Multiple gamma function: Difference between revisions

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{{Infobox knot theory
| name=              Square knot
| practical name=    Reef knot|Square knot
| image=            Blue Square Knot.png
| caption=          Three-dimensional view
| arf invariant=   
| bridge number=   
| crossing number=  6
| linking number=   
| stick number=     
| unknotting number=
| conway_notation= 
| ab_notation=     
| dowker notation= 
| thistlethwaite=   
| last knot=       
| next knot=       
| alternating=     
| class=            composite
| fibered=         
| pretzel=          pretzel
| slice=            slice
| symmetry=        amphichiral
| tricolorable=    tricolorable
}}
[[File:Square ribbon knot.svg|thumb|240px|The square knot, drawn as a [[ribbon knot]]]]
 
In [[knot theory]], the '''square knot''' is a [[composite knot]] obtained by taking the [[Connected sum#Connected sum of knots|connected sum]] of a [[trefoil knot]] with its [[Reflection (mathematics)|reflection]]. It is closely related to the [[granny knot (mathematics)|granny knot]], which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots.
 
The square knot is the mathematical version of the common [[reef knot]].
 
==Construction==
The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise.  The resulting connected sum is the square knot.
 
It is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot.
 
==Properties==
The square knot is [[amphichiral knot|amphichiral]], meaning that it is indistinguishable from its own mirror image. The [[crossing number (knot theory)|crossing number]] of a square knot is six, which is the smallest possible crossing number for a composite knot.
 
The [[Alexander polynomial]] of the square knot is
 
:<math>\Delta(t) = (t - 1 + t^{-1})^2, \, </math>
 
which is simply the [[square (algebra)|square]] of the Alexander polynomial of a trefoil knot. Similarly, the [[Alexander–Conway polynomial]] of a square knot is
 
:<math>\nabla(z) = (z^2+1)^2.</math>
 
These two polynomials are the same as those for the granny knot. However, the [[Jones polynomial]] for the square knot is
 
:<math>V(q) = (q^{-1} + q^{-3} - q^{-4})(q + q^3 - q^4) = -q^3 + q^2 - q + 3 - q^{-1} + q^{-2} - q^{-3}. \, </math>
 
This is the product of the Jones polynomials for the right-handed and left-handed trefoil knots, and is different from the Jones polynomial for a granny knot.
 
The [[knot group]] of the square knot is given by the presentation
 
:<math>\langle x, y, z \mid x y x = y x y, x z x = z x z \rangle. \, </math><ref>{{MathWorld|title=Square Knot|urlname=SquareKnot}}</ref>
 
This is [[group isomorphism|isomorphic]] to the knot group of the granny knot, and is the simplest example of two different knots with isomorphic knot groups.
 
Unlike the granny knot, the square knot is a [[ribbon knot]], and it is therefore also a [[slice knot]].
 
==References==
{{reflist}}
 
{{Knot theory|state=collapsed}}

Latest revision as of 14:21, 5 January 2015

It can be very expensive to attend the dentist now ,. If you only get your teeth cleaned and a general checkup, you're still looking at bill that's often over $100. Of course, neglecting your teeth is even more in the long run, as certainly you'll have deal with the problems. It is cut back on dental costs by using some sensible guidelines for oral wellbeing. They don't take much time and are simple to do. If you loved this article and also you would like to obtain more info about Bradenton Dentists kindly visit our own web-site. Getting your teeth into better shape isn't hard if you follow some basic factors. What's important is that you make these part of your everyday schedule.

You may possibly heard of yet one other popular option called porcelain veneers. When obtain veneers ascribed to your teeth, the porcelain is designed give the front side of your teeth your own surface. Advertising have seen someone with that gorgeous movie star smile, is actually an a pretty good chance that mention have had porcelain veneers applied a few time time of life.

Composite veneers are just a little bit cheaper, as they do not require the impression and custom-made coverings. Instead, your dentist will make the new surface of your teeth while you've got the office to suit your teeth. Furthermore, it does n't need as much of the enamel to be removed. Instead, the dentist will provide simple facts of pearly white's to increase the strength of the link. Then, a composite material is designed into the lips. Once it is dry, your teeth will once again be sparkling white.

Fixing any of these problems is now far easier than it ever was before. In fact, the majority of can be helped with a painless for you to reshape and permanently whiten your teeth called porcelain veneers. One kind of veneer is termed as Lumineers.

Now you might be lucky this may let you spouse that's still employed by a company or organization that has a dental plan that also covers girlfriend. In which case the pressure is off finding your dental insurance plan. However, if you spouse's plan isn't particularly generous, he/she does not a plan or are usually single, you are heading to want to do some research to find dental plans.

Vets. For the dog, in order to the dreaded vets is not different a person having a dentist visit. Putting the collar about them the day before and within visit on the vet great this should help keep these things calm. Alternatively put the diffuser on a day before and inside previous evening to help soothe puppy before their trip into the vets.

Try stuffing my face with some cheese after you eat, it'll help your enamel. Studies have shown that the calcium in cheese can help rebuild the enamel on your teeth. Strong, healthy enamel means beautiful white teeth, so ensure that to eat cheese up to a few times a time.

Walnut, a dark and exclusive wood, is among Bradenton Dentists the types found making hardwood panels. Its color comes from light brown to dark chocolate, and sometimes purple. It gains luster as it ages and also a straight graining. Is actually also 20% softer than Pine.

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