11a
132
(K11a
132
)
A knot diagram
1
Linearized knot diagam
6 1 8 10 2 4 11 3 5 7 9
Solving Sequence
2,5 6,9
10 1 3 4 8 11 7
c
5
c
9
c
1
c
2
c
4
c
8
c
11
c
7
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−116u
48
− 310u
47
+ ··· + 2304b − 8416, −4487u
49
− 19250u
48
+ ··· + 71424a − 445724,
u
50
+ 4u
49
+ ··· + 255u + 62i
I
u
2
= hb
2
− bu + u, a − u + 1, u
2
− u + 1i
I
u
3
= hb + u, a − 2, u
2
− u + 1i
I
u
4
= hb
2
au + b
3
+ bu + au + b + u − 1, u
2
− u + 1i
I
v
1
= ha, b
3
+ b − 1, v − 1i
* 4 irreducible components of dim
C
= 0, with total 59 representations.
* 1 irreducible components of dim
C
= 1
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−116u
48
− 310u
47
+ · · · + 2304b − 8416, −4487u
49
− 19250u
48
+
· · · + 71424a − 445724, u
50
+ 4u
49
+ · · · + 255u + 62i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
9
=
0.0628220u
49
+ 0.269517u
48
+ ··· + 24.7408u + 6.24054
0.0503472u
48
+ 0.134549u
47
+ ··· + 9.09288u + 3.65278
a
10
=
0.0628220u
49
+ 0.319864u
48
+ ··· + 33.8336u + 9.89331
0.0503472u
48
+ 0.134549u
47
+ ··· + 9.09288u + 3.65278
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
5
+ u
3
+ u
a
4
=
0.0753528u
49
+ 0.332661u
48
+ ··· + 20.5942u + 5.51184
0.0104167u
49
+ 0.0559896u
48
+ ··· + 3.65495u + 0.393229
a
8
=
0.176103u
49
+ 0.671427u
48
+ ··· + 38.7086u + 8.87769
0.0625000u
49
+ 0.159722u
48
+ ··· + 4.80382u + 1.71528
a
11
=
−0.0727487u
49
− 0.228495u
48
+ ··· − 18.8299u − 4.29049
0.0625000u
49
+ 0.190104u
48
+ ··· + 3.84115u − 0.151042
a
7
=
0.0408546u
49
+ 0.0874636u
48
+ ··· − 6.34781u − 1.55343
0.0156250u
49
− 0.0447049u
48
+ ··· − 12.4293u − 3.52170
a
7
=
0.0408546u
49
+ 0.0874636u
48
+ ··· − 6.34781u − 1.55343
0.0156250u
49
− 0.0447049u
48
+ ··· − 12.4293u − 3.52170
(ii) Obstruction class = −1
(iii) Cusp Shapes =
625
864
u
49
+
5359
1728
u
48
+ ··· +
43991
216
u +
193
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
50
+ 4u
49
+ ··· + 255u + 62
c
2
u
50
+ 20u
49
+ ··· + 27851u + 3844
c
3
, c
8
9(9u
50
+ 9u
49
+ ··· − 6u + 1)
c
4
, c
9
9(9u
50
+ 9u
49
+ ··· + 8u + 1)
c
6
16(16u
50
− 32u
49
+ ··· − 26298u + 5463)
c
7
, c
10
u
50
− 6u
49
+ ··· − 7031u + 1274
c
11
16(16u
50
− 16u
49
+ ··· + 612u + 63)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
50
+ 20y
49
+ ··· + 27851y + 3844
c
2
y
50
+ 20y
49
+ ··· + 97271135y + 14776336
c
3
, c
8
81(81y
50
+ 2457y
49
+ ··· − 2y + 1)
c
4
, c
9
81(81y
50
+ 2781y
49
+ ··· − 2y + 1)
c
6
256(256y
50
− 1408y
49
+ ··· − 8.62735 × 10
7
y + 2.98444 × 10
7
)
c
7
, c
10
y
50
− 34y
49
+ ··· + 15979843y + 1623076
c
11
256(256y
50
− 1920y
49
+ ··· + 153522y + 3969)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.092353 + 1.013550I
a = −0.004651 − 0.358199I
b = 0.717415 + 0.391005I
−2.36463 + 4.68098I −6.72560 − 6.42701I
u = 0.092353 − 1.013550I
a = −0.004651 + 0.358199I
b = 0.717415 − 0.391005I
−2.36463 − 4.68098I −6.72560 + 6.42701I
u = −0.777689 + 0.592166I
a = 1.48088 + 0.63093I
b = −1.077960 − 0.011537I
3.41598 + 5.21199I 0.12705 − 3.56799I
u = −0.777689 − 0.592166I
a = 1.48088 − 0.63093I
b = −1.077960 + 0.011537I
3.41598 − 5.21199I 0.12705 + 3.56799I
u = −0.813463 + 0.649972I
a = −1.136810 − 0.407519I
