11n
23
(K11n
23
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 8 10 4 11 6 1 9
Solving Sequence
9,11 1,4
2 5 8 7 3 10 6
c
11
c
1
c
4
c
8
c
7
c
3
c
10
c
6
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−68646u
21
− 209213u
20
+ ··· + 105421b − 71719,
− 179395u
21
− 675073u
20
+ ··· + 210842a − 1064489, u
22
+ 4u
21
+ ··· + 8u + 1i
I
u
2
= h−u
4
− u
3
+ b + u, u
4
+ u
3
+ a − u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
I
u
3
= hb + 1, a
2
− a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 30 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h−6.86 × 10
4
u
21
− 2.09 × 10
5
u
20
+ · · · + 1.05 × 10
5
b − 7.17 × 10
4
, −1.79 ×
10
5
u
21
−6.75×10
5
u
20
+· · · +2.11×10
5
a−1.06×10
6
, u
22
+4u
21
+· · · +8u +1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
1
=
1
−u
2
a
4
=
0.850850u
21
+ 3.20180u
20
+ ··· + 4.49725u + 5.04875
0.651161u
21
+ 1.98455u
20
+ ··· + 2.21543u + 0.680310
a
2
=
0.315042u
21
+ 1.37186u
20
+ ··· + 0.488432u + 3.18691
0.182947u
21
+ 0.441795u
20
+ ··· − 0.201113u + 0.0840297
a
5
=
0.146826u
21
+ 1.01918u
20
+ ··· + 1.95979u + 3.15823
0.351163u
21
+ 0.794481u
20
+ ··· − 1.67248u + 0.112710
a
8
=
−u
u
a
7
=
−0.112710u
21
− 0.0996765u
20
+ ··· − 0.646560u − 2.57416
0.431873u
21
+ 0.691897u
20
+ ··· + 1.98362u − 0.146826
a
3
=
0.688814u
21
+ 2.58804u
20
+ ··· + 1.89815u + 3.21457
−0.166694u
21
− 0.538261u
20
+ ··· − 2.05866u − 0.194885
a
10
=
−u
2
+ 1
u
4
a
6
=
−0.317366u
21
− 1.15089u
20
+ ··· − 1.03139u − 2.97993
−0.180623u
21
− 0.662766u
20
+ ··· + 0.744069u − 0.291009
a
6
=
−0.317366u
21
− 1.15089u
20
+ ··· − 1.03139u − 2.97993
−0.180623u
21
− 0.662766u
20
+ ··· + 0.744069u − 0.291009
(ii) Obstruction class = −1
(iii) Cusp Shapes =
98871
105421
u
21
+
399034
105421
u
20
+ ··· +
1542002
105421
u −
1060356
105421
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
22
− 8u
21
+ ··· − 10u + 1
c
2
u
22
+ 36u
21
+ ··· + 6u + 1
c
3
, c
7
u
22
+ 2u
21
+ ··· + 128u
2
+ 64
c
5
u
22
− 3u
21
+ ··· − u + 1
c
6
, c
9
u
22
+ 2u
21
+ ··· + 28u + 4
c
8
, c
11
u
22
+ 4u
21
+ ··· + 8u + 1
c
10
u
22
− 8u
21
+ ··· − 64u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
22
− 36y
21
+ ··· − 6y + 1
c
2
y
22
− 92y
21
+ ··· + 3898y + 1
c
3
, c
7
y
22
− 42y
21
+ ··· + 16384y + 4096
c
5
y
22
− 49y
21
+ ··· − 17y + 1
c
6
, c
9
y
22
− 18y
21
+ ··· − 264y + 16
c
8
, c
11
y
22
− 8y
21
+ ··· − 64y + 1
c
10
y
22
+ 16y
21
+ ··· − 3112y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.949302 + 0.242875I
a = 0.659054 + 0.118437I
b = −0.606013 − 0.510709I
1.72824 + 0.76607I 3.12936 − 1.22783I
u = 0.949302 − 0.242875I
a = 0.659054 − 0.118437I
b = −0.606013 + 0.510709I
1.72824 − 0.76607I 3.12936 + 1.22783I
u = 1.06873
a = 2.88944
b = −2.44319
0.373053 −36.4230
u = −0.611771 + 0.692060I
a = 0.790435 − 0.254867I
