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{{Infobox knot theory
| name=              Three-twist knot
| practical name=   
| image=            Blue Three-Twist Knot.png
| caption=         
| arf invariant=    0
| braid length=      6
| braid number=      3
| bridge number=    2
| crosscap number=  2
| crossing number=  5
| hyperbolic volume= 2.82812
| linking number=   
| stick number=      8
| unknotting number= 1
| conway_notation=  [32]
| ab_notation=      5<sub>2</sub>
| dowker notation=  4, 8, 10, 2, 6
| thistlethwaite=   
| last crossing=    5
| last order=        1
| next crossing=    6
| next order=        1
| alternating=      alternating
| class=            hyperbolic
| fibered=         
| prime=            prime
| slice=           
| symmetry=        reversible
| tricolorable=   
  | twist=            twist
}}
 
In [[knot theory]], the '''three-twist knot''' is the [[twist knot]] with three-half twists. It is listed as the '''5<sub>2</sub> knot''' in the [[Alexander-Briggs notation]], and is one of two knots with [[crossing number (knot theory)|crossing number]] five, the other being the [[cinquefoil knot]].
 
The three-twist knot is a [[prime knot]], and it is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral]]. Its [[Alexander polynomial]] is
 
:<math>\Delta(t) = 2t-3+2t^{-1}, \,</math>
 
its [[Conway polynomial]]{{dn|date=January 2014}} is
 
:<math>\nabla(z) = 2z^2+1, \, </math>
 
and its [[Jones polynomial]] is
 
:<math>V(q) = q^{-1} - q^{-2} + 2q^{-3} - q^{-4} + q^{-5} - q^{-6}. \, </math><ref>{{Knot Atlas|5_2}}</ref>
 
Because the Alexander polynomial is not [[monic polynomial|monic]], the three-twist knot is not [[fibered knot|fibered]].
 
The three-twist knot is a [[hyperbolic knot]], with its complement having a [[Hyperbolic volume (knot)|volume]] of approximately 2.82812.
 
==References==
{{reflist}}
 
{{Knot theory|state=collapsed}}
 
{{knottheory-stub}}

Latest revision as of 06:52, 6 December 2013

Template:Infobox knot theory

In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

Δ(t)=2t−3+2t−1,

its Conway polynomialTemplate:Dn is

∇(z)=2z2+1,

and its Jones polynomial is

V(q)=q−1−q−2+2q−3−q−4+q−5−q−6.[1]

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.

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

Template:Knottheory-stub

  1. ↑ Template:Knot Atlas
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