12a
0679
(K12a
0679
)
A knot diagram
1
Linearized knot diagam
3 7 11 10 9 2 12 6 5 4 1 8
Solving Sequence
5,10
4
1,11
12 3 2 9 6 8 7
c
4
c
10
c
11
c
3
c
1
c
9
c
5
c
8
c
7
c
2
, c
6
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
15
+ 2u
14
+ ··· + b − 1, u
18
+ 3u
17
+ ··· + 2a − 4, u
19
+ 3u
18
+ ··· − 6u − 2i
I
u
2
= h−7u
12
a + 12u
12
+ ··· − 4a − 9, u
10
a + u
11
+ ··· + a − 1,
u
13
− u
12
+ 10u
11
− 9u
10
+ 37u
9
− 29u
8
+ 62u
7
− 40u
6
+ 46u
5
− 22u
4
+ 12u
3
− 3u
2
+ u − 1i
I
u
3
= hb + u − 2, 3a + 2u − 3, u
2
+ 3i
I
u
4
= hb − u, a − 1, u
2
+ 1i
I
v
1
= ha, b + 1, v + 1i
* 5 irreducible components of dim
C
= 0, with total 50 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
15
+2u
14
+· · ·+b−1, u
18
+3u
17
+· · ·+2a−4, u
19
+3u
18
+· · ·−6u−2i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
1
=
−
1
2
u
18
−
3
2
u
17
+ ··· +
7
2
u + 2
−u
15
− 2u
14
+ ··· + 2u + 1
a
11
=
−u
u
3
+ u
a
12
=
1
2
u
18
+
3
2
u
17
+ ··· −
9
2
u − 2
u
16
+ 2u
15
+ ··· − u − 1
a
3
=
u
2
+ 1
−u
4
− 2u
2
a
2
=
−
1
2
u
18
−
3
2
u
17
+ ··· +
7
2
u + 1
−u
16
− 2u
15
+ ··· + 2u + 1
a
9
=
u
u
a
6
=
u
2
+ 1
u
2
a
8
=
u
3
+ 2u
u
3
+ u
a
7
=
1
2
u
18
+
3
2
u
17
+ ··· −
7
2
u − 1
−u
14
− 2u
13
+ ··· + u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
18
+ 6u
17
+ 36u
16
+ 86u
15
+ 264u
14
+ 506u
13
+ 1022u
12
+ 1566u
11
+ 2248u
10
+
2704u
9
+ 2794u
8
+ 2526u
7
+ 1806u
6
+ 1106u
5
+ 464u
4
+ 122u
3
− 16u
2
− 28u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
19
+ 7u
18
+ ··· + 13u + 1
c
2
, c
6
, c
7
c
12
u
19
− u
18
+ ··· + u + 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
19
− 3u
18
+ ··· − 6u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
19
+ 17y
18
+ ··· + 37y −1
c
2
, c
6
, c
7
c
12
y
19
− 7y
18
+ ··· + 13y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
19
+ 27y
18
+ ··· + 24y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.273151 + 0.815941I
a = −0.390307 − 0.220948I
b = −0.608791 + 0.652934I
1.82620 − 1.77807I −6.92820 + 2.34039I
u = −0.273151 − 0.815941I
a = −0.390307 + 0.220948I
b = −0.608791 − 0.652934I
1.82620 + 1.77807I −6.92820 − 2.34039I
u = 0.077265 + 0.786380I
a = −0.374812 − 0.399284I
b = −0.419169 + 0.348987I
1.76915 − 1.42092I −5.09647 + 5.54292I
u = 0.077265 − 0.786380I
a = −0.374812 + 0.399284I
b = −0.419169 − 0.348987I
1.76915 + 1.42092I −5.09647 − 5.54292I
u = −0.246012 + 1.236810I
a = −1.197950 + 0.527520I
b = 0.428066 − 0.067299I
5.58686 + 10.56180I −7.44680 − 7.73425I
u = −0.246012 − 1.236810I
a = −1.197950 − 0.527520I
b = 0.428066 + 0.067299I
5.58686 − 10.56180I −7.44680 + 7.73425I
u = −0.487418 + 0.522631I
a = 0.993393 − 0.218685I
b = 1.221660 + 0.201552I
−0.06851 + 8.01058I −10.52385 − 9.28102I
u = −0.487418 − 0.522631I
a = 0.993393 + 0.218685I
