INTRODUCTION 11

of elements of H-[n] have components in both summands. For

x,y

E

H-[n], we

have

xy

=

~~~

+

ixyi

E H+[n] EB H-[n], where

~~~

=

~(xy

+

yx):

H-[n] x H-[n]-+ H+[n]

is a symmetric H+[n]-bilinear form on H-[n] and where

1

ixyi

=

2(xy- yx):

H-[n] x H-[n]-+ H-[n]

is a skew-symmetric Lie algebra product on H- [n], over H+ [n]. It is relatively easy

to see that whenever R

®

H[n] is a free rank 4 algebra over R then the form

lXI

on

H+~

y

the 3-dimensional Lie algebra

R

®

H-[n] coincides with the (slightly modified)

H+[7r]

Killing form ~tr(ad(x)ad(y)), where

ad(x)(z)

=

ixzi.

Quaternion algebras again play a major role in Chapter 7 and the Addendum

to Chapter 7. Free module quaternion algebras

H(r, s; R)

appear in two ways. For

any group

1r

and any elements x, y

E

H[n], let r

=

~~~,

s

=

~~

;I

E

H+[n]. Then

H[n][r-

1

,s-

1]

=

H(r,s;H+[n][r-l,s-

1]).

The quaternion basis is

{1,i,j,k} = {1,

lxl,

ixyi,

lxllxy!}.

Again, it is necessary to

find a pair of elements

x,

y so that

~~~~~

;I

is not nilpotent in H+[n], to get a non-

zero localization. For a free group

1r

n,

one can take x and y to be two generators.

Secondly, if

x, y

E

H[n] are two elements with

T(x)

=

T(y)

and

N(x)

=

N(y),

for

example two meridian generators of a knot group, then, again with s

=

~~

;I,

H[n][s-

1]

=

H(I,

J;

H+[n][s-

1]).

Here, the quaternion basis is {1, i, j, k}

=

{1,

lxl - IYI, lxl + IYI, 2lxyl}

and I

=

i

2

=

2

(~~~-~~~),

J

=

P

=

2

(1~1 +~~I)·

The assumption

T(x)

=

T(y), N(x)

=

N(y)

implies

lxl

2

=

~~~

=

~~~

=

IYI

2

,

so i

=

lxi-IYI

and j

=

lxl + IYI

anticommute.

As in Chapter 4, once we have quaternion algebra localizations of H[n], it

is easy to construct irreducible representations of

1r.

Also, in the Addendum to

Chapter 7, we show how to write down very efficient presentations of H+ [

1r ][

s -

1]

and H+[n][d-

1],

where s

=

lgP gql

and d

=

lgrgsgtl,

using quaternion algebras and

gq gp

our knowledge of the localizations H+[nn][s-

1]

and H+[nn][d-

1]

when

1fn

is free.

We obtain some algorithms for describing, say, all conjugate classes of absolutely

irreducible representations p :

1r

-+ SL(2, K), where

1r

is a knot group and K is a

field, but in practice one would face serious computational problems carrying out

these algorithms and making use of the results.

Following

the rather formidable algebraic developments in chapters 5,6, and

7, we make a completely fresh, elementary start in Chapter 8. The idea, roughly,

is to develop the classical view of representations of

1r

as points of an algebraic