Knot theory is a
branch of algebraic topology where one studies what is known
as the placement problem, or the embedding of one
topological space into another. The simplest form of knot
theory involves the embedding of the unit circle into
three-dimensional space. For the purposes of this document a
knot is defined to be a closed piecewise linear curve in
three-dimensional Euclidean space R^{3}. Two or more
knots together are called a link. Thus a mathematical knot
is somewhat different from the usual idea of a knot, that
is, a piece of string with free ends. The knots studied in
knot theory are (almost) always considered to be closed
loops.
Two knots or links are considered
equivalent if one can be smoothly deformed into the other,
or equivalently, if there exists a
*homeomorphism*
on R^{3} which maps
the image of the first knot onto the second. Cutting the
knot or allowing it to pass through itself are not
permitted. In general it is very difficult problem to decide
if two given knots are equivalent, and much of knot theory
is devoted to developing techniques to aid in answering this
question. Knots that are equivalent to polygonal paths in
three-dimensional space are called *tame.* All other
knots are known as *
wild.*
Most of knot theory concerns only tame knots, and these are
the only knots examined here. Knots that are equivalent to
the unit circle are considered to be unknotted or trivial.
The simplest
non-trivial knot is the trefoil knot which comes in a left
and a right handed form.
It is not too difficult to see (but
slightly more difficult to prove) that the trefoil is not
equivalent to the unknot. Also, the right and left handed
versions of the trefoil are only equivalent if the
homeomorphism mapping one into the other includes a
reflection (other knots, such as the Figure-8 knot *are*
equivalent to their mirror images, these knots are known as
achiral knots).
**
**
# Mathematical Institute News
# Whitehead Prize
A Whitehead
prize is awarded to Marc Lackenby of St. Catherine's
College and the University of Oxford for his
contributions to three dimensional topology and to
combinatorial group theory.
He has
proved two unexpected results about Dehn surgery, which
is a much used method to construct a three-dimensional
manifold M_{2} from another one M_{1}
based on a knot K
M_{1}
and a twisting coefficient p/q. The first is a
uniqueness result: If one performs a surgery that is
'far' from the trivial one on a knot K
M_{1}
which is a null-homotopic and H_{2} (M_{1})
nontrivial, then the homeomorphism class of M_{2}
determines M_{1}, K and p/q uniquely. The second
result is that there is a constant c depending only on M_{1}
such that if M_{2} is 'exceptional' then |q| <
c.
With Daryl
Cooper he also proved a remarkable finiteness result
that for a given M_{2} there are only finitely
many hyperbolic knots K
S^{3}
such that M_{2} can be obtained by a p/q surgery
if q > 22.
He has found
other remarkable results about hyperbolic three
dimensional manifolds. One is a simple algorithm
enhancing Thurston's famous result giving the existence
of hyperbolic structures on a large class of three
dimensional manifolds. The algorithm allows one to
calculate (up to explicit bounds) the volume of the (hyperbolic)
complement of a class of knots. Another of his theorems
is related to the famous 2p theorem of Gromov and
Thurston that a Dehn filling of a cusped hyperbolic
manifold M^{3} along a curve of length more than
2p always gives rise to a negatively curved manifold.
Using new methods Lackenby has shown that if 2p is
replaced by 6 then the fundamental group of the
resulting manifold is Gromov hyperbolic. A consequence
is that at most 12 manifolds obtained by surgery on a
hyperbolic knot can have non-negatively curved
fundamental group. This is close to the best possible
general result since the figure eight knot has ten
exceptional surgeries.
His recent work on the Heegaard genus of coverings has
opened up new relations with other areas of mathematics.
By using comparatively elementary methods, he has found
novel connections between the isoperimetric value of a
Cayley graph of a finite group and the Betti numbers of
a 2-complex associated with the presentation of the
group. There are exciting possible consequences of this
work in combinatorial group theory.
**
** |