14 INTRODUCTION
when two such objects are not isomorphic. Thus one would like to extract simpler
invariants, such as numbers and polynomials, from the algebras
H+[1r]
and
H[1r].
The Alexander module of
1r,
and hence the Alexander polynomial, is an invariant of
a rather small quotient algebra of
H[1r].
We regard this as evidence that
H[1r]
is a
powerful invariant. Certainly
H[1r]
belongs to a category of algebras more accessible
than the group
1r
itself, or the group ring
k[1r],
but, even so, its secrets are hard
to unlock. The
A
and D-polynomials mentioned above look difficult to compute.
They seem to lack the type of quickly stated iterative algorithm for computation
which makes the Conway and Jones polynomials so attractive. Nonetheless, we
optimistically believe that some persistence will reveal
H[1r]
to be a knot invariant
of both theoretical and practical significance.
In this Introduction, we have emphasized the relevance of recent work in invari-
ant theory due to Procesi and others,
[6], [7], [13], [19],
[20]. A referee has pointed
out that many results concerning conjugate classes of S£(2, C) representations of
free groups can be found in 19th century work of Poincare, Vogt, Fricke, and Klein.
This classical work has been developed further by Magnus and his students,
[15],
[16], [9], [10], [24].
The survey paper,
[16],
of Magnus contains historical com-
ments, an extensive bibliography, and a discussion of a number of applications of
the theory of S£(2) representations to knot theory and abstract group theory. In
the paper
[15],
many key identities involving S£(2,
q
matrices and their traces
are systematically presented. The conjugate classes of irreducible complex repre-
sentations of a free group are parametrized. Recently, the paper [8] of Gonzalez
and Montesinos studies trace identities, with the goal of
obtain~ng
explicit formulas
which define the complex character variety of a finitely generated group as an affine
algebraic set. For the most part, all this work deals with S£(2, C) representations,
or at least representations over a field. In the pages above, we have tried to explain
how our
H[1r]
approach differs from the classical approaches.
Even more recently, a paper of K. Saito,
[22],
studies certain "trace rings"
associated to representations of groups into 2 x 2 matrix algebras over arbitrary
commutative rings. Although his techniques are different from ours, we find the
spirit of his work close to our philosophy. Among other things, Saito succeeds in
giving a certain kind of solution to the problem of describing conjugate classes of
irreducible representations of any group into 2 x 2 matrices over commutative rings.
After studying Saito's paper, as well as the earlier Magnus papers, we saw how some
of the results about localizations of
H[1r]
and H+
[1r]
which we prove in Chapter 7
and Section A.7*, where we assume
~ E
k, can just as easily be established for any
ground ring k. Among other things, this leads to quite a lot of information about
SL(2, K) representations, where K is a field of characteristic two. In an Afterword
to this manuscript, Appendix B, we organize these extended results.
Our work on
H[1r]
began in the Autumn of 1990 in H. Hilden's Hawaii Hy-
perbolic Geometry Seminar. Some of the first results were worked out by Amelia
Jones, and we thank her for her interest in the project. Additionally, we are very
grateful to Jose Montesinos and Maite Lozano for their helpful insights. We also
thank the University of Hawaii for their support and we thank Sharlene Pereira and
Elizabeth Katznelson of Stanford for their excellent preparation of this manuscript.
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