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{{DISPLAYTITLE:7<sub>1</sub> knot}}
{{short description|Mathematical knot with crossing number 7}}
{{Infobox knot theory
| name=              7<sub>1</sub> 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
| genus=             3
| 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]].

==Properties==
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>

==Example==
[[File:7₁ knot.webm |thumb |400px |center |Assembling of 7<sub>1</sub> knot.]]


==See also==
*[[Heptagram]]

==References==
{{reflist}}

{{Knot theory|state=collapsed}}

{{DEFAULTSORT:7 1 knot}}