This book is a survey of current topics in the mathematical theory of knots. For a mathematician, a knot is a closed loop in 3-dimensional space: imagine knotting an extension cord and then closing it up by inserting its plug into its outlet. Table of Contents for the Handbook of Knot Theory. William W. Menasco and Morwen B. Thistlethwaite, Editors. (1) Colin Adams, Hyperbolic Knots. (2) Joan S. It discusses the foundations of knot theory, focusing on virtual knot theory and topological quantum field theory. Reidemeister's theorem states that two diagrams represent ambient isotopic knots (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other.
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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.
Handbook of Knot Theory - Google книги
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. The unlink takes a bit of sneakiness: Putting all this together will show: Since the Alexander—Conway polynomial is a knot invariant, handbook of knot theory shows that the trefoil is not equivalent to the unknot.
Mathematics > Geometric Topology
So the trefoil really is "knotted". The left-handed trefoil knot.
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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 Dehnbefore 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 handbook of knot theory 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 knotsmeaning that the knot complement i. The hyperbolic structure depends only on the handbook of knot theory so any quantity computed handbook of knot theory 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.
SnapPea 's cusp view: 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 handbook of knot theory 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 handbook of knot theory of one color as there are infinitely many light rays from the observer to the link component.
Knot theory - Wikipedia
The fundamental parallelogram which is indicated in the picturetiles 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 handbook of knot theory fundamental parallelogram, length of shortest geodesic, and volume.
Modern knot and link tabulation efforts have utilized these invariants effectively.
[math/] Computation of Hyperbolic Structures in Knot Theory
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 handbook of knot theory 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.