7₁ knot

7₁ knot

Arf invariant	0
Braid length	7
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	7
Genus	3
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

Knot theory (knots and links)

Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Whitehead (52 1) Borromean rings (63 2) L10a140

Satellite	Composite knots Granny Square Knot sum

Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (02 1) Hopf (22 1) Solomon's (42 1)

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

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

Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe

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