Celebratio Mathematica

Joan S. Birman

Correspondence with Vaughan Jones

Editor’s Note

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.


[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