74 knot
{{DISPLAYTITLE:74 knot}}
{{short description|Mathematical knot with crossing number 7}}
{{Infobox knot theory
| name= 74
| practical name=
| image= Blue 7 4 Knot.png
| caption=
| arf invariant= 0
| braid length= 9
| braid number= 4
| bridge number= 2
| crosscap number= 3
| crossing number= 7
| genus= 1
| hyperbolic volume= 5.13794
| linking number=
| stick number= 9
| unknotting number= 2
| conway_notation= [313]
| ab_notation= 74
| dowker notation= 6, 10, 12, 14, 4, 2, 8
| thistlethwaite=
| last crossing= 7
| last order= 3
| next crossing= 7
| next order= 5
| alternating= alternating
| class= hyperbolic
| fibered=
| pretzel=
| prime= prime
| slice=
| symmetry= reversible
| tricolorable= 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.{{Knot Atlas|7_4}} (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{{cite journal | last1 = Lamm | first1 = C. | year = 1997 | title = There are infinitely many Lissajous knots | journal = Manuscripta Mathematica | volume = 93 | pages = 29–37 | doi=10.1007/BF02677455| s2cid = 123288245 }}
:
\begin{aligned}
x &= \cos (2t+0.22)\\
y &= \cos (3t+1.10)\\
z &= \cos 7t
\end{aligned}
File:EndlessKnot03d.png|One form of the Endless knot of Buddhism
File:Interwoven unicursal hexagram.svg|Interwoven unicursal hexagram.
File:Celtic-knot-linear-7crossings.svg|74 knot in Celtic artistic form, also found in some Hausa embroideries.Celtic Art: The Methods of Construction by George Bain, p. 27 ({{ISBN|0-486-22923-8}})
File:Logo of the National Coordination Commitee.svg|A 74 knot combined with the Syrian flag is used as a logo by the National Coordination Committee for Democratic Change.
{{Commonscat}}