b = 0.746378 − 0.288180I
6.17950 + 0.24137I 3.95341 + 1.60515I
u = −0.813463 − 0.649972I
a = −1.136810 + 0.407519I
b = 0.746378 + 0.288180I
6.17950 − 0.24137I 3.95341 − 1.60515I
u = 0.736653 + 0.765266I
a = −0.584317 − 1.167250I
b = 0.120988 + 1.046830I
−3.34886 + 2.77656I −9.46596 − 3.37700I
u = 0.736653 − 0.765266I
a = −0.584317 + 1.167250I
b = 0.120988 − 1.046830I
−3.34886 − 2.77656I −9.46596 + 3.37700I
u = 0.941593 + 0.502895I
a = 0.691476 − 0.854824I
b = −0.51438 + 1.34602I
−0.83094 − 10.80580I −3.00000 + 5.61837I
u = 0.941593 − 0.502895I
a = 0.691476 + 0.854824I
b = −0.51438 − 1.34602I
−0.83094 + 10.80580I −3.00000 − 5.61837I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.447528 + 0.991757I
a = 0.195342 + 1.153440I
b = 0.109550 − 0.264828I
−0.55196 + 1.45362I 0.849896 + 0.307578I
u = 0.447528 − 0.991757I
a = 0.195342 − 1.153440I
b = 0.109550 + 0.264828I
−0.55196 − 1.45362I 0.849896 − 0.307578I
u = −0.672195 + 0.888770I
a = −0.642949 + 1.017540I
b = 0.177110 − 0.103647I
−0.22553 − 2.62229I 0.95524 + 3.73688I
u = −0.672195 − 0.888770I
a = −0.642949 − 1.017540I
b = 0.177110 + 0.103647I
−0.22553 + 2.62229I 0.95524 − 3.73688I
u = 1.019770 + 0.461213I
a = −0.478079 + 0.501193I
b = 0.402304 − 1.083860I
3.76854 − 4.51039I 0. + 5.29534I
u = 1.019770 − 0.461213I
a = −0.478079 − 0.501193I
b = 0.402304 + 1.083860I
3.76854 + 4.51039I 0. − 5.29534I
u = −0.757822 + 0.403687I
a = −0.635336 − 1.216680I
b = 0.53792 + 1.33201I
−5.06249 + 5.26813I −6.22611 − 3.60097I
u = −0.757822 − 0.403687I
a = −0.635336 + 1.216680I
b = 0.53792 − 1.33201I
−5.06249 − 5.26813I −6.22611 + 3.60097I
u = −0.289618 + 0.801801I
a = −1.43783 + 0.89732I
b = 0.224512 + 0.714675I
−2.47998 − 1.07634I −10.74745 − 1.48815I
u = −0.289618 − 0.801801I
a = −1.43783 − 0.89732I
b = 0.224512 − 0.714675I
−2.47998 + 1.07634I −10.74745 + 1.48815I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.117739 + 1.161840I
a = −0.382470 − 0.726071I
b = −0.28062 − 1.44993I
−10.17260 + 2.94051I −12.16356 − 1.23495I
u = −0.117739 − 1.161840I
a = −0.382470 + 0.726071I
b = −0.28062 + 1.44993I
−10.17260 − 2.94051I −12.16356 + 1.23495I
u = −0.563809 + 1.077480I
a = −1.70743 − 0.59740I
b = 0.427612 − 1.164930I
−2.46832 − 6.35887I 0. + 6.52323I
u = −0.563809 − 1.077480I
a = −1.70743 + 0.59740I
b = 0.427612 + 1.164930I
−2.46832 + 6.35887I 0. − 6.52323I
u = −1.121410 + 0.484988I
a = 0.399214 − 0.478349I
b = −0.210445 + 1.103920I
−0.91613 − 5.10341I 0. + 10.19626I
u = −1.121410 − 0.484988I
a = 0.399214 + 0.478349I
b = −0.210445 − 1.103920I
−0.91613 + 5.10341I 0. − 10.19626I
u = −0.686258 + 1.016050I
a = 0.805119 + 0.489405I
b = −0.883363 − 0.123330I
5.05351 − 5.86110I 0
u = −0.686258 − 1.016050I
a = 0.805119 − 0.489405I
b = −0.883363 + 0.123330I
5.05351 + 5.86110I 0
u = −0.660397 + 1.034530I
a = −1.17709 − 0.84449I
b = 1.218970 − 0.134344I
2.08583 − 10.64970I 0. + 8.66549I
u = −0.660397 − 1.034530I
a = −1.17709 + 0.84449I
b = 1.218970 + 0.134344I
2.08583 + 10.64970I 0. − 8.66549I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.718487 + 0.241043I
a = 0.617067 + 0.760645I
b = −0.443255 + 0.080226I
1.86829 + 2.54813I 2.68166 − 3.38783I
u = 0.718487 − 0.241043I
