4
INTRODUCTION
(Z/2)d-graded algebras. The idea is that link groups can be generated by meridian
elements, and isomorphisms of link groups induced by geometric link equivalences
carry meridians to meridians. The relations between meridians are such that one
has a well-defined (Z/2)d-length vector for all elements of
1r
written as words in
meridians. Other results in the Addendum to Chapter 1 are some formulas for
powers, conjugates, and commutators in H[n] and a brief discussion of PSL(2)
representations.
In Chapter 2, we introduce extremely useful extra structure in H[n]
when~ E
k.
We define and study two operators on H[n]
x+
=
x
+
L(x) and x-
=
x- L(x).
2 2
Then x
=
x+
+
x- and we get a decomposition H[n]
=
H+ [n] EB H- [n]. The
summand H+ [n] is a ring which is central in H[n] and which coincides with T H[n]
when
~ E
k. The summand H- [n] is an H+ [n]-module. With respect to this direct
sum decomposition, the involution
L
on H[n] is multiplication by
(1, -1).
There is
a parallel decomposition of the matrix ring M(2, A) if
~ E
A.
Namely
(
~ ~)
=
cr
Q )
(a2d
a+d
+
-2-
c
d~a)
which shows M (2, A)
=
AI EB
M~,
where AI denotes the diagonal matrices and
M~
denotes the matrices of trace zero.
It is convenient to have the following notation for certain elements of H[n] when
~
E
k.
lxl
We prove a few identities involving the (
+)
and (-)-operators, for example
~~~
is
a symmetric bilinear form,
Jxyj
is skew-symmetric, and
Jxyzj
is an alternating tri-
linear form, and arrive at the following results. Assume
1r
is generated by elements
{gi},i:::;
n.
Then H+[n]
=
k[gt,
1;~1,
l9r9s9tl], j k,
r s
t.
Again, these
generators are not algebraically independent if
n
~
3. Also, H- [n] is spanned as an
H+[n]-module by the elements
{JgiJ,
l9j9kl},j
k. This is a partial improvement
of the finiteness statement proved in Chapter
1,
but we go even further in Chapter
5. In chapters 3 and 4, we study two-generator groups, as discussed below, for
which the baby results in chapters 1 and 2 are adequate.
It should be emphasized that all the identities we prove in H[n] are more
or less familiar identities for 2 x 2 matrices. Obviously, if we proved first that
H[n] can be embedded in a matrix ring, we would have an alternate method of
establishing identities in H[n]. Since any H[n] is a quotient of H[7T], where 1T is a
free group, it would even suffice to prove H[7T] embeds in a matrix ring, in order to
establish identities in H[n]. But the existence of a matrix embedding of H[7T] is a
sophisticated result, so we think our very elementary development of arithmetic in
Previous Page Next Page