Solubility equilibrium

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. Template:Infobox knot theory

Figure-eight knot of practical knot-tying, with ends joined.

In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot. The figure-eight knot is a prime knot.

Origin of name

The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.

Description

A simple parametric representation of the figure-eight knot is as the set of all points (x,y,z) where

${\begin{aligned}x&=\left(2+\cos {(2t)}\right)\cos {(3t)}\\y&=\left(2+\cos {(2t)}\right)\sin {(3t)}\\z&=\sin {(4t)}\end{aligned}}$

for t varying over the real numbers (see 2D visual realization at bottom right).

The figure-eight knot is prime, alternating, rational with an associated value of 5/2, and is achiral. The figure-eight knot is also a fibered knot. This follows from other, less simple (but very interesting) representations of the knot:

(1) It is a homogeneous closed braid (namely, the closure of the 3-string braid σ₁σ₂^-1σ₁σ₂^-1), and a theorem of John Stallings shows that any closed homogeneous braid is fibered.

(2) It is the link at (0,0,0,0) of an isolated critical point of a real-polynomial map F: R⁴→R², so (according to a theorem of John Milnor) the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely,

$F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!$

where

${\begin{aligned}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\end{aligned}}$

Mathematical properties

Simple squared depiction of figure-eight configuration.

The figure-eight knot has played an important role historically (and continues to do so) in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction to other knots and links.

The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume, 2.02988... according to the work of Chun Cao and Robert Meyerhoff. From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds.

The figure-eight knot and the (−2,3,7) pretzel knot are the only two hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6.

Symmetric depiction generated by parametric equations.

Invariants

The Alexander polynomial of the figure-eight knot is

$\Delta (t)=-t+3-t^{-1},\$

the Conway polynomial is

$\nabla (z)=1-z^{2},\$

and the Jones polynomial is

$V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\$

The symmetry between $q$ and $q^{-1}$ in the Jones polynomial reflects the fact that the figure-eight knot is achiral.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4 (2000), 431–449. Template:MathSciNet
• Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. Template:MathSciNet
• Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140 (2000), no. 2, 243–282. Template:MathSciNet
• Marc Lackenby and Robert Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176
• Robion Kirby, Problems in low-dimensional topology, (see problem 1.77, due to Cameron Gordon, for exceptional slopes)
• William Thurston, The Geometry and Topology of Three-Manifolds, Princeton University lecture notes (1978–1981).

External links

• Template:Knot Atlas
• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.