Page 260 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 251

modular Hopf algebra [76].
The theory of quantum invariants had led to the discovery of tangle cate-
gories. By looking at the most generalized form of `knot', which is a graph with
`knotted ribboned parts', one can construct so-called ribbon categories. The
finite dimensional representations for quantum groups comprise such a ribbon
category. This connection could thus be used to find the new invariants for
3-manifolds. Other results made Reidemeister's theorem a covariant functor
between the categories of links and diagrams, while quantum invariants be-
came covariant functors from tangle categories to categories of modules. On
the whole, category theory has been able to cut a lot of cake, as the language
was quite effective to formulate and extend very general ideas about the central
link polynomials.
So far, the story has been one of `simple' hierarchical generalizations.
An exciting change of perspective comes from V. A. Vassiliev [91]. He
considers the so-called knot space M, which is the space of all embeddings
ry : S1 '- S3. This allows one to study more than just a single knot and
the ways in which distinct knots relate to each other. The object of utmost
interest is the natural stratification of M . Deforming knots to the level where
we permit self-intersections of the cord in which they are realised leads to the
notion of chambers in M. The discriminant C of M is defined to be the set
of mappings which are not embeddings. This is a singular hypersurface in
M. The components of M - C are clearly in one-to-one correspondence with
the knot types. Thinking of a numerical knot invariant as a function on the
components of M - C one is led to study the cohomology of M. Vassiliev
introduced a system of subgroups of H°(M - C) :

0=G1CG2CG3C...CH°(M-C)

where H° is reduced cohomology with integer coefficients and Gi is free abelian
of finite rank. The evaluation of an element in Gi/Gi_1 on the component of
M - C corresponding to an oriented knot type K yields a rational number
ui(K) associated to K. This is a Vassiliev invariant of order i. Like the
Jones invariants, one computes Vassiliev invariants from a diagram by changing
crossings. However, the combinatorics of the computation are very much more
difficult.
Vassiliev's invariants are rational numbers, while generalized Jones invari-
ants are Laurent polynomials over Z[q, q-1]. Joan Birman and Xiao-Song Lin
showed that there is a relation between them [16]. Let a knot K have Jones
polynomial VK(t). Set UU(K) = VK(ex) and express it as a power series in x:
00
Q,(K) = Eui(K)xi
i=0