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{{Infobox knot theory
| name=              6₂ knot
| practical name=   
| image=            Blue 6 2 Knot.png
| caption=         
| arf invariant=    1
| braid length=      6
| braid number=      3
| bridge number=    2
| crosscap number=  2
| crossing number=  6
| hyperbolic volume= 4.40083
| linking number=   
| stick number=      8
| unknotting number= 1
| conway_notation=  [312]
| ab_notation=      6<sub>2</sub>
| dowker notation=  4, 8, 10, 12, 2, 6
| thistlethwaite=   
| last crossing=    6
| last order=        1
| next crossing=    6
| next order=        3
| alternating=      alternating
| class=            hyperbolic
| fibered=          fibered
| prime=            prime
| slice=           
  | symmetry=        reversible
| tricolorable=   
| twist=            twist<!--?-->
}}
 
In [[knot theory]], the '''6<sub>2</sub> knot''' is one of three [[prime knot]]s with [[crossing number (knot theory)|crossing number]] six, the others being the [[Stevedore knot (mathematics)|stevedore knot]] and the [[6₃ knot|6<sub>3</sub> knot]].  This knot is sometimes referred to as the '''Miller Institute knot''',<ref>{{MathWorld|title=Miller Institute Knot|urlname=MillerInstituteKnot}}</ref> because is appears in the logo<ref>[http://millerinstitute.berkeley.edu Miller Institute - Home Page]</ref> of the [[Miller Institute]] for Basic Research in Science at the [[University of California, Berkeley]].
 
The 6<sub>2</sub> knot is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral]]. Its [[Alexander polynomial]] is
 
:<math>\Delta(t) = -t^2 + 3t - 3 - 1 + 3t^{-1} - t^{-2}, \, </math>
 
its [[Conway polynomial]]{{dn|date=January 2014}} is
 
:<math>\nabla(z) = -z^4 - z^2 + 1, \, </math>
 
and its [[Jones polynomial]] is
 
:<math>V(q) = q - 1 + 2q^{-1} - 2q^{-2} + 2q^{-3} - 2q^{-4} + q^{-5}. \, </math><ref>{{Knot Atlas|6_2}}</ref>
 
The 6<sub>2</sub> knot is a [[hyperbolic knot]], with its [[knot complement|complement]] having a [[Hyperbolic volume (knot)|volume]] of approximately 4.40083.
 
==References==
{{reflist}}
 
{{Knot theory|state=collapsed}}
 
{{DEFAULTSORT:6 2 knot}}
 
{{knottheory-stub}}

Revision as of 04:00, 26 January 2014

Template:Infobox knot theory

In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes referred to as the Miller Institute knot,[1] because is appears in the logo[2] of the Miller Institute for Basic Research in Science at the University of California, Berkeley.

The 62 knot is invertible but not amphichiral. Its Alexander polynomial is

Δ(t)=−t2+3t−3−1+3t−1−t−2,

its Conway polynomialTemplate:Dn is

∇(z)=−z4−z2+1,

and its Jones polynomial is

V(q)=q−1+2q−1−2q−2+2q−3−2q−4+q−5.[3]

The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083.

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.

Template:Knot theory


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  1. ↑

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  2. ↑ Miller Institute - Home Page
  3. ↑ Template:Knot Atlas
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