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{{Infobox knot theory
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Here is more information in regards to [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] have a look at the website.
| name=             
| practical name=   
| image=            Hopf Link.png
| caption=         
| arf invariant=   
| braid length=      2
| braid number=      2
| bridge number=   
| crossing number=  2
| hyperbolic volume= 0
| linking number=    1
| stick number=      6
| unknotting number= 1
| conway_notation= 
| ab_notation=      2{{sup sub|2|1}}
| dowker notation= 
| thistlethwaite=    L2a1
| other=           
| alternating=      alternating
| class=            torus
| fibered=          fibered
| slice=           
| symmetry=       
| tricolorable=    tricolorable
| last link=        L0
| next link=        L4a1
}}
[[Image:Skein-relation-link22-plus-sm.png|thumb|right|[[Skein relation]] for the Hopf link.]]
 
In [[mathematical]] [[knot theory]], the '''Hopf link''' is the simplest nontrivial [[link (knot theory)|link]] with more than one component.<ref>{{citation|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots|first=Colin Conrad|last=Adams|publisher=American Mathematical Society|year=2004|isbn=9780821836781|page=151|url=http://books.google.com/books?id=M-B8XedeL9sC&pg=PA151}}.</ref> It consists of two [[circle]]s linked together exactly once,<ref name="ks98"/> and is named after [[Heinz Hopf]].<ref name="ps97"/>
 
==Geometric realization==
A concrete model consists of two [[unit circle]]s in perpendicular planes, each passing through the center of the other.<ref name="ks98">{{citation
| last1 = Kusner | first1 = Robert B.
| last2 = Sullivan | first2 = John M.
| contribution = On distortion and thickness of knots
| doi = 10.1007/978-1-4612-1712-1_7
| location = New York
| mr = 1655037
| pages = 67–78
| publisher = Springer
| series = IMA Vol. Math. Appl.
| title = Topology and geometry in polymer science (Minneapolis, MN, 1996)
| volume = 103
| year = 1998}}. See in particular [http://books.google.com/books?id=FIPcAxs29ikC&pg=PA77 p.&nbsp;77].</ref> This model minimizes the [[ropelength]] of the link and until 2002 the Hopf link was the only link whose ropelength was known.<ref>{{citation
| last1 = Cantarella | first1 = Jason
| last2 = Kusner | first2 = Robert B.
| last3 = Sullivan | first3 = John M.
| arxiv = math/0103224
| doi = 10.1007/s00222-002-0234-y
| issue = 2
| journal = Inventiones Mathematicae
| mr = 1933586
| pages = 257–286
| title = On the minimum ropelength of knots and links
| volume = 150
| year = 2002}}.</ref> The [[convex hull]] of these two circles forms a shape called an [[oloid]].<ref>{{citation
| last1 = Dirnböck | first1 = Hans
| last2 = Stachel | first2 = Hellmuth
| issue = 2
| journal = Journal for Geometry and Graphics
| mr = 1622664
| pages = 105–118
| title = The development of the oloid
| url = http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf
| volume = 1
| year = 1997}}.</ref>
 
==Properties==
Depending on the relative [[Orientation (geometry)|orientation]]s of the two components the [[linking number]] of the Hopf link is &plusmn;1.<ref>{{harvtxt|Adams|2004}}, [http://books.google.com/books?id=M-B8XedeL9sC&pg=PA21 p.&nbsp;21].</ref>
 
The Hopf link is a (2,2)-[[torus link]]<ref>{{citation|title=On Knots|volume=115|series=Annals of Mathematics Studies|first=Louis H.|last=Kauffman|publisher=Princeton University Press|year=1987|isbn=9780691084350|page=373|url=http://books.google.com/books?id=BLvGkIY8YzwC&pg=PA373}}.</ref> with the [[braid word]]<ref>{{harvtxt|Adams|2004}}, Exercise 5.22, [http://books.google.com/books?id=M-B8XedeL9sC&pg=PA133 p.&nbsp;133].</ref>
:<math>\sigma_1^2.\,</math>
 
