12a
0689
(K12a
0689
)
A knot diagram
1
Linearized knot diagam
3 7 12 11 8 9 2 6 1 5 4 10
Solving Sequence
5,11
4 12
3,7
2 8 10 1 9 6
c
4
c
11
c
3
c
2
c
7
c
10
c
12
c
9
c
6
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
61
− u
60
+ ··· + b − 1, u
64
− 2u
63
+ ··· + a + 2, u
65
− 2u
64
+ ··· + u − 1i
I
u
2
= hb − 1, u
3
+ u
2
+ a + 3u + 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 69 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
61
− u
60
+ · · · + b − 1, u
64
− 2u
63
+ · · · + a + 2, u
65
− 2u
64
+ · · · + u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
12
=
−u
u
3
+ u
a
3
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
−u
64
+ 2u
63
+ ··· − 4u − 2
−u
61
+ u
60
+ ··· + 2u + 1
a
2
=
−u
11
− 6u
9
− 12u
7
− 8u
5
− u
3
− 2u
u
13
+ 7u
11
+ 17u
9
+ 16u
7
+ 6u
5
+ 5u
3
+ u
a
8
=
−u
64
+ 2u
63
+ ··· − 3u − 1
−u
61
+ u
60
+ ··· + u + 1
a
10
=
u
u
a
1
=
u
5
+ 2u
3
− u
u
5
+ 3u
3
+ u
a
9
=
u
9
+ 4u
7
+ 3u
5
− 2u
3
+ u
u
9
+ 5u
7
+ 7u
5
+ 2u
3
+ u
a
6
=
−u
64
+ 2u
63
+ ··· − 3u − 1
−u
61
+ u
60
+ ··· + 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
64
+ 2u
63
+ ··· − 10u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
65
+ 27u
64
+ ··· − 1984u − 256
c
2
, c
7
u
65
+ u
64
+ ··· + 24u + 16
c
3
, c
4
, c
10
c
11
u
65
− 2u
64
+ ··· + u − 1
c
5
, c
6
, c
8
u
65
− 5u
64
+ ··· + u + 1
c
9
, c
12
u
65
+ 12u
64
+ ··· + 1405u + 131
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
65
+ 15y
64
+ ··· − 1257472y −65536
c
2
, c
7
y
65
+ 27y
64
+ ··· − 1984y −256
c
3
, c
4
, c
10
c
11
y
65
+ 72y
64
+ ··· + 5y −1
c
5
, c
6
, c
8
y
65
− 57y
64
+ ··· − 33y −1
c
9
, c
12
y
65
+ 36y
64
+ ··· + 294081y −17161
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.176961 + 0.834517I
a = 1.40512 + 0.84371I
b = 0.99803 + 1.05009I
−1.61201 + 5.84427I −3.29017 − 6.41076I
u = −0.176961 − 0.834517I
a = 1.40512 − 0.84371I
b = 0.99803 − 1.05009I
−1.61201 − 5.84427I −3.29017 + 6.41076I
u = 0.587464 + 0.605812I
a = 1.99776 + 1.35416I
b = 1.11944 − 1.68323I
−6.38469 − 11.44690I −7.68442 + 9.11849I
u = 0.587464 − 0.605812I
a = 1.99776 − 1.35416I
b = 1.11944 + 1.68323I
−6.38469 + 11.44690I −7.68442 − 9.11849I
u = 0.562023 + 0.589176I
a = −1.70682 − 1.57560I
b = −1.43980 + 1.35021I
−1.08419 − 7.25640I −4.14361 + 8.68494I
u = 0.562023 − 0.589176I
a = −1.70682 + 1.57560I
b = −1.43980 − 1.35021I
−1.08419 + 7.25640I −4.14361 − 8.68494I
u = −0.434655 + 0.677483I
a = −0.76935 − 1.40090I
b = 0.761684 − 0.558455I
−3.21792 − 0.04263I −6.95293 − 1.05086I
u = −0.434655 − 0.677483I
a = −0.76935 + 1.40090I
b = 0.761684 + 0.558455I
−3.21792 + 0.04263I −6.95293 + 1.05086I
u = −0.559917 + 0.567694I
a = 0.51071 − 1.73794I
b = 0.802552 + 0.895254I
−4.08533 + 5.19997I −6.95238 − 6.01151I
