lissajous toric knot
Lissajous-toric knot GudangMovies21 Rebahinxxi LK21
In knot theory, a Lissajous-toric knot is a knot defined by parametric equations of the form:
x
(
t
)
=
(
2
+
sin
q
t
)
cos
N
t
,
y
(
t
)
=
(
2
+
sin
q
t
)
sin
N
t
,
z
(
t
)
=
cos
p
(
t
+
ϕ
)
,
{\displaystyle x(t)=(2+\sin qt)\cos Nt,\qquad y(t)=(2+\sin qt)\sin Nt,\qquad z(t)=\cos p(t+\phi ),}
where
N
{\displaystyle N}
,
p
{\displaystyle p}
, and
q
{\displaystyle q}
are integers, the phase shift
ϕ
{\displaystyle \phi }
is a real number
and the parameter
t
{\displaystyle t}
varies between 0 and
2
π
{\displaystyle 2\pi }
.
For
p
=
q
{\displaystyle p=q}
the knot is a torus knot.
Braid and billiard knot definitions
In braid form these knots can be defined in a square solid torus (i.e. the cube
[
−
1
,
1
]
3
{\displaystyle [-1,1]^{3}}
with identified top and bottom) as
x
(
t
)
=
sin
2
π
q
t
,
y
(
t
)
=
cos
2
π
p
(
t
+
ϕ
)
,
z
(
t
)
=
2
(
N
t
−
⌊
N
t
⌋
)
−
1
,
t
∈
[
0
,
1
]
{\displaystyle x(t)=\sin 2\pi qt,\qquad y(t)=\cos 2\pi p(t+\phi ),\qquad z(t)=2(Nt-\lfloor Nt\rfloor )-1,\qquad t\in [0,1]}
.
The projection of this Lissajous-toric knot onto the x-y-plane is a Lissajous curve.
Replacing the sine and cosine functions in the parametrization by a triangle wave transforms a Lissajous-toric
knot isotopically into a billiard curve inside the solid torus. Because of this property Lissajous-toric knots are also called billiard knots in a solid torus.
Lissajous-toric knots were first studied as billiard knots and they share many properties with billiard knots in a cylinder.
They also occur in the analysis of singularities of minimal surfaces with branch points and in the study of
the Three-body problem.
The knots in the subfamily with
p
=
q
⋅
l
{\displaystyle p=q\cdot l}
, with an integer
l
≥
1
{\displaystyle l\geq 1}
, are known as ′Lemniscate knots′. Lemniscate knots have period
q
{\displaystyle q}
and are fibred. The knot shown on the right is of this type (with
l
=
5
{\displaystyle l=5}
).
Properties
Lissajous-toric knots are denoted by
K
(
N
,
q
,
p
,
ϕ
)
{\displaystyle K(N,q,p,\phi )}
. To ensure that the knot is traversed only once in the parametrization
the conditions
gcd
(
N
,
q
)
=
gcd
(
N
,
p
)
=
1
{\displaystyle \gcd(N,q)=\gcd(N,p)=1}
are needed. In addition, singular values for the phase, leading to self-intersections, have to be excluded.
The isotopy class of Lissajous-toric knots surprisingly does not depend on the phase
ϕ
{\displaystyle \phi }
(up to mirroring).
If the distinction between a knot and its mirror image is not important, the notation
K
(
N
,
q
,
p
)
{\displaystyle K(N,q,p)}
can be used.
The properties of Lissajous-toric knots depend on whether
p
{\displaystyle p}
and
q
{\displaystyle q}
are coprime or
d
=
gcd
(
p
,
q
)
>
1
{\displaystyle d=\gcd(p,q)>1}
. The main properties are:
Interchanging
p
{\displaystyle p}
and
q
{\displaystyle q}
:
K
(
N
,
q
,
p
)
=
K
(
N
,
p
,
q
)
{\displaystyle K(N,q,p)=K(N,p,q)}
(up to mirroring).
Ribbon property:
If
p
{\displaystyle p}
and
q
{\displaystyle q}
are coprime,
K
(
N
,
q
,
p
)
{\displaystyle K(N,q,p)}
is a symmetric union and therefore a ribbon knot.
Periodicity:
If
d
=
gcd
(
p
,
q
)
>
1
{\displaystyle d=\gcd(p,q)>1}
, the Lissajous-toric knot has period
d
{\displaystyle d}
and the factor knot is a ribbon knot.
Strongly positive amphicheirality:
If
p
{\displaystyle p}
and
q
{\displaystyle q}
have different parity, then
K
(
N
,
q
,
p
)
{\displaystyle K(N,q,p)}
is strongly positive amphicheiral.
Period 2:
If
p
{\displaystyle p}
and
q
{\displaystyle q}
are both odd, then
K
(
N
,
q
,
p
)
{\displaystyle K(N,q,p)}
has period 2 (for even
N
{\displaystyle N}
) or is freely 2-periodic (for odd
N
{\displaystyle N}
).
= Example
=The knot T(3,8,7), shown in the graphics, is a symmetric union and a ribbon knot (in fact, it is the composite knot
5
1
♯
−
5
1
{\displaystyle 5_{1}\sharp -5_{1}}
).
It is strongly positive amphicheiral: a rotation by
π
{\displaystyle \pi }
maps the knot to its mirror image, keeping its orientation.
An additional horizontal symmetry occurs as a combination of the vertical symmetry and the rotation (′double palindromicity′ in Kin/Nakamura/Ogawa).
′Classification′ of billiard rooms
In the following table a systematic overview of the possibilities to build billiard rooms from the interval and the circle (interval with identified boundaries) is given:
In the case of Lissajous knots reflections at the boundaries occur in all of the three cube's dimensions.
In the second case reflections occur in two dimensions and we have a uniform movement in the third dimension.
The third case is nearly equal to the usual movement on a torus, with an additional triangle wave movement in the first dimension.