Page 261 - J. C. Turner "History and Science of Knots"

252 History and Science of Knots

then uo(K) = 1 and each ui(K), for i > 1, is a Vassiliev invariant of order
i. Although this scheme so far works only for knots, the Vassiliev invariants
seem to offer at least part of the topological framework we seek for the quan-
tum invariants. Hence there are speculations abounding. For instance, it is
well-known that quantum groups do not detect invertibility of knots, but Vas-
siliev's invariants just might. This would make them stronger than quantum
invariants. The research with ribbon categories and Vassiliev invariants have
caused the singular braid monoid to place itself permanently on the scene.
Moreover it threatens to become just as fundamental a mathematical tool as
the braid groups. However, amongst all of these recent developments, mainly
involving the braid groups, the knot complement group is far from forgotten.
David Joyce, Colin Rourke and Roger Fenn have concocted an algebraic struc-
ture, which dates back to John Conway and van Brieskorn in the 1960s, and
is now called a rack. It generalizes the knot group, but also captures aspects
of the peripheral system. This completely classifying invariant seems to be a
promising and exciting new part of the overall picture [34], [51].

15. The Future?
In the foregoing sections we have seen how vague, intuitive notions about
knotted structures, beginning with the work of Listing, Gauss, Kirkman, Tait
and Little of last century, were gradually developed until they reached, by the
last decade of this century, extremely high levels of abstraction and complexity.
Concerning the future of this process, one can only speculate on how far, and
in which directions, the current attempts to solve a variety of outstanding
problems will take us.
The prime knots with up to 13 crossings have been distinguished and
tabled; and these knots are relatively simple objects. Attacks on the classifi-
cation of 14- and 15-fold crossing knots are in progress; there are very many
more of such knots, and no doubt it will require combinations of several of the
available invariants to distinguish them all. It will be difficult, and perhaps
not sensible, to produce diagrams for these vast numbers of knots; most will
be `known' only by their corresponding invariant values, arranged in classes
and stored in some digitised form. The baffling problem of finding a single,
complete, knot invariant (if one exists) still remains.
Future research will certainly be affected by the amazing developments
stemming from Jones' discovery. They continue unrelentingly; yet many simple
questions remain unanswered. It is still not known whether a non-trivial link
can have the same V-, P- or F-polynomial as has the trivial link of the same
number of components; we know this can happen for A and V. Resolution of
this question would lead to significant conceptual progress. No generalization
of it to knots and links of higher dimension has yet been achieved.