a table of alternating knots through 11 crossings and nonalternating knots through 10 crossings”; the quotation is from [e16], which describes the work leading to a table of all prime knots with 16 or fewer crossings.

The Jones polynomial was very soon generalized. This generalization
was so natural that it was discovered simultaneously and
independently by five sets of authors. Four research announcements
were submitted to the Bulletin of the AMS within a few days in late
September and early October 1984. They were published as one paper,
with an introductory section written “by a disinterested party”
[e5].
The work of the fifth group was slow in reaching
the United States, because of poor mail service (Poland was ruled
under martial law between 13 December 1981 and 22 July 1983), and was
published later
[e12].
Jones’ own account appeared as
[7].
The generalization associates to an oriented link __\( L \)__
a Laurent polynomial in two variables __\( P_L(\ell, m) \)__; the polynomial
__\( P_L \)__ has specializations __\( V_L \)__ and the Alexander polynomial of __\( L \)__,
but contains strictly more information than these two
[e8].

Knot polynomials had also impact on biology. As Andrzej Stasiak remembers:

The fame of Jones’ and others’ polynomials coincided with the first biological studies of DNA knots in which precise knot types were determined; see for example [e4] […]. Biologists and mathematicians started to attend the same meetings, as mathematicians liked to see real life knots and wanted to understand how they form and what may be their function. More recently though, when we can work with coordinates of thousands of proteins, we can search for linear knots in them and characterize knots resulting from their closure. As this work is done with thousands or even millions of configurations (each subchain is randomly closed hundreds of time) all configurations are analyzed by computers and polynomials are calculated to determine the knot type. However in such a work we used HOMFLYPT polynomials; see for example [e18].

There have been further fascinating results showing these interactions between operator algebras and low-dimensional topology. We will only allude to the foundation of the theory of planar algebras [8] and an analogue of Alexander theorem showing that “the Thompson group is in fact as good as the braid groups at producing unoriented knots and links” [12]. For more on the importance of these results, see [e19] and [e20].

Vaughan Jones has had a deep influence on people in Geneva. For me in particular, after the time of his thesis, I had the chance of being near him on many other occasions: when he spent sabbatical periods in Geneva; in IHÉS; in large groups after lectures by Alain Connes in Paris around Leffe beers; in Berkeley and MSRI; cowriting a book with Fred Goodman and him; or hiking around. Moreover, it is amazing to realize that he was so influential at the same time in so many other places — New Zealand and the US of course, but also France, England, Wales, and other countries. To all of us, his visits were gifts.

It was Vaughan Jones’ pleasure to draw a picture of his beloved type
II__\( _1 \)__ factor showing a big fat black point (see
[6]).
In
Geneva and elsewhere, all his colleagues and friends miss him, and
will remember him as a large, bright, friendly sun.

I am grateful to Jean-Pierre Eckmann, David Gauld, Fred Goodman, André Haefliger, Didier Hatt-Arnold, Paul Hafner, Flavio da Silveira, Andrzej Stasiak, and Claude Weber for having shared their memories with me, and checked mine.