12a
0178
(K12a
0178
)
A knot diagram
1
Linearized knot diagam
3 5 9 6 2 11 12 4 1 8 7 10
Solving Sequence
6,11
7 12
2,8
5 3 1 4 9 10
c
6
c
11
c
7
c
5
c
2
c
1
c
4
c
8
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
76
− u
75
+ ··· + 2b − 5u, 2u
78
− 4u
77
+ ··· + 2a − 1, u
79
− 3u
78
+ ··· + u − 1i
I
u
2
= hu
3
a + u
3
− au + b − a − u − 1, −u
3
a − u
4
+ a
2
+ 2au + 2u
2
+ 2a, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 89 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
= hu
76
−u
75
+· · ·+2b−5u, 2u
78
−4u
77
+· · ·+2a−1, u
79
−3u
78
+· · ·+u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
2
=
−u
78
+ 2u
77
+ ··· −
3
2
u +
1
2
−
1
2
u
76
+
1
2
u
75
+ ··· −
11
2
u
2
+
5
2
u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
−2u
78
+ 4u
77
+ ··· + 5u −
1
2
3u
78
− 5u
77
+ ··· +
3
2
u + 1
a
3
=
−5u
78
+ 8u
77
+ ··· +
17
2
u −
5
2
5u
78
− 8u
77
+ ··· +
3
2
u + 2
a
1
=
u
9
− 4u
7
+ 5u
5
− 2u
3
+ u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
4
=
u
78
− u
77
+ ··· +
13
2
u +
1
2
3u
78
− 5u
77
+ ··· +
3
2
u + 1
a
9
=
u
13
− 6u
11
+ 13u
9
− 12u
7
+ 6u
5
− 4u
3
+ u
−u
15
+ 7u
13
− 18u
11
+ 19u
9
− 6u
7
+ 2u
5
− 4u
3
− u
a
10
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
11
2
u
78
−
19
2
u
77
+ ··· − 22u +
3
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
79
+ 24u
78
+ ··· − 14u − 1
c
2
, c
5
u
79
+ 6u
78
+ ··· + 2u + 1
c
3
, c
8
u
79
+ u
78
+ ··· − 2048u − 1024
c
6
, c
7
, c
11
u
79
− 3u
78
+ ··· + u − 1
c
9
, c
12
u
79
− 11u
78
+ ··· − 117u − 73
c
10
u
79
+ 9u
78
+ ··· − 2487u + 851
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
79
+ 68y
78
+ ··· + 270y −1
c
2
, c
5
y
79
+ 24y
78
+ ··· − 14y −1
c
3
, c
8
y
79
+ 55y
78
+ ··· − 9437184y −1048576
c
6
, c
7
, c
11
y
79
− 75y
78
+ ··· − 23y −1
c
9
, c
12
y
79
+ 73y
78
+ ··· − 70115y −5329
c
10
y
79
− 31y
78
+ ··· − 2637999y −724201
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.960687 + 0.153679I
a = 0.507114 + 0.208480I
b = −0.750208 + 0.893053I
4.67325 − 2.88323I 0
u = 0.960687 − 0.153679I
a = 0.507114 − 0.208480I
b = −0.750208 − 0.893053I
4.67325 + 2.88323I 0
u = 0.827370 + 0.170912I
a = 0.960753 − 0.367241I
b = −0.764781 − 0.869204I
4.74752 + 2.85194I 5.96777 − 3.55563I
u = 0.827370 − 0.170912I
a = 0.960753 + 0.367241I
b = −0.764781 + 0.869204I
4.74752 − 2.85194I 5.96777 + 3.55563I
u = −0.431098 + 0.703014I
a = −0.011403 + 1.045270I
b = −0.906120 − 0.737961I
9.87956 − 5.56873I 5.48276 + 3.81380I
u = −0.431098 − 0.703014I
a = −0.011403 − 1.045270I
b = −0.906120 + 0.737961I
9.87956 + 5.56873I 5.48276 − 3.81380I
u = −0.417770 + 0.709165I
a = 2.06985 + 1.21835I
b = −0.786589 + 1.032560I
8.9584 − 11.8257I 3.97751 + 8.54937I
u = −0.417770 − 0.709165I