u = −0.559917 − 0.567694I
a = 0.51071 + 1.73794I
b = 0.802552 − 0.895254I
−4.08533 − 5.19997I −6.95238 + 6.01151I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618882 + 0.492399I
a = −0.690536 − 0.334295I
b = −0.479177 − 0.103237I
−11.32390 − 2.09934I −11.90160 + 3.33110I
u = 0.618882 − 0.492399I
a = −0.690536 + 0.334295I
b = −0.479177 + 0.103237I
−11.32390 + 2.09934I −11.90160 − 3.33110I
u = 0.541988 + 0.545165I
a = 1.05538 + 1.71117I
b = 1.65899 − 0.67000I
−3.16680 − 2.60582I −8.31028 + 4.78950I
u = 0.541988 − 0.545165I
a = 1.05538 − 1.71117I
b = 1.65899 + 0.67000I
−3.16680 + 2.60582I −8.31028 − 4.78950I
u = −0.083204 + 0.762987I
a = −1.47229 − 0.86393I
b = −1.103520 − 0.489689I
3.00423 + 2.37974I 3.15751 − 4.54472I
u = −0.083204 − 0.762987I
a = −1.47229 + 0.86393I
b = −1.103520 + 0.489689I
3.00423 − 2.37974I 3.15751 + 4.54472I
u = −0.491902 + 0.580057I
a = −0.057631 + 1.353840I
b = −0.650246 − 0.287216I
0.50199 + 2.34229I −0.41176 − 4.10900I
u = −0.491902 − 0.580057I
a = −0.057631 − 1.353840I
b = −0.650246 + 0.287216I
0.50199 − 2.34229I −0.41176 + 4.10900I
u = 0.628694 + 0.357461I
a = −0.979961 − 0.460853I
b = 0.98668 + 1.61826I
−7.11475 + 7.31767I −9.69803 − 3.17801I
u = 0.628694 − 0.357461I
a = −0.979961 + 0.460853I
b = 0.98668 − 1.61826I
−7.11475 − 7.31767I −9.69803 + 3.17801I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.573521 + 0.396534I
a = −1.36868 + 0.72006I
b = 0.700052 − 0.740111I
−4.58689 − 1.30630I −8.69876 − 0.72424I
u = −0.573521 − 0.396534I
a = −1.36868 − 0.72006I
b = 0.700052 + 0.740111I
−4.58689 + 1.30630I −8.69876 + 0.72424I
u = 0.547064 + 0.428484I
a = −0.648498 + 0.866630I
b = 1.39053 + 0.84768I
−3.51243 − 1.15687I −10.08026 + 3.04392I
u = 0.547064 − 0.428484I
a = −0.648498 − 0.866630I
b = 1.39053 − 0.84768I
−3.51243 + 1.15687I −10.08026 − 3.04392I
u = 0.584549 + 0.365784I
a = 1.077210 − 0.038246I
b = −1.19511 − 1.33848I
−1.73587 + 3.32691I −6.23751 − 2.52670I
u = 0.584549 − 0.365784I
a = 1.077210 + 0.038246I
b = −1.19511 + 1.33848I
−1.73587 − 3.32691I −6.23751 + 2.52670I
u = 0.061107 + 0.669375I
a = 1.86977 + 0.92766I
b = 1.256900 − 0.293507I
−0.052808 − 1.067730I 0.522323 + 0.688585I
u = 0.061107 − 0.669375I
a = 1.86977 − 0.92766I
b = 1.256900 + 0.293507I
−0.052808 + 1.067730I 0.522323 − 0.688585I
u = 0.12768 + 1.41720I
a = −0.219186 + 0.941667I
b = 0.74968 + 1.50855I
−1.52933 + 4.66645I 0
u = 0.12768 − 1.41720I
a = −0.219186 − 0.941667I
b = 0.74968 − 1.50855I
−1.52933 − 4.66645I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.549260 + 0.140873I
a = −0.771450 − 0.335665I
b = 0.703771 + 0.825353I
−4.80826 + 3.38006I −10.74337 − 4.24525I
u = −0.549260 − 0.140873I
a = −0.771450 + 0.335665I
b = 0.703771 − 0.825353I
−4.80826 − 3.38006I −10.74337 + 4.24525I