a = 0.617067 − 0.760645I
b = −0.443255 − 0.080226I
1.86829 − 2.54813I 2.68166 + 3.38783I
u = −0.601973 + 1.087870I
a = 1.90278 + 0.36506I
b = −0.61493 + 1.48243I
−7.03977 − 10.39490I 0
u = −0.601973 − 1.087870I
a = 1.90278 − 0.36506I
b = −0.61493 − 1.48243I
−7.03977 + 10.39490I 0
u = −0.045175 + 1.293490I
a = −0.187844 − 0.818979I
b = 0.313266 − 1.365130I
−7.73894 − 8.36358I 0
u = −0.045175 − 1.293490I
a = −0.187844 + 0.818979I
b = 0.313266 + 1.365130I
−7.73894 + 8.36358I 0
u = −0.445173 + 1.219660I
a = 0.768913 + 0.623393I
b = −0.017515 + 1.152480I
−4.31677 − 1.36210I 0
u = −0.445173 − 1.219660I
a = 0.768913 − 0.623393I
b = −0.017515 − 1.152480I
−4.31677 + 1.36210I 0
u = −0.599521 + 0.343884I
a = 0.830282 + 0.552154I
b = −0.337189 − 0.937740I
−0.46677 + 1.70599I −3.14456 − 3.69461I
u = −0.599521 − 0.343884I
a = 0.830282 − 0.552154I
b = −0.337189 + 0.937740I
−0.46677 − 1.70599I −3.14456 + 3.69461I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.691337 + 1.130820I
a = −1.87160 + 0.33845I
b = 0.54431 + 1.42511I
−2.7651 + 16.7788I 0
u = 0.691337 − 1.130820I
a = −1.87160 − 0.33845I
b = 0.54431 − 1.42511I
−2.7651 − 16.7788I 0
u = 0.209377 + 0.627431I
a = 0.907953 + 0.723087I
b = −0.282187 − 0.534606I
0.051801 + 1.349280I −0.36388 − 5.58397I
u = 0.209377 − 0.627431I
a = 0.907953 − 0.723087I
b = −0.282187 + 0.534606I
0.051801 − 1.349280I −0.36388 + 5.58397I
u = 0.701751 + 1.159990I
a = 1.47811 − 0.36399I
b = −0.458469 − 1.232360I
1.61095 + 10.69360I 0
u = 0.701751 − 1.159990I
a = 1.47811 + 0.36399I
b = −0.458469 + 1.232360I
1.61095 − 10.69360I 0
u = −0.304509 + 1.329570I
a = 0.428450 + 0.613658I
b = −0.012890 + 1.171490I
−4.37509 − 1.36766I 0
u = −0.304509 − 1.329570I
a = 0.428450 − 0.613658I
b = −0.012890 − 1.171490I
−4.37509 + 1.36766I 0
u = 0.89791 + 1.14735I
a = −0.799502 − 0.240995I
b = 0.092862 + 1.140070I
−3.45411 + 3.73566I 0
u = 0.89791 − 1.14735I
a = −0.799502 + 0.240995I
b = 0.092862 − 1.140070I
−3.45411 − 3.73566I 0
9
II. I
u
2
= hb
2
− bu + u, a − u + 1, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u + 1
a
9
=
u − 1
b
a
10
=
b + u − 1
b
a
1
=
−u
u − 1
a
3
=
1
0
a
4
=
2bu − b − u + 1
bu − u
a
8
=
b + u − 1
b
a
11
=
−b − u + 1
−b
a
7
=
b + u − 1
b
a
7
=
b + u − 1
b
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
− u + 1)
2
c
2
(u
2
+ u + 1)
2
c
3
, c
4
, c
6
c
8
, c
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
7
, c
10
u
4
c
11
u
4
+ 3u
3
+ 2u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y
2
+ y + 1)
2
c
3
, c
4
, c
6
c
8
, c
9
y
4
+ 3y
3
+ 2y
2
+ 1
c
7
, c
10
y
4
c
11
y
4
− 5y
3
+ 6y
2
+ 4y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 0.621744 − 0.440597I
2.02988I 0. − 3.46410I
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −0.121744 + 1.306620I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 0.621744 + 0.440597I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −0.121744 − 1.306620I
− 2.02988I 0. + 3.46410I
13
III. I
u
3
= hb + u, a − 2, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u + 1
a
9
=
2
−u
a
10
=
−u + 2
−u
a
1
=
−u
u − 1
a
3
=
1
0
a
4
=
−u
u − 1
a
8
=
−u + 2
−u
a
11
=
u − 2
u
a
7
=
−u + 2
−u
a
7
=
−u + 2