a = 2.06985 − 1.21835I
b = −0.786589 − 1.032560I
8.9584 + 11.8257I 3.97751 − 8.54937I
u = −0.541357 + 0.617925I
a = 1.260830 + 0.092818I
b = −0.902271 + 0.758090I
10.28590 + 1.15255I 6.41028 + 2.08924I
u = −0.541357 − 0.617925I
a = 1.260830 − 0.092818I
b = −0.902271 − 0.758090I
10.28590 − 1.15255I 6.41028 − 2.08924I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.554854 + 0.602818I
a = 0.818220 + 0.193142I
b = −0.794943 − 1.020680I
9.46345 + 7.42771I 5.21274 − 2.74453I
u = −0.554854 − 0.602818I
a = 0.818220 − 0.193142I
b = −0.794943 + 1.020680I
9.46345 − 7.42771I 5.21274 + 2.74453I
u = −0.460232 + 0.639959I
a = −0.538335 − 0.631341I
b = 0.819242 − 0.023424I
5.68856 − 2.11060I 6.83378 + 3.31832I
u = −0.460232 − 0.639959I
a = −0.538335 + 0.631341I
b = 0.819242 + 0.023424I
5.68856 + 2.11060I 6.83378 − 3.31832I
u = 0.446094 + 0.644815I
a = −2.25296 + 0.31798I
b = 0.776837 + 0.897829I
4.79648 + 5.02800I 3.77738 − 5.92269I
u = 0.446094 − 0.644815I
a = −2.25296 − 0.31798I
b = 0.776837 − 0.897829I
4.79648 − 5.02800I 3.77738 + 5.92269I
u = 1.211160 + 0.128664I
a = 1.28344 − 1.04569I
b = 0.018725 − 0.877436I
0.410093 + 0.569090I 0
u = 1.211160 − 0.128664I
a = 1.28344 + 1.04569I
b = 0.018725 + 0.877436I
0.410093 − 0.569090I 0
u = 0.465338 + 0.625622I
a = −0.819821 + 1.071380I
b = 0.783132 − 0.871429I
4.87790 − 0.84564I 4.12547 − 0.52207I
u = 0.465338 − 0.625622I
a = −0.819821 − 1.071380I
b = 0.783132 + 0.871429I
4.87790 + 0.84564I 4.12547 + 0.52207I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.416187 + 0.649698I
a = −1.47061 − 1.27515I
b = 0.281669 − 1.127160I
1.81369 − 5.81158I 1.46444 + 7.37025I
u = −0.416187 − 0.649698I
a = −1.47061 + 1.27515I
b = 0.281669 + 1.127160I
1.81369 + 5.81158I 1.46444 − 7.37025I
u = −0.467486 + 0.588976I
a = 0.271507 + 0.612178I
b = 0.318145 + 1.112630I
2.05435 + 1.75127I 2.62331 − 0.74126I
u = −0.467486 − 0.588976I
a = 0.271507 − 0.612178I
b = 0.318145 − 1.112630I
2.05435 − 1.75127I 2.62331 + 0.74126I
u = 1.26366
a = 0.931332
b = −0.238856
2.75626 0
u = 0.174701 + 0.686237I
a = −0.699194 + 0.373449I
b = −0.764897 + 0.812003I
2.54707 + 0.60070I 2.16281 − 1.57893I
u = 0.174701 − 0.686237I
a = −0.699194 − 0.373449I
b = −0.764897 − 0.812003I
2.54707 − 0.60070I 2.16281 + 1.57893I
u = 0.140244 + 0.692618I
a = 0.66990 − 1.89405I
b = −0.742698 − 0.939246I
2.15562 + 6.31590I 0.72997 − 6.99064I
u = 0.140244 − 0.692618I
a = 0.66990 + 1.89405I
b = −0.742698 + 0.939246I
2.15562 − 6.31590I 0.72997 + 6.99064I
u = 0.331444 + 0.619754I
a = 1.70156 − 0.90801I
b = −0.146875 − 0.718790I
−0.39859 + 2.41679I −2.51815 − 2.50379I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.331444 − 0.619754I
a = 1.70156 + 0.90801I
b = −0.146875 + 0.718790I
