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
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