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{{DISPLAYTITLE:6<sub>3</sub> knot}}
{{short description|Mathematical knot with crossing number 6}}
{{Infobox knot theory
| name=              6<sub>3</sub> knot
| practical name=    
| image=             Blue 6 3 Knot.png
| caption=           
| arf invariant=     1
| braid length=      6
| braid number=      3
| bridge number=     2
| crosscap number=   3
| crossing number=   6
| genus=             2
| hyperbolic volume= 5.69302
| linking number=    
| stick number=      8
| unknotting number= 1
| conway_notation=   [2112]
| ab_notation=       6<sub>3</sub>
| dowker notation=   4, 8, 10, 2, 12, 6
| thistlethwaite=    
| last crossing=     6
| last order=        2
| next crossing=     7
| next order=        1
 | alternating=      alternating
 | class=            hyperbolic
 | fibered=          fibered
 | prime=            prime
 | slice=            
 | symmetry=         fully amphichiral
 | tricolorable=     
 | twist=            
}}
In [[knot theory]], the '''6<sub>3</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 [[6₂ knot|6<sub>2</sub> knot]]. It is [[alternating knot|alternating]], [[hyperbolic knot|hyperbolic]], and [[amphichiral knot|fully amphichiral]].
It can be written as the [[braid word]]
:<math>\sigma_1^{-1}\sigma_2^2\sigma_1^{-2}\sigma_2. \, </math><ref>{{Cite web|url=https://www.wolframalpha.com/input/?i=6_3+knot|title = 6_3 knot - Wolfram&#124;Alpha}}</ref>

==Symmetry==
Like the [[figure-eight knot (mathematics)|figure-eight knot]], the 6<sub>3</sub> knot is [[amphichiral knot|fully amphichiral]]. This means that the 6<sub>3</sub> knot is [[amphichiral knot|amphichiral]],<ref>{{MathWorld|title=Amphichiral Knot|id=AmphichiralKnot}} Accessed: May 12, 2014.</ref> meaning that it is indistinguishable from its own mirror image. In addition, it is also [[invertible knot|invertible]], meaning that orienting the curve in either direction yields the same oriented knot.

==Invariants==
The [[Alexander polynomial]] of the 6<sub>3</sub> knot is
:<math>\Delta(t) = t^2 - 3t + 5 - 3t^{-1} + t^{-2}, \, </math>

[[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]] is
:<math>\nabla(z) = z^4 + z^2 + 1, \, </math>

[[Jones polynomial]] is
:<math>V(q) = -q^3 + 2q^2 - 2q + 3 - 2q^{-1} + 2q^{-2} - q^{-3}, \, </math>

and the [[Kauffman polynomial]] is
:<math>L(a,z) = az^5 + z^5a^{-1} + 2a^2z^4 + 2z^4a^{-2} + 4z^4 + a^3z^3 + az^3 + z^3a^{-1} + z^3a^{-3} - 3a^2z^2 - 3z^2a^{-2} - 6z^2 - a^3z - 2az - 2za^{-1} - za^{}-3 + a^2 + a^{-2} +3. \, </math> <ref>{{Knot Atlas|6_3}}</ref>

The 6<sub>3</sub> knot is a [[hyperbolic knot]], with its complement having a [[Hyperbolic volume (knot)|volume]] of approximately 5.69302.

==References==
{{reflist}}

{{Knot theory|state=collapsed}}

{{DEFAULTSORT:6 3 knot}}
[[Category:Double torus knots and links]]