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The monograph gives a systematic treatment of 3-dimensional topological quantum field theories TQFTs based on the work of the author with N. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.
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Quantum Invariants of Knots and 3-Manifolds
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Knot theory
You are also entitled to have goods repaired or replaced if the goods fail to be of acceptable quality and the failure does not amount to a major failure. Please refer to the ACL official website for details. For any questions, feel free to contact us. We will answer your enquiries via eBay messages within 2 business days. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology. These topologists in the early part of the 20th century— Max Dehn , J.
Alexander , and others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject. In the late s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots , enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in Sossinsky , pp.
A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology. In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral has a "handedness" or not Simon Tangles , strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA Flapan Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation Collins A knot is created by beginning with a one- dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop Adams Sossinsky Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.
The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. Two knots K 1 and K 2 are equivalent if there exists a continuous mapping H: Such a function H is known as an ambient isotopy. These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of R 3 to itself is the final stage of an ambient isotopy starting from the identity.
The basic problem of knot theory, the recognition problem , is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late s Hass Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is Hass The special case of recognizing the unknot , called the unknotting problem , is of particular interest Hoste A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall.
A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings , where the "shadow" of the knot crosses itself once transversely Rolfsen At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand.
This is often done by creating a break in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing. These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link. Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space. A reduced diagram is a knot diagram in which there are no reducible crossings also nugatory or removable crossings , or in which all of the reducible crossings have been removed.
In , working with this diagrammatic form of knots, J.
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Alexander and Garland Baird Briggs, and independently Kurt Reidemeister , demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves , are:. The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another.
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The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point. A close inspection will show that complicated events can be eliminated, leaving only the simplest events: These are precisely the Reidemeister moves Sossinsky , ch.
A knot invariant is a "quantity" that is the same for equivalent knots Adams Lickorish Rolfsen For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability. In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones and Alexander polynomials.
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A variant of the Alexander polynomial, the Alexander—Conway polynomial , is a polynomial in the variable z with integer coefficients Lickorish The Alexander—Conway polynomial is actually defined in terms of links , which consist of one or more knots entangled with each other. The concepts explained above for knots, e. Consider an oriented link diagram, i. The second rule is what is often referred to as a skein relation. 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. Applying the relation to the Hopf link where indicated,.
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The unlink takes a bit of sneakiness:. 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". 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 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 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.