12a
0423
(K12a
0423
)
A knot diagram
1
Linearized knot diagam
3 6 10 7 2 5 11 12 1 4 9 8
Solving Sequence
9,11
12 8 1
4,7
5 6 10 3 2
c
11
c
8
c
12
c
7
c
4
c
6
c
10
c
3
c
1
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
67
+ 29u
66
+ ··· + 2b − 5, −27u
67
− 95u
66
+ ··· + 4a + 37, u
68
+ 4u
67
+ ··· − 4u − 1i
I
u
2
= hb, a
2
− au + 2u
2
− u + 3, u
3
− u
2
+ 2u − 1i
I
u
3
= hb, −u
2
+ a + u − 1, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 77 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
= h7u
67
+ 29u
66
+ · · · + 2b − 5, −27u
67
− 95u
66
+ · · · + 4a + 37, u
68
+
4u
67
+ · · · − 4u − 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
−u
u
3
+ u
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
4
=
27
4
u
67
+
95
4
u
66
+ ··· −
51
2
u −
37
4
−
7
2
u
67
−
29
2
u
66
+ ··· + 11u +
5
2
a
7
=
−u
3
− 2u
u
3
+ u
a
5
=
9
2
u
67
+
31
2
u
66
+ ··· − 19u − 6
−
9
4
u
67
− 10u
66
+ ··· +
31
4
u +
3
4
a
6
=
u
16
+ 7u
14
+ ··· − 6u + 1
1
4
u
67
+
3
4
u
66
+ ··· −
1
2
u −
1
4
a
10
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
−
3
4
u
67
−
3
4
u
66
+ ··· −
19
2
u −
19
4
5
2
u
67
+
21
2
u
66
+ ··· − 12u −
9
2
a
2
=
−
1
4
u
67
−
3
4
u
66
+ ··· + 5u +
1
2
u
17
+ 7u
15
+ ··· − 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
11
2
u
67
−
51
2
u
66
+ ··· + 13u −
1
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
68
+ 16u
67
+ ··· + 28u + 1
c
2
, c
5
u
68
+ 4u
67
+ ··· + 4u − 1
c
3
, c
10
u
68
+ u
67
+ ··· + 1536u + 512
c
7
, c
9
u
68
− 4u
67
+ ··· + 298u − 193
c
8
, c
11
, c
12
u
68
+ 4u
67
+ ··· − 4u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
68
+ 76y
67
+ ··· − 468y + 1
c
2
, c
5
y
68
− 16y
67
+ ··· − 28y + 1
c
3
, c
10
y
68
− 49y
67
+ ··· − 3276800y + 262144
c
7
, c
9
y
68
− 56y
67
+ ··· + 843772y + 37249
c
8
, c
11
, c
12
y
68
+ 56y
67
+ ··· + 12y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.883196 + 0.110879I
a = 2.28702 + 0.20938I
b = −1.54451 + 0.55845I
14.5227 − 3.6785I 9.66151 + 1.29736I
u = −0.883196 − 0.110879I
a = 2.28702 − 0.20938I
b = −1.54451 − 0.55845I
14.5227 + 3.6785I 9.66151 − 1.29736I
u = −0.878083 + 0.122555I
a = −2.27299 − 0.23480I
b = 1.50934 − 0.61168I
14.0779 − 10.2341I 8.93592 + 6.05514I
u = −0.878083 − 0.122555I
a = −2.27299 + 0.23480I
b = 1.50934 + 0.61168I
14.0779 + 10.2341I 8.93592 − 6.05514I
u = −0.842036 + 0.041084I
a = 2.47425 + 0.12172I
b = −1.380970 + 0.196665I
7.31201 − 1.61336I 10.65283 + 0.71053I
u = −0.842036 − 0.041084I
a = 2.47425 − 0.12172I
b = −1.380970 − 0.196665I
7.31201 + 1.61336I 10.65283 − 0.71053I
u = 0.836342 + 0.008912I
a = −0.017629 + 1.041540I
b = 0.042391 − 1.358950I
9.34458 + 3.20678I 8.45402 − 2.50897I
u = 0.836342 − 0.008912I
a = −0.017629 − 1.041540I
b = 0.042391 + 1.358950I
9.34458 − 3.20678I 8.45402 + 2.50897I
u = −0.820055 + 0.078945I
a = −2.47808 − 0.26474I
b = 1.263950 − 0.365738I
5.17218 − 6.11204I 6.64053 + 6.47258I
u = −0.820055 − 0.078945I
a = −2.47808 + 0.26474I
b = 1.263950 + 0.365738I
