We are pleased to publish in Celebratio for the first time
holograph letters from
Vaughan Jones
to Joan Birman.
The bulk of these were sent over a period from 1984 to mid-1990s.
A few of the letters are undated. Such letters remind us that the speed
and abundance of email can obscure one kind of experience: the chance
to see how ideas written out by hand on paper embody a thought
process, not merely a result.

Among these letters is one
[1]
(dated 31 May 1984, from Metuchen, New Jersey)
in which Jones describes a “new knot
invariant” — what we now know as the Jones polynomial.

Letters

[1]V. Jones:
Letter to J. Birman of 31 May 1984.
As a follow-up to their May 22, 1984 meeting, Jones explains to Birman, who
was not familiar with his work on type \( \text{II}_1 \) factors, how that work had lead him to a formula
for a 1-variable polynomial invariant of a classical link in \( \mathbb{R}^3 \). He calls his invariant \( V (t) \).
Starting on page 5, he works out some of its basic elementary properties.misc

[2]V. Jones:
Notes for J. Birman, undated.
Undated, but refers to the May 31, 1984 letter as “yesterday’s letter”. Works out additional
properties of \( V (t) \).misc

[3]V. Jones:
Letter to J. Birman of 14 November 1984.
Birman and Jones had met at a conference at MSRI October 10–16, and
discussed, among other things, forming knots and links from braids, but using the connections
needed to get plat and bridge presentations.misc

[4]V. Jones:
Letter to J. Birman of 21 November 1984.
About another topic that had been discussed at the October 10–16
gathering, i.e., representations of the mapping class group of a
surface of genus 2, using 6-plats.misc

[5]V. Jones:
Letter to J. Birman of 26 February 1985.
About the formula for closed 3-braids that are knots.misc

[6]V. Jones:
Letter to J. Birman of 15 May 1985.
About his observation that the plat representation of the 1-variable Jones
polynomial satisfies a skein relation.misc

[7]V. Jones:
Letter to J. Birman of 31 January 1986.
A letter that told Birman about the submission
of the “first draft” of
“Hecke algebra representations of braid groups and link
polynomials” for publication. Essentially everything that had been discussed in the
letters that preceded this one (and more) appeared in the published paper.misc

[8]V. Jones:
Copy of Letter to L. Kauffman of 3 October 1986.
About a
states model for the two-variable Jones polynomial.misc

[9]V. Jones:
Email to J. Birman of 12 June 1990.
An e-mail from V Jones to J. Birman, about calculating the braid index of a
knot. To understand its content, note that near the end of Jones’
paper “Hecke algebra representations of braid groups and link
polynomials”,
there is a table
that assigns braid indices to the 84 knots from the table at the end of Rolfsen’s classic book
Knots and Links.
Birman had asked Jones whether he discovered new tricks for changing knots into braids,
and if not, how he had the patience to do it on so many knots? Read this 12 June 1990 e-mail
to learn his answer.misc