−0.39859 − 2.41679I −2.51815 + 2.50379I
u = −1.284980 + 0.179165I
a = −0.74042 − 1.23710I
b = 0.156711 − 1.014790I
1.15848 − 4.83085I 0
u = −1.284980 − 0.179165I
a = −0.74042 + 1.23710I
b = 0.156711 + 1.014790I
1.15848 + 4.83085I 0
u = −1.300270 + 0.074806I
a = −0.754795 + 0.685987I
b = 0.494882 + 1.001490I
3.08759 + 1.25662I 0
u = −1.300270 − 0.074806I
a = −0.754795 − 0.685987I
b = 0.494882 − 1.001490I
3.08759 − 1.25662I 0
u = −1.311810 + 0.262394I
a = 1.97615 + 0.97895I
b = −0.742423 + 0.968481I
6.68975 − 9.77430I 0
u = −1.311810 − 0.262394I
a = 1.97615 − 0.97895I
b = −0.742423 − 0.968481I
6.68975 + 9.77430I 0
u = 1.330220 + 0.148111I
a = −2.77349 + 0.42700I
b = 0.643454 + 0.948330I
4.02017 + 5.12137I 0
u = 1.330220 − 0.148111I
a = −2.77349 − 0.42700I
b = 0.643454 − 0.948330I
4.02017 − 5.12137I 0
u = 1.345390 + 0.105400I
a = −0.77119 + 1.72286I
b = 0.672238 − 0.709217I
4.74837 + 0.02263I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.345390 − 0.105400I
a = −0.77119 − 1.72286I
b = 0.672238 + 0.709217I
4.74837 − 0.02263I 0
u = −1.354000 + 0.125122I
a = −0.577117 + 0.093747I
b = 0.589160 − 0.295718I
5.02105 − 2.84369I 0
u = −1.354000 − 0.125122I
a = −0.577117 − 0.093747I
b = 0.589160 + 0.295718I
5.02105 + 2.84369I 0
u = −1.336080 + 0.257261I
a = 0.095226 + 0.877787I
b = −0.779371 − 0.771197I
7.28961 − 4.01466I 0
u = −1.336080 − 0.257261I
a = 0.095226 − 0.877787I
b = −0.779371 + 0.771197I
7.28961 + 4.01466I 0
u = 0.055935 + 0.584888I
a = 0.34048 + 2.64860I
b = 0.118978 + 0.935231I
−2.96390 + 2.06663I −7.76438 − 4.75095I
u = 0.055935 − 0.584888I
a = 0.34048 − 2.64860I
b = 0.118978 − 0.935231I
−2.96390 − 2.06663I −7.76438 + 4.75095I
u = 0.372135 + 0.444695I
a = 0.571797 + 0.154582I
b = −0.064395 + 0.528805I
0.152685 + 0.966028I 0.58728 − 4.99432I
u = 0.372135 − 0.444695I
a = 0.571797 − 0.154582I
b = −0.064395 − 0.528805I
0.152685 − 0.966028I 0.58728 + 4.99432I
u = −1.42465 + 0.19612I
a = 0.757172 + 0.386330I
b = −0.274557 − 0.523897I
5.85268 − 3.48752I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.42465 − 0.19612I
a = 0.757172 − 0.386330I
b = −0.274557 + 0.523897I
5.85268 + 3.48752I 0
u = −1.42914 + 0.23677I
a = 1.75980 + 0.17758I
b = −0.210954 + 0.737939I
5.24909 − 5.55916I 0
u = −1.42914 − 0.23677I
a = 1.75980 − 0.17758I
b = −0.210954 − 0.737939I
5.24909 + 5.55916I 0
u = −1.45767 + 0.01095I
a = 1.81232 − 0.84748I
b = −0.830455 + 0.896981I
11.59420 − 3.09088I 0
u = −1.45767 − 0.01095I
a = 1.81232 + 0.84748I
b = −0.830455 − 0.896981I
11.59420 + 3.09088I 0
u = 1.46442 + 0.23981I
a = −1.64516 + 0.18165I
b = 0.275942 + 1.152230I
7.87641 + 9.06816I 0
u = 1.46442 − 0.23981I
a = −1.64516 − 0.18165I
b = 0.275942 − 1.152230I
7.87641 − 9.06816I 0