5.17218 + 6.11204I 6.64053 − 6.47258I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.074714 + 1.188830I
a = 0.891353 − 1.049750I
b = −0.638314 − 0.584251I
0.28440 + 1.43816I 0
u = −0.074714 − 1.188830I
a = 0.891353 + 1.049750I
b = −0.638314 + 0.584251I
0.28440 − 1.43816I 0
u = −0.789953
a = −2.69327
b = 1.14488
2.31303 4.51890
u = 0.576977 + 0.530503I
a = 0.824531 − 0.923493I
b = −1.359680 − 0.151800I
8.22238 + 5.29147I 7.35195 − 5.78163I
u = 0.576977 − 0.530503I
a = 0.824531 + 0.923493I
b = −1.359680 + 0.151800I
8.22238 − 5.29147I 7.35195 + 5.78163I
u = 0.593054 + 0.504319I
a = −0.807717 + 0.928418I
b = 1.357810 + 0.068287I
8.30625 − 1.12217I 7.67861 − 0.76629I
u = 0.593054 − 0.504319I
a = −0.807717 − 0.928418I
b = 1.357810 − 0.068287I
8.30625 + 1.12217I 7.67861 + 0.76629I
u = −0.448687 + 1.144800I
a = 0.709101 + 0.607422I
b = −1.54402 − 0.50394I
10.94400 + 5.49512I 0
u = −0.448687 − 1.144800I
a = 0.709101 − 0.607422I
b = −1.54402 + 0.50394I
10.94400 − 5.49512I 0
u = −0.081643 + 1.234740I
a = −0.87255 + 1.30038I
b = 0.594137 + 0.582932I
−0.15037 − 4.30964I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.081643 − 1.234740I
a = −0.87255 − 1.30038I
b = 0.594137 − 0.582932I
−0.15037 + 4.30964I 0
u = −0.357510 + 1.191950I
a = 1.211980 + 0.542043I
b = −1.229020 − 0.227598I
1.76607 + 1.85694I 0
u = −0.357510 − 1.191950I
a = 1.211980 − 0.542043I
b = −1.229020 + 0.227598I
1.76607 − 1.85694I 0
u = −0.449847 + 1.160780I
a = −0.740080 − 0.666202I
b = 1.57548 + 0.44356I
11.30140 − 1.07993I 0
u = −0.449847 − 1.160780I
a = −0.740080 + 0.666202I
b = 1.57548 − 0.44356I
11.30140 + 1.07993I 0
u = 0.273567 + 1.231890I
a = 0.610435 − 0.305888I
b = −0.123364 − 0.850513I
−1.79566 + 1.87689I 0
u = 0.273567 − 1.231890I
a = 0.610435 + 0.305888I
b = −0.123364 + 0.850513I
−1.79566 − 1.87689I 0
u = 0.114749 + 1.267340I
a = 0.241747 − 0.755643I
b = −0.563150 − 0.524155I
−3.19195 + 1.95948I 0
u = 0.114749 − 1.267340I
a = 0.241747 + 0.755643I
b = −0.563150 + 0.524155I
−3.19195 − 1.95948I 0
u = 0.718407 + 0.054666I
a = −0.136876 + 0.792692I
b = 0.229382 − 0.831076I
1.79469 + 1.70423I 5.68933 − 4.03428I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.718407 − 0.054666I
a = −0.136876 − 0.792692I
b = 0.229382 + 0.831076I
1.79469 − 1.70423I 5.68933 + 4.03428I
u = −0.386268 + 1.231950I
a = −1.17649 − 0.83959I
b = 1.372460 + 0.057138I
3.63723 − 2.79864I 0
u = −0.386268 − 1.231950I
a = −1.17649 + 0.83959I
b = 1.372460 − 0.057138I
3.63723 + 2.79864I 0
u = 0.015638 + 1.299180I
a = −0.263537 + 1.187850I
b = 0.654251 + 0.629382I
−5.82782 − 0.26682I 0
u = 0.015638 − 1.299180I
a = −0.263537 − 1.187850I
b = 0.654251 − 0.629382I
−5.82782 + 0.26682I 0
u = 0.380086 + 1.262700I
a = 0.897471 − 0.203783I
b = 0.103923 − 1.353660I
5.45713 + 1.16252I 0
u = 0.380086 − 1.262700I
a = 0.897471 + 0.203783I
b = 0.103923 + 1.353660I
5.45713 − 1.16252I 0
u = −0.344177 + 1.273700I
a = 1.53531 + 1.04902I
b = −1.156260 + 0.164863I
−1.64536 − 4.08494I 0
u = −0.344177 − 1.273700I
a = 1.53531 − 1.04902I
b = −1.156260 − 0.164863I
