Misplaced Pages

74
Arf invariant	0
Braid length	9
Braid no.	4
Bridge no.	2
Crosscap no.	3
Crossing no.	7
Genus	1
Hyperbolic volume	5.13794
Stick no.	9
Unknotting no.	2
Conway notation
A–B notation	74
Dowker notation	6, 10, 12, 14, 4, 2, 8
Last / Next	73 / 75
Other
alternating, hyperbolic, prime, reversible, tricolorable

In mathematical knot theory, 7₄ is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures.

Visual representations

The interlaced version of the simplest form of the Endless knot symbol of Buddhism is topologically equivalent to the 7₄ knot (though it appears to have nine crossings), as is the interlaced version of the unicursal hexagram of occultism. (However, the endless knot symbol has more complex forms not equivalent to 7₄, and both the endless knot and unicursal hexagram can appear in non-interlaced versions, in which case they are not knots at all.)

The 7₄ knot is a Lissajous knot, representable for example by the parametric equation

${\displaystyle {\begin{aligned}x&=\cos(2t+0.22)\\y&=\cos(3t+1.10)\\z&=\cos 7t\end{aligned}}}$

• One form of the Endless knot of Buddhism
• Interwoven unicursal hexagram.
• 7₄ knot in Celtic artistic form, also found in some Hausa embroideries.
• A 7₄ knot combined with the Syrian flag is used as a logo by the National Coordination Committee for Democratic Change.

Example

Sources

1. "7_4", The Knot Atlas.
2. Lamm, C. (1997). "There are infinitely many Lissajous knots". Manuscripta Mathematica. 93: 29–37. doi:10.1007/BF02677455. S2CID 123288245.
3. Celtic Art: The Methods of Construction by George Bain, p. 27 (ISBN 0-486-22923-8)

• Knot theory
• Alternating knots and links
• Hyperbolic knots and links
• Unfibered knots and links
• Prime knots and links
• Reversible knots and links
• Tricolorable knots and links
• Double torus knots and links