Page 259 - J. C. Turner "History and Science of Knots"

250 History and Science of Knots

structures describing symmetry properties encountered in solvable statistical
models. However, they have not much to do with quantum mechanics, and
are not groups either! Drinfel'd, who introduced the term quantum group,
defined the structure as a Hopf algebra, essentially a bialgebra with an an-
tipode. Quantum groups constitute an exciting generalization of the concept
of symmetry. In this context, the parametrised YBE becomes the Quantum
YBE (i.e. QYBE).
The invariants from the quantum group setting are polynomials. They
are gathered under the heading quantum invariants, also called generalized
Jones invariants. Since transfer matrices satisfy the YBE if and only if they
are a representation for Bn, every quantum invariant is obtained from a trace
function on an R-matrix representation for B.
The renaissance in the interactions between between physics and knot
theory (recall Gauss's use of properties of knots in the solution of an electro-
magnetic problem, and Thompson's plans to describe chemistry in terms of
knots) was due to statistical mechanics studies. But nowadays there are at
least three other, different, ways in which physics and knot theory are related.
Not only is there topological quantum field theory, but the theory of quantum
invariants has also proved to be closely related to conformal field theories.
In this connection one should mention Edward Witten's papers [96], where
it is shown that Jones' polynomial and its generalizations are related to the
topological Chern Simons actions.
So, order is emerging from chaos, and new results are being achieved con-
tinuously. The order appears to be part of an even larger order, which involves
the physics of conformal field theory, and leads to further invariants, now in
arbitrary 3-manifolds. As we have seen, the theory of R-matrices gives a sys-
tematic description for the quantum invariants. It has been known for a long
time that any compact orientable 3-manifold arises (up to PL-homeomorphism
or diffeomorphism, depending on the choice of category) as the boundary of
a 4-manifold M, where M is obtained by attaching 2-handles to the 4-ball,
along some framed link in S3, i.e. by surgery along framed links. Lickorish
and Wallace proved this at the beginning of the 1960s [61], [92]. One can think
of framing as the thickening of the knot into a ribbon-like object. Any closed
oriented 3-manifold may thus be obtained by performing surgery along differ-
ent framed links in the 3-sphere. This yields an equivalence relation on framed
links. Robion Kirby proposed a set of moves which generated this equivalence
relation [55]. Roger Fenn and Colin Rourke simplified them. Their moves may
be described by means of tangle generators [33]. By using Kirby, Fenn and
Rourke calculus, Nicolai Reshetikhin and Vladimir Turaev in 1991 defined 3-
manifold invariants using the theory of quantum groups. They produced new
3-manifold invariants, which can be defined from any simple Lie algebra, pro-
vided the associated quantum groups have the structure of a finite dimensional