−1.64536 + 4.08494I 0
u = 0.378704 + 1.277060I
a = −0.901537 + 0.167233I
b = −0.183952 + 1.350690I
5.34738 + 7.57130I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.378704 − 1.277060I
a = −0.901537 − 0.167233I
b = −0.183952 − 1.350690I
5.34738 − 7.57130I 0
u = 0.299464 + 1.300730I
a = −0.671330 + 0.054559I
b = −0.290086 + 0.868205I
−2.44438 + 5.38952I 0
u = 0.299464 − 1.300730I
a = −0.671330 − 0.054559I
b = −0.290086 − 0.868205I
−2.44438 − 5.38952I 0
u = −0.380669 + 1.300130I
a = −1.27591 − 1.24442I
b = 1.369760 − 0.318562I
3.12774 − 6.00570I 0
u = −0.380669 − 1.300130I
a = −1.27591 + 1.24442I
b = 1.369760 + 0.318562I
3.12774 + 6.00570I 0
u = 0.217986 + 1.345040I
a = −0.350229 − 0.323689I
b = −0.564549 + 0.170572I
−3.80220 + 2.41064I 0
u = 0.217986 − 1.345040I
a = −0.350229 + 0.323689I
b = −0.564549 − 0.170572I
−3.80220 − 2.41064I 0
u = 0.093295 + 1.367930I
a = 0.160591 + 0.969459I
b = 0.835398 + 0.483273I
−5.23627 + 4.58427I 0
u = 0.093295 − 1.367930I
a = 0.160591 − 0.969459I
b = 0.835398 − 0.483273I
−5.23627 − 4.58427I 0
u = −0.364104 + 1.323200I
a = 1.35385 + 1.40669I
b = −1.271370 + 0.466726I
0.77875 − 10.37800I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.364104 − 1.323200I
a = 1.35385 − 1.40669I
b = −1.271370 − 0.466726I
0.77875 + 10.37800I 0
u = −0.395915 + 1.350660I
a = −1.13840 − 1.49595I
b = 1.49290 − 0.63680I
9.93330 − 8.26771I 0
u = −0.395915 − 1.350660I
a = −1.13840 + 1.49595I
b = 1.49290 + 0.63680I
9.93330 + 8.26771I 0
u = −0.390304 + 1.356620I
a = 1.15307 + 1.53169I
b = −1.45794 + 0.68098I
9.4275 − 14.7877I 0
u = −0.390304 − 1.356620I
a = 1.15307 − 1.53169I
b = −1.45794 − 0.68098I
9.4275 + 14.7877I 0
u = 0.341611 + 0.461012I
a = 0.861805 − 0.885756I
b = −0.878881 − 0.293282I
0.40677 + 3.17475I 3.07710 − 8.65765I
u = 0.341611 − 0.461012I
a = 0.861805 + 0.885756I
b = −0.878881 + 0.293282I
0.40677 − 3.17475I 3.07710 + 8.65765I
u = 0.498530 + 0.267549I
a = −0.756073 + 0.717769I
b = 0.795737 − 0.123613I
1.109940 − 0.236478I 7.83765 + 0.33377I
u = 0.498530 − 0.267549I
a = −0.756073 − 0.717769I
b = 0.795737 + 0.123613I
1.109940 + 0.236478I 7.83765 − 0.33377I
u = 0.15691 + 1.43163I
a = −0.531513 − 0.823143I
b = −1.167860 − 0.205420I
2.05830 + 1.37245I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.15691 − 1.43163I
a = −0.531513 + 0.823143I
b = −1.167860 + 0.205420I
2.05830 − 1.37245I 0
u = 0.14259 + 1.43508I
a = 0.516840 + 0.882774I
b = 1.175310 + 0.290911I
1.86781 + 7.64221I 0
u = 0.14259 − 1.43508I
a = 0.516840 − 0.882774I
b = 1.175310 − 0.290911I
1.86781 − 7.64221I 0
u = 0.459924
a = −0.727898
b = 0.446382
0.791286 12.9510
u = −0.327604 + 0.031735I
a = −0.07580 − 2.80687I
b = −0.005736 − 0.498706I
3.56841 − 2.90168I −2.97092 + 3.54606I
u = −0.327604 − 0.031735I
a = −0.07580 + 2.80687I
b = −0.005736 + 0.498706I
3.56841 + 2.90168I −2.97092 − 3.54606I
u = −0.048081 + 0.224645I
a = 0.94797 − 1.88047I
b = −0.308184 − 0.394663I
−1.259210 − 0.317933I −6.24438 + 0.78722I
u = −0.048081 − 0.224645I