u = −0.397665 + 0.374585I
a = 0.879636 − 0.312320I
b = −0.225733 + 0.046292I
−0.161016 + 0.965316I −1.89057 − 5.10054I
u = −0.397665 − 0.374585I
a = 0.879636 + 0.312320I
b = −0.225733 − 0.046292I
−0.161016 − 0.965316I −1.89057 + 5.10054I
u = 0.10989 + 1.46208I
a = 0.189540 − 1.294750I
b = −0.77445 − 1.43903I
4.08220 + 1.01313I 0
u = 0.10989 − 1.46208I
a = 0.189540 + 1.294750I
b = −0.77445 + 1.43903I
4.08220 − 1.01313I 0
u = −0.12743 + 1.47880I
a = −0.875843 + 0.046170I
b = 0.504534 − 0.571892I
1.48337 + 1.06097I 0
u = −0.12743 − 1.47880I
a = −0.875843 − 0.046170I
b = 0.504534 + 0.571892I
1.48337 − 1.06097I 0
u = 0.13453 + 1.49947I
a = 0.60759 + 1.59619I
b = 1.18048 + 1.16499I
2.82349 − 3.48654I 0
u = 0.13453 − 1.49947I
a = 0.60759 − 1.59619I
b = 1.18048 − 1.16499I
2.82349 + 3.48654I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.18092 + 1.50611I
a = −0.976025 − 0.476688I
b = −0.478652 − 0.330158I
−4.77664 − 4.96073I 0
u = 0.18092 − 1.50611I
a = −0.976025 + 0.476688I
b = −0.478652 + 0.330158I
−4.77664 + 4.96073I 0
u = −0.10169 + 1.52949I
a = 0.709783 − 0.051260I
b = 0.019341 + 0.265567I
6.41258 + 2.64230I 0
u = −0.10169 − 1.52949I
a = 0.709783 + 0.051260I
b = 0.019341 − 0.265567I
6.41258 − 2.64230I 0
u = 0.15680 + 1.54612I
a = 2.43920 + 0.89187I
b = 1.88545 − 0.56925I
3.81275 − 5.11770I 0
u = 0.15680 − 1.54612I
a = 2.43920 − 0.89187I
b = 1.88545 + 0.56925I
3.81275 + 5.11770I 0
u = −0.352512 + 0.264358I
a = 0.901458 − 0.344587I
b = −0.244395 − 0.205512I
−0.166994 + 0.943418I −4.05068 − 5.99237I
u = −0.352512 − 0.264358I
a = 0.901458 + 0.344587I
b = −0.244395 + 0.205512I
−0.166994 − 0.943418I −4.05068 + 5.99237I
u = −0.16566 + 1.55185I
a = 1.25278 − 0.66968I
b = 0.885611 + 1.024030I
2.98284 + 7.83055I 0
u = −0.16566 − 1.55185I
a = 1.25278 + 0.66968I
b = 0.885611 − 1.024030I
2.98284 − 7.83055I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14397 + 1.55990I
a = −0.870671 + 0.798870I
b = −0.814944 − 0.452840I
7.68999 + 4.65543I 0
u = −0.14397 − 1.55990I
a = −0.870671 − 0.798870I
b = −0.814944 + 0.452840I
7.68999 − 4.65543I 0
u = 0.16814 + 1.55987I
a = −2.69089 − 0.22163I
b = −1.62903 + 1.34795I
6.09318 − 9.92009I 0
u = 0.16814 − 1.55987I
a = −2.69089 + 0.22163I
b = −1.62903 − 1.34795I
6.09318 + 9.92009I 0
u = 0.17903 + 1.56537I
a = 2.62610 − 0.20365I
b = 1.23470 − 1.72300I
0.8595 − 14.2592I 0
u = 0.17903 − 1.56537I
a = 2.62610 + 0.20365I
b = 1.23470 + 1.72300I
0.8595 + 14.2592I 0
u = 0.00918 + 1.57703I
a = 2.49998 + 0.14603I
b = 1.54857 − 0.46319I
7.58996 − 1.27405I 0
u = 0.00918 − 1.57703I
a = 2.49998 − 0.14603I
b = 1.54857 + 0.46319I
7.58996 + 1.27405I 0
u = −0.12188 + 1.58644I
a = 0.279040 − 1.317600I
b = 0.885103 − 0.421580I
4.43394 + 1.99633I 0
