INTRODUCTION 13

The set of n-tuples of arbitrary 2 x 2 matrices over a ring is just a

4n

dimensional

affine space, with polynomial affine coordinate ring A[n]

=

k[A;, B;, C;, Di]

i

:S:

n.

Again, GL(2,

k)

acts on A[n] and on M(2,

A[n]),

and the rings of invariants

T[n]

=

A[n]GL(2,k)

and S[n]

=

M(2,

A[n])GL(2,k)

are classical objects of study. The references [13] and [19], for example, deal

specifically with these objects, which are rings of invariants of n-tuples (X1

...

Xn)

of generic 2 x 2 matrices, under simultaneous conjugation by GL(2,

k).

Here

X;

= (

~: ~:

) .

In these references, it is observed that both T[n] and S[n]

are polynomial rings over subalgebras T[n] and S[n],

T[n]

=

T[n][tr(X;)] and S[n]

=

S[n][tr(X;)].

Our work in Chapter 9 gives isomorphisms

if+[rrn]

=

T[n] and

H[rrn]

S[n],

where

H[rrn]

C

H[rrn]

is the k-subalgebra generated by the elements

IY;I,

which we

introduced in Chapter 5. Therefore, our finite presentations of

if+[rrn]

and

H[rrn]

in

the Addendum to Chapter 5 yield presentations of T[n] and S[n] which are perhaps

new for

n

~

4.

The Addendum to Chapter 9 gives some applications of the main theorem of

Chapter 9 to the algebras

H[rrn],

and more generally to

H[rr].

The point is, we

can translate known results from invariant theory back to

H[rr].

The Addendum

to Chapter 8 is rather different from the results of Chapter 8, although there is a

connection. Roughly, in the Addendum to Chapter 8 we introduce certain natural

quotient algebras of

H[rr]

and

H+[rr],

which are also invariants of

rr,

and which

are related to reducible representations or other representations of special type.

These special representations can be viewed as algebraic subsets of the set of all

representations of

rr,

which is the connection with Chapter 8. For knot groups

rr,

we identify the Alexander module

rr' /rr"

as a natural subquotient of

H[rr].

We also

use one of these new invariants to distinguish the first two knots in the knot table

which have the same Alexander polynomial, as we mentioned earlier.

The final chapter of the paper, Chapter 10, discusses some of the additional

structure of the rings

H+[rr]

and

H[rr],

regarded as invariants of a knot with group

rr.

Choice of a commuting meridian and longitude pair

m,

l

Err

gives

H[rr]

the struc-

ture of an algebra over k[m±l, t±I]. An oriented equivalence of two oriented knots

induces an isomorphism of the corresponding

H[rr]

algebras over k[m±

1

,

t±

1].

Thus,

in principle,

H[rr]

can be used to detect non-amphicheirality or non-invertibility of

a knot, by showing no automorphism of

H[rr]

exists with appropriate behavior on

the elements

m,

l. Also, the trace of a meridian in

H+[rr]

is an element, say

M,

invariant under isomorphisms corresponding to unoriented knot equivalences. Thus

H+[rr],

as an invariant of unoriented knots, is an algebra over k[M].

Certain polynomial invariants can be extracted from. this extra structure on

H[rr]

and H+

[rr]

for knot groups. The A-polynomial of [4] is readily seen to be

closely related to the structure map

a:

k[m±

1

,t± 1]---

H[rr].

For many knots, the

A-polynomial is essentially kernel

(a).

There is also a discriminant type polynomial

invariant of the extension k[M]---

H+[rr],

which we call the D-polynomial. We have

already mentioned that even though H+

[rr]

is a finitely presented commutative ring,

and

H[rr]

is a finitely presented module over

H+[rr],

it is still difficult to determine