Knot theory

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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 R3. 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 R3 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).

University of Oxford

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 M2 from another one M1 based on a knot K subset symbol M1 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 subset symbol M1 which is a null-homotopic and H2 (M1) nontrivial, then the homeomorphism class of M2 determines M1, K and p/q uniquely. The second result is that there is a constant c depending only on M1 such that if M2 is ‘exceptional’ then |q| < c.

With Daryl Cooper he also proved a remarkable finiteness result that for a given M2 there are only finitely many hyperbolic knots K subset symbol S3 such that M2 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 M3 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.

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