7₁ knot

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7₁ knot

Arf invariant	0
Braid length	7
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	7
Hyperbolic volume	0
Stick no.	9
Unknotting no.	3
Conway notation	[7]
A-B notation	7₁
Dowker notation	8, 10, 12, 14, 2, 4, 6
Last /Next	6₃ / 7₂
Other
alternating, torus, fibered, prime, reversible

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.

$\Delta (t)=t^{3}-t^{2}+t-1+t^{{-1}}-t^{{-2}}+t^{{-3}},\,$

$\nabla (z)=z^{6}+5z^{4}+6z^{2}+1,\,$

$V(q)=q^{{-3}}+q^{{-5}}-q^{{-6}}+q^{{-7}}-q^{{-8}}+q^{{-9}}-q^{{-10}}.\,$ ^[1]

References

v t e Knot theory (knots and links)

Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Borromean rings (63 2)

Satellite	Composite knots Granny Square

Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Carrick mat (8₁₈) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Whitehead (52 1) L10a140

Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no.

Notation & operations	Alexander–Briggs notation Conway notation Dowker notation Mutation Reidemeister move Skein relation

Other	Berge Braid theory Complement Double torus Fibered Framed Knot List of knots and links Ribbon Slice Sum Twist Wild

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