In Preparation
The information below is preliminary and cannot be trusted! (v)
A HOMFLY Braidor
The Algebra
Let
be the free associative (but non-commutative) algebra generated by the elements of the symmetric group
on
and by formal variables
and
, and let
be the quotient of
by the following "HOMFLY" relations:
commutes with everything else.
- The product of permutations is as in the symmetric group
.
- If
is a permutation then
.
, where
is the transposition of
and
.
Finally, declare that
while
for every
and every
, and let
be the graded completion of
.
We say that an element of
is "sorted" if it is written in the form
where
is a permutation and
and the
's are all non-negative integer. The HOMFLY relations imply that every element of
is a linear combinations of sorted elements. Thus as a vector space,
can be identified with the ring
of power series in the variables
tensored with the group ring of
. The product of
is of course very different than that of
.
The Equations
The Equations in Functional Form
A Solution