b = 0.122623 + 0.098224I
−2.04648 + 0.07308I −6.61841 + 0.32192I
u = −0.611771 − 0.692060I
a = 0.790435 + 0.254867I
b = 0.122623 − 0.098224I
−2.04648 − 0.07308I −6.61841 − 0.32192I
u = −0.831560
a = −0.842263
b = −0.991565
−7.60774 −21.1720
u = −1.040460 + 0.605021I
a = 0.213779 − 0.135252I
b = −0.431084 + 0.709735I
−0.69505 − 5.13446I −2.61215 + 4.09914I
u = −1.040460 − 0.605021I
a = 0.213779 + 0.135252I
b = −0.431084 − 0.709735I
−0.69505 + 5.13446I −2.61215 − 4.09914I
u = −0.837414 + 0.879532I
a = −0.820247 + 0.387550I
b = 0.79848 − 1.81489I
−6.52288 − 1.21996I −10.21847 + 1.61822I
u = −0.837414 − 0.879532I
a = −0.820247 − 0.387550I
b = 0.79848 + 1.81489I
−6.52288 + 1.21996I −10.21847 − 1.61822I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.908141 + 0.818565I
a = 1.68712 + 1.20696I
b = −0.40292 − 2.59906I
−12.75240 + 3.06700I −8.27093 − 2.19380I
u = 0.908141 − 0.818565I
a = 1.68712 − 1.20696I
b = −0.40292 + 2.59906I
−12.75240 − 3.06700I −8.27093 + 2.19380I
u = −0.522764 + 1.140680I
a = 1.47562 − 0.18862I
b = −0.27191 + 1.80152I
−18.8826 + 3.9450I −10.61812 − 1.00166I
u = −0.522764 − 1.140680I
a = 1.47562 + 0.18862I
b = −0.27191 − 1.80152I
−18.8826 − 3.9450I −10.61812 + 1.00166I
u = −0.982971 + 0.827014I
a = −1.20538 + 1.02888I
b = −0.113345 − 1.400720I
−6.06672 − 5.11915I −9.50515 + 3.92885I
u = −0.982971 − 0.827014I
a = −1.20538 − 1.02888I
b = −0.113345 + 1.400720I
−6.06672 + 5.11915I −9.50515 − 3.92885I
u = 0.569732 + 0.260828I
a = −2.62584 − 0.99559I
b = 1.40111 + 0.75463I
−0.990371 + 0.924237I −8.66470 − 0.43219I
u = 0.569732 − 0.260828I
a = −2.62584 + 0.99559I
b = 1.40111 − 0.75463I
−0.990371 − 0.924237I −8.66470 + 0.43219I
u = −1.22965 + 0.77380I
a = 0.90635 − 1.49914I
b = 0.11518 + 2.75329I
−16.6374 − 10.8083I −8.69774 + 4.99684I
u = −1.22965 − 0.77380I
a = 0.90635 + 1.49914I
b = 0.11518 − 2.75329I
−16.6374 + 10.8083I −8.69774 − 4.99684I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.49238
a = −0.327992
b = −1.18058
−10.9429 −8.28670
u = −0.133848
a = 4.11903
b = 0.391099
−0.845350 −11.9660
7
II. I
u
2
= h−u
4
− u
3
+ b + u, u
4
+ u
3
+ a − u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
1
=
1
−u
2
a
4
=
−u
4
− u
3
+ u
u
4
+ u
3
− u
a
2
=
−u
4
− u
3
+ u + 1
u
4
+ u
3
− u
2
− u
a
5
=
−1
u
2
a
8
=
−u
u
a
7
=
−u
u
a
3
=
−u
4
− u
3
+ u
u
4
+ u
3
− u
a
10
=
−u
2
+ 1
u
4
a
6
=
−u
4
+ u
2
− 1
u
4
a
6
=
−u
4
+ u
2
− 1
u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
+ 2u
3
+ 3u
2
+ 2u − 11
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
7
u
6
c
5
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
6
, c
11
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
8
, c
9
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
10
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
10
