#### by Roger Fenn

The following fragments are my personal take on my good friend and mathematical collaborator. I first saw Colin striding along King’s Parade, Cambridge, in his duffel coat. It was about 1961, “Between the end of the Chatterley ban // And the Beatle’s first LP.”1 I was a student at Clare College and Colin was a year or two ahead of me at Trinity College. We were both mathematics students but we didn’t really get to know each other until I was a PhD student at King’s College, University of London, under John Reeve and he was a junior lecturer at Queen Mary, University of London. Reeve ran a topology seminar for his and other students and because geometric topology was not common in London at the time, Colin was tasked with giving a course of lectures on geometric topology and we sat at his feet, lapping it up. We then formed a nonmathematical bond since we were both young parents and keen on DIY. Later Colin went to the Institute for Advanced Study at Princeton and a new position at the University of Warwick’s mathematics department. This was an exciting time for Colin and the new department, and although we often met because of our shared love of DIY, it wasn’t until the late 70s that we finally got round to writing mathematics instead of just talking about it. We wrote a paper on spines of 3-manifolds [2] published in the proceedings of a conference I organised in 1997. This didn’t make many waves at the time but sometime later an extension of this result was used in the doctoral thesis of Ruben Vigara, a student of Montesinos. The result was a number of related papers; see [12], [e8], [e9], [e10], [e11], [e12], [e13], [e14].

There followed a paper on the Kirby calculus and the so called K moves
which was published in the journal, *Topology*
[1].
Colin clearly
recalls one important conversation when he explained the K-move
(twisting around an unknotted curve) and the other Kirby moves
including handle slides. Apparently I pointed out at one crucial point
that if you twisted around a curve parallel to the one being used for
the K-move, then the effect was a handle slide across the K-move. With
a fair bit more work, we used this to prove that all handle slides can
be achieved by K-moves and the calculus reduced to a single move. This
is a key result because it makes the calculus local — handle slides
are global moves. The first real application of the calculus (to the
construction of the Reshetikhin–Turaev invariant) used this local
version of the calculus.

Rather than just write a short note on the K-move and handle slides,
we decided to write a complete proof of the calculus, setting it in
the PL category, rather than the smooth setting that
Kirby
used.
Because there is both a triangulation theorem and a
Hauptvermutung
theorem in dimension 3, there is no difference between smooth and
PL topology for 3-manifolds, so there should be a simpler proof using
PL methods rather than the “Cerf theory” used in Kirby’s proof.
We
almost achieved this in our paper on the calculus
[1].
However, there is no proof in the literature that shows that
two PL handle decompositions of a given manifold are equivalent by
handle moves (introduction and cancellation of complementary pairs and
handle slides) though one could be readily given using PL handle
theory. So to avoid a lengthy digression at this point,
we moved
into the smooth category just for this result. The paper contains the
“local” version of the calculus. However the importance of
our
contribution often escapes the reader. See, for example,
the review of this paper by
Wilbur Whitten
in *Mathematical Reviews*.
The fundamental
theorem that Whitten states in the first paragraph is this local
version of the calculus and he needs the Fenn–Rourke paper for its
proof. Rourke ruefully remarks that instead of calling the local move
a K-move (for Kirby) we should perhaps have called
it an FR-move!
However I am not that bothered.

Our paper “Racks and links in codimension two” [3], published in the early 90s, has consistently been the most quoted of all our publications. At the time it appeared, the Reshetikhin–Turaev invariant had had a galvanising effect on geometric topologists. There are many examples of an invariant established using moves and many topologists started a new set of quests for invariants using moves. In particular we stumbled upon an interesting (and we thought new) invariant of knots and links invariant under just two of the Reidemeister moves (R2 and R3) which, after some notes of Conway and Wraith on the conjugacy relation in groups, we called the fundamental “rack” (a simplification of the name “wrack” used by Conway and Wraith). The genesis of this paper is interesting. Email had not been invented and I was at the Autonomous University of Madrid as a visiting academic. So our ideas were communicated by the agency of a fax machine. I noticed at the time that a rack had a natural cubical structure which defined the classifying space for racks. It was three faxes sent from Madrid which defined this classifying space: the first was the 1-section, the second the 2-section and the third was general [e6].

Racks have a natural algebraic structure analogous to groups with which they were closely associated. Every semiframed link in a 3-manifold has a fundamental rack which, in the irreducible case, classifies both the link and the 3-manifold. There are several standard algebraic results similar to ones in group theory for example there is a Tietze theorem and an analogue of Nielsen theory. Unfortunately soon after its birth, we quickly discovered that racks were not a new concept, having been anticipated by Joyce, [e3] Matveev [e4], and Brieskorn [e5], as well as Conway and Wraith [e2] and the Japanese kei published during the second world war [e1]. Thus the big paper that we wrote on the subject is partly expository of this previous work. Despite this (or perhaps because of this) this joint paper is the most widely cited of all of our papers.

The homology of the cubical classifying space is closely associated to
the invariants constructed by
Scott Carter
et al.
[e7].
It turns
out that the homotopy type of this space is not usually interesting
but the combinatorial structure carries the full information of the
underlying link. Cubical sets of this type have interesting
homotopy-theoretic properties. There is an associated hierarchy of
bundles called *James bundles* because of their intimate connection
with the James–Hopf invariants of loops __\( (S^2) \)__. The first James bundle
can be interpreted as the classifying bundle for the link. The
Fenn–Rourke–Sanderson papers on the rack space are published in
*Applied Categorical Structures*
[4],
*Proceedings of the London Mathematical Society*
[8],
and
*Transactions of the American Mathematical Society*
[9].
Because of the connection with the
Carter et al. work area
[e7],
these papers also are very widely
cited. The paper on racks also let to many spin-offs including
invaluable joint work with
Brian Sanderson,
which are listed in
Brian’s contribution.

We also published a paper on the ideas of Klyachko which nearly proves the Kervaire conjecture [5].

At this time, the late 70s, I was getting very interested in generalised knot and braid theories and the paper on the braid-permutation group introduced welded braids and knots which are a rarer form of virtual knots and braids introduced by Kauffman [6], [10].

At a conference at the Isle of Thorns which I organised in 1998 and hosted by the University of Sussex, Dale Rolfsen announced that Patrick Dehornoy had shown that the braid groups were left orderable using esoteric methods from logic theory. This was a direct challenge to the topologists there and by the late summer a gang of us had formulated a topological proof [7]. This was the last (so far) of our collaborations, as Colin decided it was time to concentrate on cosmology in joint work with Robert MacKay [11]. He wrote a book [12] on a proper understanding of the “geometry of the universe” without either big bang or dark matter.