u = 1.46930 + 0.21252I
a = −0.105941 + 0.396384I
b = 0.341084 − 1.138020I
8.28626 + 1.18360I 0
u = 1.46930 − 0.21252I
a = −0.105941 − 0.396384I
b = 0.341084 + 1.138020I
8.28626 − 1.18360I 0
u = −1.47358 + 0.23322I
a = −2.85449 + 0.44307I
b = 0.790272 − 0.913410I
10.99410 − 8.23765I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.47358 − 0.23322I
a = −2.85449 − 0.44307I
b = 0.790272 + 0.913410I
10.99410 + 8.23765I 0
u = −1.47618 + 0.22313I
a = −1.48000 − 1.76752I
b = 0.805517 + 0.864477I
11.14650 − 2.25375I 0
u = −1.47618 − 0.22313I
a = −1.48000 + 1.76752I
b = 0.805517 − 0.864477I
11.14650 + 2.25375I 0
u = 1.47734 + 0.22877I
a = −1.30565 + 0.55825I
b = 0.851193 + 0.043651I
11.94830 + 5.28265I 0
u = 1.47734 − 0.22877I
a = −1.30565 − 0.55825I
b = 0.851193 − 0.043651I
11.94830 − 5.28265I 0
u = 1.47342 + 0.26226I
a = 2.75426 − 0.27890I
b = −0.787624 − 1.043110I
15.0608 + 15.3708I 0
u = 1.47342 − 0.26226I
a = 2.75426 + 0.27890I
b = −0.787624 + 1.043110I
15.0608 − 15.3708I 0
u = 1.47783 + 0.25743I
a = 0.82024 − 1.67408I
b = −0.918969 + 0.728773I
16.0459 + 9.0741I 0
u = 1.47783 − 0.25743I
a = 0.82024 + 1.67408I
b = −0.918969 − 0.728773I
16.0459 − 9.0741I 0
u = 1.49805 + 0.19160I
a = 1.56162 − 1.06045I
b = −0.811404 + 1.021460I
16.1357 − 4.5825I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.49805 − 0.19160I
a = 1.56162 + 1.06045I
b = −0.811404 − 1.021460I
16.1357 + 4.5825I 0
u = 1.49830 + 0.20045I
a = 2.02683 + 0.60119I
b = −0.917511 − 0.773257I
16.9160 + 1.7936I 0
u = 1.49830 − 0.20045I
a = 2.02683 − 0.60119I
b = −0.917511 + 0.773257I
16.9160 − 1.7936I 0
u = −0.108953 + 0.459074I
a = −1.93875 − 2.39147I
b = 0.573629 − 0.929349I
−0.49704 − 2.88847I −5.04267 + 1.92432I
u = −0.108953 − 0.459074I
a = −1.93875 + 2.39147I
b = 0.573629 + 0.929349I
−0.49704 + 2.88847I −5.04267 − 1.92432I
u = 0.247554 + 0.377813I
a = 0.560959 + 0.199573I
b = 0.193812 + 0.327293I
0.102843 + 1.008930I 1.78319 − 6.38352I
u = 0.247554 − 0.377813I
a = 0.560959 − 0.199573I
b = 0.193812 − 0.327293I
0.102843 − 1.008930I 1.78319 + 6.38352I
u = −0.152462 + 0.276606I
a = 1.19363 − 1.28731I
b = 0.511852 + 0.767170I
0.09111 + 1.51758I −2.07514 − 4.93514I
u = −0.152462 − 0.276606I
a = 1.19363 + 1.28731I
b = 0.511852 − 0.767170I
0.09111 − 1.51758I −2.07514 + 4.93514I
12
II. I
u
2
= hu
3
a + u
3
− au + b − a − u − 1, −u
3
a − u
4
+ a
2
+ 2au + 2u
2
+
2a, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
2
=
a
−u
3
a − u
3
+ au + a + u + 1
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
u
3
a − au + u + 1
−u
3
a − u
3
+ au + a + u
a
3
=
−u
3
+ a + 2u + 1
−u
3
a − u
3
+ au + a + u
a
1
=
−1
0
a
4
=
−u
3
+ a + 2u + 1
−u
3
a − u
3
+ au + a + u
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
