8

INTRODUCTION

The polynomial

Q(I, J)

itself, and thus the presentations above of the rings H+

[1r]

and

H[1r],

depend on a presentation of the group

1r.

This brings up an important

general point. Although H+

[1r]

and

H[1r]

look like powerful invariants of knots,

there are clearly formidable algebraic difficulties involved in deciding whether, say,

the rings H+

[1r1]

and H+

[1r2

]

are isomorphic, if

1r1

and

1r2

are two finitely presented

groups. At least this is a question about finitely presented commutative rings,

rather than finitely presented non-abelian groups. For knot groups, both H+

[1r]

and

H[1r]

have additional invariant structure which is useful. In the case of the

two-bridge knots, the first distinct pair on the knot table which share the same

Conway polynomial (the knots 74 and 92

,

or in Schubert's notation (15,11) and

(15,7), with Conway polynomial 1

+

4z

2

)

can be distinguished by their respective

rings

H+[7r]/(I3

).

The elements i

E

H[1r]

and

I=

i

2

E

H+[1r]

in the presentations

above are not invariants, but the ideals

((i))

C

H[1r]

and

(I)

c

H+[7r]

turn out

to be invariants. We establish these facts in the Addendum to Chapter 8, where

we also explain how the Alexander module of any knot is a natural subquotient of

H[7r].

In chapters 5, 6, and 7 and the addenda to these chapters, we make a detailed

study of H+

[7rn]

and

H[7rnJ,

where

7rn

is the free group on generators

{g;},

1 ::; i ::; n.

Basically, we obtain some finite presentations of the ring H+

[1r n]

and of the algebra

H[7rn] = H+[7rn]

ffiH~[7rn]

as a module over

H+[7rn]·

We

assume~

E k throughout

these chapters. First of all, in Chapter 5 we establish a large number of new

relations between the elements defined earlier

In fact, we establish universal Polynomial Identities involving elements of this type,

that is, the gi can be replaced by any elements Xi

E

H[7rn]·

Here is a brief list of

the most important identities. We need a little more notation.

IX]

Y1

X21

Y2

det

(lx'l)

E

H+[1r]

YJ

I::i.j::;2

IX]

X2

X:! I

(-1)det

(I~;

I)

I;i.j;:l

E

H+[1r]

Y1 Y2

Y:l

Determinant Identity

Rank Three Identity

o

=

det

(lxil)

Yi

I

::z.J::

I

Tetrad Identity

0

=

jx,x2x:dlx.,j-jx,x2x.dlx:ll +

lxJX:Jxdlx21-lx2X:Jx.dlxii