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
6
, c
8
, c
9
c
11
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = −0.23185 − 1.65564I
b = 0.23185 + 1.65564I
0.245672 + 0.924305I −6.22669 + 0.83820I
u = 1.002190 − 0.295542I
a = −0.23185 + 1.65564I
b = 0.23185 − 1.65564I
0.245672 − 0.924305I −6.22669 − 0.83820I
u = −0.428243 + 0.664531I
a = −0.659772 + 0.298454I
b = 0.659772 − 0.298454I
−3.53554 + 0.92430I −10.88169 − 1.11590I
u = −0.428243 − 0.664531I
a = −0.659772 − 0.298454I
b = 0.659772 + 0.298454I
−3.53554 − 0.92430I −10.88169 + 1.11590I
u = −1.073950 + 0.558752I
a = −0.108378 + 0.818891I
b = 0.108378 − 0.818891I
−1.64493 − 5.69302I −8.89162 + 7.09196I
u = −1.073950 − 0.558752I
a = −0.108378 − 0.818891I
b = 0.108378 + 0.818891I
−1.64493 + 5.69302I −8.89162 − 7.09196I
11
III. I
u
3
= hb + 1, a
2
− a − 1, u − 1i
(i) Arc colorings
a
9
=
0
1
a
11
=
1
0
a
1
=
1
−1
a
4
=
a
−1
a
2
=
2
−a
a
5
=
−a + 2
0
a
8
=
−1
1
a
7
=
0
−a + 2
a
3
=
a
−2a + 2
a
10
=
0
1
a
6
=
0
−a + 2
a
6
=
0
−a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
+ u − 1
c
2
u
2
+ 3u + 1
c
4
, c
7
u
2
− u − 1
c
5
u
2
− 3u + 1
c
6
, c
9
u
2
c
8
, c
10
(u + 1)
2
c
11
(u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
2
− 3y + 1
c
2
, c
5
y
2
− 7y + 1
c
6
, c
9
y
2
c
8
, c
10
, c
11
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.618034
b = −1.00000
−7.23771 1.00000
u = 1.00000
a = 1.61803
b = −1.00000
0.657974 1.00000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
2
+ u − 1)(u
22
− 8u
21
+ ··· − 10u + 1)
c
2
((u + 1)
6
)(u
2
+ 3u + 1)(u
22
+ 36u
21
+ ··· + 6u + 1)
c
3
u
6
(u
2
+ u − 1)(u
22
+ 2u
21
+ ··· + 128u
2
+ 64)
c
4
((u + 1)
6
)(u
2
− u − 1)(u
22
− 8u
21
+ ··· − 10u + 1)
c
5
(u
2
− 3u + 1)(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
· (u
22
− 3u
21
+ ··· − u + 1)
c
6
u
2
(u
6
+ u
5
+ ··· + u + 1)(u
22
+ 2u
21
+ ··· + 28u + 4)
c
7
u
6
(u
2
− u − 1)(u
22
+ 2u
21
+ ··· + 128u
2
+ 64)
c
8
((u + 1)
2
)(u
6
− u
5
+ ··· − u + 1)(u
22
+ 4u
21
+ ··· + 8u + 1)
c
9
u
2
(u
6
− u
5
+ ··· − u + 1)(u
22
+ 2u
21
+ ··· + 28u + 4)
c
10
(u + 1)
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
· (u
22
− 8u
21
+ ··· − 64u + 1)
c
11
((u − 1)
2
)(u
6
+ u
5
+ ··· + u + 1)(u
22
+ 4u
21
+ ··· + 8u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
2
− 3y + 1)(y
22
− 36y
21
+ ··· − 6y + 1)
c
2
((y −1)
6
)(y
2
− 7y + 1)(y
22
− 92y
21
+ ··· + 3898y + 1)
c
3
, c
7
y
6
(y
2
− 3y + 1)(y
22
− 42y
21
+ ··· + 16384y + 4096)
c
5
(y
2
− 7y + 1)(y
6
+ y
5
+ ··· + 3y + 1)(y
22
− 49y
21
+ ··· − 17y + 1)
c
6
, c
9
y
2
(y
6
− 3y
5
+ ··· − y + 1)(y
22
− 18y
21
+ ··· − 264y + 16)
c
8
, c
11
(y −1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
22
− 8y
21
+ ··· − 64y + 1)
c
10
((y −1)
2
)(y
6
+ y
5
+ ··· + 3y + 1)(y
22
+ 16y
21
+ ··· − 3112y + 1)
17
