Summary: Representation Theory and the Apolynomial of a knot.
D. Cooper D.D. Long \Lambda
1 The motivation and basic idea.
A knot is a smooth simple closed curve, K; in the threesphere, S 3 : The knot exterior is the compact
threemanifold X = S 3 \Gamma j(K) where j(K) is an open tubular neighborhood. The group ß 1 (X) is
called the knot group associated to the knot. We recall that the boundary of X is a torus T and there
are two simple closed curves on X called a longitude and meridian which intersect transversely in a
single point. These two curves generate ß 1 (T ) ¸ = Z \Theta Z which is usually referred to as the peripheral
subgroup. In the sixties, Waldhausen showed that the data consisting of the knot group plus the peripheral
subgroup is a complete knot invariant. The difficulty is that although a powerful invariant it is
extremely difficult to work with this data directly. One usually makes a tradeoff and at the expense
of some loss of information constructs a usable invariant. The Alexander polynomial is an example
of this. A classical way to study groups is to look at their linear representations and this suggests that
one studies representations of knot groups into linear groups. Waldhausen's results suggest that one
should also try and keep track of some of the peripheral data.
What is actually studied in this context is the totality of linear representations into SL(2; C).
(One can work with more general Lie groups but a good deal less is known. The isomorphism
PSL(2; C) ¸ = Isom + (H 3 ) makes this choice occupy a special place in the current theory.) As we
indicate below, given a finitely generated group, G; the set of representations of G into SL(2; C) is
an affine algebraic set which we will refer to as the representation variety even though it is typically