a + 4u
3
a + u
4
− 2u
2
a + u
3
− 4au − 2u
2
− 5a + 3u + 1
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
3
, c
8
u
10
c
6
, c
7
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
9
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
10
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
5
c
3
, c
8
y
10
c
6
, c
7
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −0.685143 + 0.545349I
b = 0.500000 + 0.866025I
2.40108 + 2.02988I 0.33682 − 4.42764I
u = −1.21774
a = −0.685143 − 0.545349I
b = 0.500000 − 0.866025I
2.40108 − 2.02988I 0.33682 + 4.42764I
u = −0.309916 + 0.549911I
a = 0.394874 + 0.200669I
b = 0.500000 + 0.866025I
0.329100 + 0.499304I −0.01046 + 1.42329I
u = −0.309916 + 0.549911I
a = −1.52365 − 1.30833I
b = 0.500000 − 0.866025I
0.32910 − 3.56046I 2.49844 + 7.77102I
u = −0.309916 − 0.549911I
a = 0.394874 − 0.200669I
b = 0.500000 − 0.866025I
0.329100 − 0.499304I −0.01046 − 1.42329I
u = −0.309916 − 0.549911I
a = −1.52365 + 1.30833I
b = 0.500000 + 0.866025I
0.32910 + 3.56046I 2.49844 − 7.77102I
u = 1.41878 + 0.21917I
a = −0.335573 + 0.598472I
b = 0.500000 − 0.866025I
5.87256 + 2.37095I 4.29156 + 0.98555I
u = 1.41878 + 0.21917I
a = −1.85051 + 0.27617I
b = 0.500000 + 0.866025I
5.87256 + 6.43072I 6.88365 − 7.29164I
u = 1.41878 − 0.21917I
a = −0.335573 − 0.598472I
b = 0.500000 + 0.866025I
5.87256 − 2.37095I 4.29156 − 0.98555I
u = 1.41878 − 0.21917I
a = −1.85051 − 0.27617I
b = 0.500000 − 0.866025I
5.87256 − 6.43072I 6.88365 + 7.29164I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
2
− u + 1)
5
)(u
79
+ 24u
78
+ ··· − 14u − 1)
c
2
((u
2
+ u + 1)
5
)(u
79
+ 6u
78
+ ··· + 2u + 1)
c
3
, c
8
u
10
(u
79
+ u
78
+ ··· − 2048u − 1024)
c
5
((u
2
− u + 1)
5
)(u
79
+ 6u
78
+ ··· + 2u + 1)
c
6
, c
7
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
79
− 3u
78
+ ··· + u − 1)
c
9
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
79
− 11u
78
+ ··· − 117u − 73)
c
10
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
79
+ 9u
78
+ ··· − 2487u + 851)
c
11
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
79
− 3u
78
+ ··· + u − 1)
c
12
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
79
− 11u
78
+ ··· − 117u − 73)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
5
)(y
79
+ 68y
78
+ ··· + 270y −1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
79
+ 24y
78
+ ··· − 14y −1)
c
3
, c
8
y
10
(y
79
+ 55y
78
+ ··· − 9437184y −1048576)
c
6
, c
7
, c
11
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
79
− 75y
78
+ ··· − 23y −1)
c
9
, c
12
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
79
+ 73y
78
+ ··· − 70115y −5329)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
79
− 31y
78
+ ··· − 2637999y −724201)
18