a = 0.94797 + 1.88047I
b = −0.308184 + 0.394663I
−1.259210 + 0.317933I −6.24438 − 0.78722I
11
II. I
u
2
= hb, a
2
− au + 2u
2
− u + 3, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
−u
u
2
− u + 1
a
1
=
u
2
+ 1
−u
2
+ u − 1
a
4
=
a
0
a
7
=
−u
2
− 1
u
2
− u + 1
a
5
=
0
au
a
6
=
−u
2
− 1
−u
2
a + au − u
a
10
=
1
0
a
3
=
a
0
a
2
=
−au + a − 1
−u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
a − 2au + 10u
2
+ 3a − 12u + 15
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
11
c
12
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
10
u
6
c
5
, c
7
, c
9
(u
3
− u
2
+ 1)
2
c
6
, c
8
(u
3
+ u
2
+ 2u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
7
c
9
(y
3
− y
2
+ 2y −1)
2
c
3
, c
10
y
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.592519 + 0.986732I
b = 0
5.65624I 2.97732 − 6.46189I
u = 0.215080 + 1.307140I
a = −0.377439 + 0.320410I
b = 0
−4.13758 + 2.82812I −5.23142 − 6.76304I
u = 0.215080 − 1.307140I
a = 0.592519 − 0.986732I
b = 0
− 5.65624I 2.97732 + 6.46189I
u = 0.215080 − 1.307140I
a = −0.377439 − 0.320410I
b = 0
−4.13758 − 2.82812I −5.23142 + 6.76304I
u = 0.569840
a = 0.28492 + 1.73159I
b = 0
4.13758 − 2.82812I 11.75410 + 2.09676I
u = 0.569840
a = 0.28492 − 1.73159I
b = 0
4.13758 + 2.82812I 11.75410 − 2.09676I
15
III. I
u
3
= hb, −u
2
+ a + u − 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
−u
u
2
− u + 1
a
1
=
u
2
+ 1
−u
2
+ u − 1
a
4
=
u
2
− u + 1
0
a
7
=
−u
2
− 1
u
2
− u + 1
a
5
=
0
−u + 1
a
6
=
−u
2
− 1
1
a
10
=
1
0
a
3
=
u
2
− u + 1
0
a
2
=
u
2
− u + 2
−u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
11
c
12
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
3
, c
10
u
3
c
5
, c
7
, c
9
u
3
− u
2
+ 1
c
6
, c
8
u
3
+ u
2
+ 2u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
7
c
9
y
3
− y
2
+ 2y −1
c
3
, c
10
y
3
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.877439 − 0.744862I
b = 0
0 1.66236 − 0.56228I
u = 0.215080 − 1.307140I
a = −0.877439 + 0.744862I
b = 0
0 1.66236 + 0.56228I
u = 0.569840
a = 0.754878
b = 0
0 −0.324720
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
3
− u
2
+ 2u − 1)
3
)(u
68
+ 16u
67
+ ··· + 28u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
68
+ 4u
67
+ ··· + 4u − 1)
c
3
, c
10
u
9
(u
68
+ u
67
+ ··· + 1536u + 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
68
+ 4u
67
+ ··· + 4u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
68
+ 16u
67
+ ··· + 28u + 1)
c
7
, c
9
((u
3
− u
2
+ 1)
3
)(u
68
− 4u
67
+ ··· + 298u − 193)
c
8
((u
3
+ u
2
+ 2u + 1)
3
)(u
68
+ 4u
67
+ ··· − 4u − 1)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
3
)(u
68
+ 4u
67
+ ··· − 4u − 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
68
+ 76y
67
+ ··· − 468y + 1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
68
− 16y
67
+ ··· − 28y + 1)
c
3
, c
10
y
9
(y
68
− 49y
67
+ ··· − 3276800y + 262144)
c
7
, c
9
((y
3
− y
2
+ 2y −1)
3
)(y
68
− 56y
67
+ ··· + 843772y + 37249)
c
8
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
68
+ 56y
67
+ ··· + 12y + 1)
21