u = −0.12188 − 1.58644I
a = 0.279040 + 1.317600I
b = 0.885103 + 0.421580I
4.43394 − 1.99633I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.01577 + 1.59335I
a = −2.24569 − 0.79934I
b = −1.46124 − 0.34322I
10.99110 + 2.69855I 0
u = −0.01577 − 1.59335I
a = −2.24569 + 0.79934I
b = −1.46124 + 0.34322I
10.99110 − 2.69855I 0
u = −0.03506 + 1.60818I
a = 2.00100 + 1.33814I
b = 1.26274 + 1.00587I
6.66598 + 6.54137I 0
u = −0.03506 − 1.60818I
a = 2.00100 − 1.33814I
b = 1.26274 − 1.00587I
6.66598 − 6.54137I 0
u = 0.266227
a = −2.91703
b = 0.922923
−2.12035 −4.69260
11
II. I
u
2
= hb − 1, u
3
+ u
2
+ a + 3u + 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
12
=
−u
u
3
+ u
a
3
=
u
2
+ 1
u
3
+ u
2
+ 2u + 1
a
7
=
−u
3
− u
2
− 3u − 1
1
a
2
=
u
2
+ 1
u
3
+ u
2
+ 2u + 1
a
8
=
−u
3
− u
2
− 3u − 1
1
a
10
=
u
u
a
1
=
u
2
+ 1
u
3
+ u
2
+ 2u + 1
a
9
=
−1
0
a
6
=
−u
3
− u
2
− 3u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
− 3u
2
− 10u − 8
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
4
c
3
, c
4
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
5
, c
6
(u − 1)
4
c
8
(u + 1)
4
c
9
u
4
− u
3
+ u
2
+ 1
c
10
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
c
12
u
4
+ u
3
+ u
2
+ 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
4
c
3
, c
4
, c
10
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
5
, c
6
, c
8
(y −1)
4
c
9
, c
12
y
4
+ y
3
+ 3y
2
+ 2y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 0.043315 − 1.227190I
b = 1.00000
−1.85594 + 1.41510I −4.47493 − 4.18840I
u = −0.395123 − 0.506844I
a = 0.043315 + 1.227190I
b = 1.00000
−1.85594 − 1.41510I −4.47493 + 4.18840I
u = −0.10488 + 1.55249I
a = 0.956685 − 0.641200I
b = 1.00000
5.14581 + 3.16396I −2.02507 − 3.47609I
u = −0.10488 − 1.55249I
a = 0.956685 + 0.641200I
b = 1.00000
5.14581 − 3.16396I −2.02507 + 3.47609I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
4
(u
65
+ 27u
64
+ ··· − 1984u − 256)
c
2
, c
7
u
4
(u
65
+ u
64
+ ··· + 24u + 16)
c
3
, c
4
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
65
− 2u
64
+ ··· + u − 1)
c
5
, c
6
((u − 1)
4
)(u
65
− 5u
64
+ ··· + u + 1)
c
8
((u + 1)
4
)(u
65
− 5u
64
+ ··· + u + 1)
c
9
(u
4
− u
3
+ u
2
+ 1)(u
65
+ 12u
64
+ ··· + 1405u + 131)
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
65
− 2u
64
+ ··· + u − 1)
c
12
(u
4
+ u
3
+ u
2
+ 1)(u
65
+ 12u
64
+ ··· + 1405u + 131)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
4
(y
65
+ 15y
64
+ ··· − 1257472y −65536)
c
2
, c
7
y
4
(y
65
+ 27y
64
+ ··· − 1984y −256)
c
3
, c
4
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
65
+ 72y
64
+ ··· + 5y −1)
c
5
, c
6
, c
8
((y −1)
4
)(y
65
− 57y
64
+ ··· − 33y −1)
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
65
+ 36y
64
+ ··· + 294081y −17161)
17
