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{{Short description|Mathematical knot with crossing number 6}}
{{DISPLAYTITLE:6<sub>2</sub> knot}}
{{Infobox knot theory
| name=              6<sub>2</sub> 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
| genus=             2
| 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=
}}
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 it 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 + 3t^{-1} - t^{-2}, \, </math>

its [[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]] 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.

==Surface==
<gallery>
File:Superfície - bordo Nó 6,2.jpg|Surface of knot 6.2
</gallery>

==Example==
Ways to assemble of knot 6.2 
<gallery>
File:6₂ knot.webm|Example 1
File:6₂ knot (2).webm|Example 2
</gallery>

If a [[bowline]] is tied and the two free ends of the rope are brought together in the simplest way, the knot obtained is the 6<sub>2</sub> knot. The sequence of necessary moves are depicted here:
<gallery>
Image:Bowline to 6 2 knot.gif|From a bowline (ends connected) to the 6₂ knot.
</gallery>

==References==
{{reflist}}

{{Knot theory|state=collapsed}}

{{DEFAULTSORT:6 2 knot}}
[[Category:Double torus knots and links]]