−u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u + 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
c
9
u
2
− u + 1
c
2
, c
11
u
2
+ u + 1
c
7
, c
10
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
11
y
2
+ y + 1
c
7
, c
10
y
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 2.00000
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = 2.00000
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
17
IV. I
u
4
= hb
2
au + b
3
+ bu + au + b + u − 1, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u + 1
a
9
=
a
b
a
10
=
b + a
b
a
1
=
−u
u − 1
a
3
=
1
0
a
4
=
b
2
+ ba + 1
b
2
a
8
=
b + a
b
a
11
=
bau + a
2
u − a
2
− u
b
2
u + bau − ba + u − 1
a
7
=
−bau − a
2
u + a
2
+ b + a + u
−b
2
u − bau + ba + b − u + 1
a
7
=
−bau − a
2
u + a
2
+ b + a + u
−b
2
u − bau + ba + b − u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u − 4
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
18
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
−1.64493 − 2.02988I −6.00000 − 3.46410I
19
V. I
v
1
= ha, b
3
+ b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
9
=
0
b
a
10
=
b
b
a
1
=
1
0
a
3
=
1
0
a
4
=
b
2
+ 1
b
2
a
8
=
b
b
a
11
=
1
−b
2
a
7
=
b + 1
−b
2
+ b
a
7
=
b + 1
−b
2
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
3
c
3
, c
4
, c
8
c
9
, c
11
u
3
+ u + 1
c
6
u
3
+ 2u
2
+ u − 1
c
7
, c
10
(u + 1)
3
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
3
c
3
, c
4
, c
8
c
9
, c
11
y
3
+ 2y
2
+ y − 1
c
6
y
3
− 2y
2
+ 5y − 1
c
7
, c
10
(y − 1)
3
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.341164 + 1.161540I
−1.64493 −6.00000
v = 1.00000
a = 0
b = −0.341164 − 1.161540I
−1.64493 −6.00000
v = 1.00000
a = 0
b = 0.682328
−1.64493 −6.00000
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
3
(u
2
− u + 1)
3
(u
50
+ 4u
49
+ ··· + 255u + 62)
c
2
u
3
(u
2
+ u + 1)
3
(u
50
+ 20u
49
+ ··· + 27851u + 3844)
c
3
, c
8
9(u
2
− u + 1)(u
3
+ u + 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (9u
50
+ 9u
49
+ ··· − 6u + 1)
c
4
, c
9
9(u
2
− u + 1)(u
3
+ u + 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (9u
50
+ 9u
49
+ ··· + 8u + 1)
c
6
16(u
2
− u + 1)(u
3
+ 2u
2
+ u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (16u
50
− 32u
49
+ ··· − 26298u + 5463)
c
7
, c
10
u
6
(u + 1)
3
(u
50
− 6u
49
+ ··· − 7031u + 1274)
c
11
16(u
2
+ u + 1)(u
3
+ u + 1)(u
4
+ 3u
3
+ 2u
2
+ 1)
· (16u
50
− 16u
49
+ ··· + 612u + 63)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
3
(y
2
+ y + 1)
3
(y
50
+ 20y
49
+ ··· + 27851y + 3844)
c
2
y
3
(y
2
+ y + 1)
3
(y
50
+ 20y
49
+ ··· + 9.72711 × 10
7
y + 1.47763 × 10
7
)
c
3
, c
8
81(y
2
+ y + 1)(y
3
+ 2y
2
+ y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (81y
50
+ 2457y
49
+ ··· − 2y + 1)
c
4
, c
9
81(y
2
+ y + 1)(y
3
+ 2y
2
+ y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (81y
50
+ 2781y
49
+ ··· − 2y + 1)
c
6
256(y
2
+ y + 1)(y
3
− 2y
2
+ 5y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (256y
50
− 1408y
49
+ ··· − 86273478y + 29844369)
c
7
, c
10
y
6
(y − 1)
3
(y
50
− 34y
49
+ ··· + 1.59798 × 10
7
y + 1623076)
c
11
256(y
2
+ y + 1)(y
3
+ 2y
2
+ y − 1)(y
4
− 5y
3
+ 6y
2
+ 4y + 1)
· (256y
50
− 1920y
49
+ ··· + 153522y + 3969)
25
