Shintani zeta function: Difference between revisions

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The first reference concerned zeta functions associated to prehomogeneous vector spaces, which are not the Shintani zeta functions described here.
 
en>David Eppstein
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{{Infobox knot theory
| name=              7₁ knot
| names=           
| image=            Blue 7 1 Knot.png
| caption=         
| arf invariant=    0
| braid length=      7
| braid number=      2
| bridge number=    2
| crosscap number=  1
| crossing number=  7
| hyperbolic volume= 0
| linking number=   
| stick number=      9
| unknotting number= 3
| conway_notation=  [7]
| ab_notation=      7<sub>1</sub>
| dowker notation=  8, 10, 12, 14, 2, 4, 6
| thistlethwaite=   
| last crossing=    6
| last order=        3
| next crossing=    7
| next order=        2
| alternating=      alternating
| class=            torus
| fibered=          fibered
| prime=            prime
| slice=           
| symmetry=        reversible
| tricolorable=   
}}
 
In [[knot theory]], the '''7<sub>1</sub> knot''', also known as the '''septoil knot''', the '''septafoil knot''', or the '''(7,&nbsp;2)-torus knot''', is one of seven [[prime knot]]s with [[crossing number (knot theory)|crossing number]] seven. It is the simplest [[torus knot]] after the [[trefoil knot|trefoil]] and [[cinquefoil knot|cinquefoil]].
 
The 7<sub>1</sub> knot is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral]]. Its [[Alexander polynomial]] is
 
:<math>\Delta(t) = t^3 - t^2 + t - 1 + t^{-1} - t^{-2} + t^{-3}, \, </math>
 
its [[Alexander–Conway polynomial|Conway polynomial]] is
 
:<math>\nabla(z) = z^6 + 5z^4 + 6z^2 + 1, \, </math>
 
and its [[Jones polynomial]] is
 
:<math>V(q) = q^{-3} + q^{-5} - q^{-6} + q^{-7} - q^{-8} + q^{-9} - q^{-10}. \, </math><ref>{{Knot Atlas|7_1}}</ref>
 
==See also==
*[[Heptagram]]
 
==References==
{{reflist}}
 
{{Knot theory|state=collapsed}}
 
{{DEFAULTSORT:7 1 knot}}
 
{{knottheory-stub}}

Revision as of 19:13, 4 October 2013

Template:Infobox knot theory

In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil.

The 71 knot is invertible but not amphichiral. Its Alexander polynomial is

Δ(t)=t3t2+t1+t1t2+t3,

its Conway polynomial is

(z)=z6+5z4+6z2+1,

and its Jones polynomial is

V(q)=q3+q5q6+q7q8+q9q10.[1]

See also

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.

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