About this book Introduction This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. Here, however, knot theory is considered as part of geometric topology. The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology. The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding.
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To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.
The following is an example of a typical computation using a skein relation. It computes the Alexander—Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied. Since the Alexander—Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot.
So the trefoil really is "knotted". The right-handed trefoil knot. Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image.
These are not equivalent to each other, meaning that they are not amphicheiral. This was shown by Max Dehn , before the invention of knot polynomials, using group theoretical methods Dehn But the Alexander—Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image.
The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots Lickorish Hyperbolic invariants Edit William Thurston proved many knots are hyperbolic knots , meaning that the knot complement i.
The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant Adams The Borromean rings are a link with the property that removing one ring unlinks the others. Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings.
The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained.
Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres of one color as there are infinitely many light rays from the observer to the link component.
The fundamental parallelogram which is indicated in the picture , tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely. This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume.
Modern knot and link tabulation efforts have utilized these invariants effectively. Higher dimensions Edit A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle the front strand having no component there ; then slide it forward, and drop it back, now in front.
Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle. In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain. Knotting spheres of higher dimension Edit Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere S2 embedded in 4-dimensional Euclidean space R4.
Such an embedding is knotted if there is no homeomorphism of R4 onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots. The mathematical technique called "general position" implies that for a given n-sphere in m-dimensional Euclidean space, if m is large enough depending on n , the sphere should be unknotted. The notion of a knot has further generalisations in mathematics, see: knot mathematics , isotopy classification of embeddings.
An n-knot is a single Sn embedded in Rm. An n-link is k-copies of Sn embedded in Rm, where k is a natural number.
An Introduction to Knot Theory
Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. The study of knots can be given some motivation in terms of applications in molecular biology or by reference to paral lels in equilibrium statistical mechanics or quantum field theory. Here, however, knot theory is considered as part of geometric topology.
W. B. R. Lickorish
To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way. The following is an example of a typical computation using a skein relation. It computes the Alexander—Conway polynomial of the trefoil knot.