### An interview with Joan Birman about her mathematics

#### Works connected to ~~Vaughan F. R. Jones~~

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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

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

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

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

:Letter to J. Birman, undated.
About matrices in __\( \mathrm{SL}(5,\mathbb{R}) \)__.
misc

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

: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

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

: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