The [[knot complement]] of the Hopf link is '''R'''&nbsp;&times;&nbsp;''S''<sup>1</sup>&nbsp;&times;&nbsp;''S''<sup>1</sup>, the [[Cylinder (geometry)|cylinder]] over a [[torus]].<ref>{{citation|title=Quantum Invariants of Knots and 3-manifolds|volume=18|series=De Gruyter studies in mathematics|first=Vladimir G.|last=Turaev|publisher=Walter de Gruyter|year=2010|isbn=9783110221831|page=194|url=http://books.google.com/books?id=w7dActmezxQC&pg=PA194}}.</ref> This space has a [[Geometrization conjecture|locally Euclidean geometry]], so the Hopf link is not a [[hyperbolic link]]. The [[knot group]] of the Hopf link (the [[fundamental group]] of its complement) is '''Z'''<sup>2</sup> (the [[free abelian group]] on two generators), distinguishing it from an unlinked pair of loops which has the [[free group]] on two generators as its group.<ref>{{citation|title=Algebraic Topology|year=2002|first=Allen|last=Hatcher|isbn= 9787302105886|page=24|url=http://books.google.com/books?id=xsIiEhRfwuIC&pg=PA24}}.</ref>
 
==Hopf bundle==
The [[Hopf fibration]] is a continuous function from the [[3-sphere]] (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar [[sphere|2-sphere]], with the property that the inverse image of each point on the 2-sphere is a circle. Thus, these images decompose the 3-sphere into a continuous family of circles, and
each two distinct circles form a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial [[fibration]]. This example began the study of [[homotopy groups of spheres]].<ref>{{citation|title=Basic Algebraic Topology|first=Anant R.|last=Shastri|publisher=CRC Press|year=2013|isbn=9781466562431|url=http://books.google.com/books?id=lYMAAQAAQBAJ&pg=PA368|page=368}}.</ref>
 
==History==
[[File:Buzanha wachigai mon.jpg|thumb|[[Shingon-shu Buzan-ha|Buzan-ha]] crest]]
The Hopf link is named after topologist [[Heinz Hopf]], who considered it in 1931 as part of his research on the [[Hopf fibration]].<ref>{{citation
|last= Hopf
|first= Heinz
|author-link= Heinz Hopf
|title= Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche
|journal= [[Mathematische Annalen]]
|publisher= [[Springer-Verlag|Springer]]
|location= Berlin
|volume= 104
|issue= 1
|pages= 637–665
|year= 1931
|url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002274760
|doi= 10.1007/BF01457962
}}.</ref> However, in mathematics, it was known to [[Carl Friedrich Gauss]] before the work of Hopf.<ref name="ps97">{{citation
| last1 = Prasolov | first1 = V. V.
| last2 = Sossinsky | first2 = A. B.
| isbn = 0-8218-0588-6
| location = Providence, RI
| mr = 1414898
| page = 6
| publisher = American Mathematical Society
| series = Translations of Mathematical Monographs
| title = Knots, links, braids and 3-manifolds: An introduction to the new invariants in low-dimensional topology
| url = http://books.google.com/books?id=znCLtJKnZXQC&pg=PA6
| volume = 154
| year = 1997}}.</ref> It has also long been used outside mathematics, for instance as the crest of [[Shingon-shu Buzan-ha|Buzan-ha]], a Japanese Buddhist sect founded in the 16th century.
 
==See also==
*[[Catenane]], a molecule with two linked loops
*[[Solomon's knot]], two loops which are doubly linked
 
==References==
{{reflist}}
 
==External links==
{{commons category|Hopf links}}
*{{MathWorld|urlname=HopfLink|title=Hopf Link}}
*{{Knot Atlas|L2a1|Hopf link}}
 
{{Knot theory|state=collapsed}}

Latest revision as of 16:02, 29 May 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

Here is more information in regards to Dentists in DC have a look at the website.