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Source: Umbilic torus

The umbilic torus or umbilic bracelet is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A cross section of the surface forms a deltoid.
The umbilic torus occurs in the mathematical subject of singularity theory, in particular in the classification of umbilical points which are determined by real cubic forms

a

x

3

+
3
b

x

2

y
+
3
c
x

y

2

+
d

y

3

{\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}

. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus. Christopher Zeeman named this set the umbilic bracelet in 1976.
The torus is defined by the following set of parametric equations.

x
=
sin
⁡
u

(

7
+
cos
⁡

(

u
3

−
2
v

)

+
2
cos
⁡

(

u
3

+
v

)

)

{\displaystyle x=\sin u\left(7+\cos \left({u \over 3}-2v\right)+2\cos \left({u \over 3}+v\right)\right)}

y
=
cos
⁡
u

(

7
+
cos
⁡

(

u
3

−
2
v

)

+
2
cos
⁡

(

u
3

+
v

)

)

{\displaystyle y=\cos u\left(7+\cos \left({u \over 3}-2v\right)+2\cos \left({u \over 3}+v\right)\right)}

z
=
sin
⁡

(

u
3

−
2
v

)

+
2
sin
⁡

(

u
3

+
v

)

{\displaystyle z=\sin \left({u \over 3}-2v\right)+2\sin \left({u \over 3}+v\right)}

for

−
π
≤
u
≤
π
,

−
π
≤
v
≤
π

{\displaystyle {\text{for }}-\pi \leq u\leq \pi ,\quad -\pi \leq v\leq \pi }