The eight knot (or eight knot ) plays a role in mathematics, especially in knot theory . It is the mathematical counterpart of the eight knot , which is used, among other things, in sailing.
Parametric representation A simple parametric representation of the figure eight node is:
x
=
(
2
+
cos
(
2
t
)
)
cos
(
3
t
)
y
=
(
2
+
cos
(
2
t
)
)
sin
(
3
t
)
z
=
sin
(
4th
t
)
{\ displaystyle {\ begin {aligned} x & = \ left (2+ \ cos {(2t)} \ right) \ cos {(3t)} \\ y & = \ left (2+ \ cos {(2t)} \ right) \ sin {(3t)} \\ z & = \ sin {(4t)} \ end {aligned}}}
The figure eight is the end of the braid .
σ
1
σ
2
-
1
σ
1
σ
2
-
1
{\ displaystyle \ sigma _ {1} \ sigma _ {2} ^ {- 1} \ sigma _ {1} \ sigma _ {2} ^ {- 1}}
Invariants The Alexander polynomial of the figure eight is
Δ
(
t
)
=
-
t
+
3
-
t
-
1
,
{\ displaystyle \ Delta (t) = - t + 3-t ^ {- 1},}
its Jones polynomial
V
(
q
)
=
q
2
-
q
+
1
-
q
-
1
+
q
-
2
.
{\ displaystyle V (q) = q ^ {2} -q + 1-q ^ {- 1} + q ^ {- 2}.}
The Kauffman polynomial is the HOMFLY polynomial , the bracket polynomial , the Conway polynomial, and the BLM polynomial .
-
1
-
1
x
2
-
x
2
-
y
x
-
x
y
+
2
y
2
+
y
2
x
2
+
x
2
y
2
+
y
3
x
+
x
y
3
{\ displaystyle -1 - {\ frac {1} {x ^ {2}}} - x ^ {2} - {\ frac {y} {x}} - xy + 2y ^ {2} + {\ frac { y ^ {2}} {x ^ {2}}} + x ^ {2} y ^ {2} + {\ frac {y ^ {3}} {x}} + xy ^ {3}}
x
2
+
1
x
2
-
y
2
-
1
{\ displaystyle x ^ {2} + {\ frac {1} {x ^ {2}}} - y ^ {2} -1}
x
8th
+
1
x
8th
-
x
4th
-
1
x
4th
+
1
{\ displaystyle x ^ {8} + {\ frac {1} {x ^ {8}}} - x ^ {4} - {\ frac {1} {x ^ {4}}} + 1}
1
-
x
2
{\ displaystyle 1-x ^ {2}}
2
x
3
+
4th
x
2
-
2
x
-
3
{\ displaystyle 2x ^ {3} + 4x ^ {2} -2x-3}
The number of intersections of the figure eight knot is 4, its gender is 1, and its Seifert matrix .
(
1
0
-
1
-
1
)
{\ displaystyle \ left ({\ begin {array} {cc} 1 & 0 \\ - 1 & -1 \ end {array}} \ right)}
The node group of the figure eight has the presentation
Γ
=
⟨
a
,
b
∣
[
a
-
1
,
b
]
a
=
b
[
a
-
1
,
b
]
⟩
{\ displaystyle \ Gamma = \ langle a, b \ mid \ left [a ^ {- 1}, b \ right] a = b \ left [a ^ {- 1}, b \ right] \ rangle}
Your character variety is the elliptical curve
X
(
Γ
)
{\ displaystyle X (\ Gamma)}
z
2
=
u
3
-
2
u
+
1
,
{\ displaystyle z ^ {2} = u ^ {3} -2u + 1,}
the A-polynomial is
-
M.
4th
+
L.
(
1
-
M.
2
-
2
M.
4th
-
M.
6th
=
M.
8th
)
-
L.
2
M.
4th
.
{\ displaystyle -M ^ {4} + L (1-M ^ {2} -2M ^ {4} -M ^ {6} = M ^ {8}) - L ^ {2} M ^ {4}. }
properties The figure eight knot is achiral (also called amphichiral), which means that it can be deformed into its mirror image. It is not a torus knot .
The figure eight knot is a hyperbolic knot , its hyperbolic volume is
v
O
l
(
S.
3
-
K
)
=
2
D.
2
(
ω
)
=
2
,
02
...
{\ displaystyle vol (S ^ {3} -K) = 2D_ {2} (\ omega) = 2 {,} 02 \ dots}
Here is the Bloch-Wigner dilogarithm and .
D.
2
{\ displaystyle D_ {2}}
ω
=
1
2
+
3
2
i
{\ displaystyle \ omega = {\ frac {1} {2}} + {\ frac {\ sqrt {3}} {2}} i}
The hyperbolic structure is given by the faithful and discrete representation
ρ
:
Γ
→
P
S.
L.
(
2
,
Z
[
ω
]
)
⊂
P
S.
L.
(
2
,
C.
)
=
I.
s
O
m
+
(
H
3
)
{\ displaystyle \ rho \ colon \ Gamma \ to PSL (2, \ mathbb {Z} \ left [\ omega \ right]) \ subset PSL (2, \ mathbb {C}) = Isom ^ {+} (H ^ {3})}
ρ
(
a
)
=
(
1
1
0
1
)
{\ displaystyle \ rho (a) = ({\ begin {array} {cc} 1 & 1 \\ 0 & 1 \ end {array}})}
ρ
(
b
)
=
(
1
0
-
ω
1
)
{\ displaystyle \ rho (b) = ({\ begin {array} {cc} 1 & 0 \\ - \ omega & 1 \ end {array}})}
The hyperbolic structure on the complement of the figure eight knot was discovered by Riley in 1975. This example motivated Thurston to search for hyperbolic structures on further node complements, which ultimately led to the geometrization conjecture .
The figure eight knot is the only arithmetic hyperbolic knot.
In 2001, Cao and Meyerhoff proved that the figure eight knot is the hyperbolic knot of smallest volume.
Simple square representation of the figure-eight configuration.
Symmetrical representation generated by parametric equations.
See also Web links Individual evidence
↑ Eric W. Weisstein : Figure Eight Knot . MathWorld
↑ Mehmet Haluk Șengün: An introduction to A-polynomials and their Mahler measures.
↑ Johannes Diemke: Torus node original Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. Retrieved May 19, 2012.@1 @ 2 Template: Webachiv / IABot / www.informatik.uni-oldenburg.de
^ Robert Riley : A quadratic parabolic group. Math. Proc. Cambridge Philos. Soc. 1975, 77: 281-288.
^ Alan Reid , Arithmeticity of Knot Complements.
↑ Chun Cao , Robert Meyerhoff : The orientable cusped hyperbolic 3-manifolds of minimum volume.
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle \ Gamma = \ langle a, b \ mid \ left [a ^ {- 1}, b \ right] a = b \ left [a ^ {- 1}, b \ right] \ rangle}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9922cf6981f3844d900f096c6f2bc5a16cb7803e)
![{\ displaystyle \ rho \ colon \ Gamma \ to PSL (2, \ mathbb {Z} \ left [\ omega \ right]) \ subset PSL (2, \ mathbb {C}) = Isom ^ {+} (H ^ {3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a436b466dbcefa53e42185c699cfa923c2fa5754)