b = 1.221660 − 0.201552I
−0.06851 − 8.01058I −10.52385 + 9.28102I
u = −0.076466 + 1.310960I
a = 0.792972 − 0.747592I
b = −0.069397 − 0.530421I
8.75335 − 0.83971I −3.29462 + 2.18721I
u = −0.076466 − 1.310960I
a = 0.792972 + 0.747592I
b = −0.069397 + 0.530421I
8.75335 + 0.83971I −3.29462 − 2.18721I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.550052 + 0.150551I
a = 0.41192 + 1.61654I
b = −0.276507 + 0.224858I
−1.18097 − 4.60134I −13.18408 + 3.99244I
u = −0.550052 − 0.150551I
a = 0.41192 − 1.61654I
b = −0.276507 − 0.224858I
−1.18097 + 4.60134I −13.18408 − 3.99244I
u = 0.308487
a = −0.641715
b = 0.245608
−0.592779 −16.7390
u = −0.01455 + 1.70648I
a = −0.003380 − 1.097050I
b = −0.23251 − 1.76936I
10.80000 − 1.26776I −5.99488 + 5.70666I
u = −0.01455 − 1.70648I
a = −0.003380 + 1.097050I
b = −0.23251 + 1.76936I
10.80000 + 1.26776I −5.99488 − 5.70666I
u = −0.06366 + 1.79454I
a = −2.87448 + 0.37712I
b = −5.52313 + 0.65502I
16.6584 + 11.9628I −6.98346 − 6.50856I
u = −0.06366 − 1.79454I
a = −2.87448 − 0.37712I
b = −5.52313 − 0.65502I
16.6584 − 11.9628I −6.98346 + 6.50856I
u = −0.02020 + 1.81069I
a = 1.96350 − 0.53377I
b = 3.85697 − 1.33560I
−19.1741 − 0.3767I −3.17814 + 1.98776I
u = −0.02020 − 1.81069I
a = 1.96350 + 0.53377I
b = 3.85697 + 1.33560I
−19.1741 + 0.3767I −3.17814 − 1.98776I
6
II.
I
u
2
= h−7u
12
a+12u
12
+· · ·−4a−9, u
10
a+u
11
+· · ·+a−1, u
13
−u
12
+· · ·+u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
1
=
a
0.189189au
12
− 0.324324u
12
+ ··· + 0.108108a + 0.243243
a
11
=
−u
u
3
+ u
a
12
=
−0.567568au
12
− 0.0270270u
12
+ ··· + 0.675676a + 0.270270
−0.0810811au
12
− 0.432432u
12
+ ··· − 0.189189a + 0.324324
a
3
=
u
2
+ 1
−u
4
− 2u
2
a
2
=
0.108108au
12
+ 0.243243u
12
+ ··· + 0.918919a − 0.432432
−0.0810811au
12
− 0.432432u
12
+ ··· − 0.189189a + 0.324324
a
9
=
u
u
a
6
=
u
2
+ 1
u
2
a
8
=
u
3
+ 2u
u
3
+ u
a
7
=
−0.108108au
12
− 0.243243u
12
+ ··· − 0.918919a + 0.432432
u
11
− 2u
10
+ ··· + au + u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
11
+ 4u
10
− 36u
9
+ 32u
8
− 116u
7
+ 88u
6
− 160u
5
+ 96u
4
− 88u
3
+ 36u
2
− 12u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
26
+ 13u
25
+ ··· + 385u + 64
c
2
, c
6
, c
7
c
12
u
26
− u
25
+ ··· − 7u + 8
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(u
13
+ u
12
+ ··· + u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
26
− y
25
+ ··· + 33151y + 4096
c
2
, c
6
, c
7
c
12
y
26
− 13y
25
+ ··· − 385y + 64
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y
13
+ 19y
12
+ ··· − 5y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.083038 + 1.167020I
a = −0.182812 + 0.329365I
b = −0.28373 + 1.50091I
1.55956 + 1.92579I −8.00122 − 3.82169I
u = −0.083038 + 1.167020I
a = −1.82248 − 0.57547I
b = 0.180439 − 0.168950I
1.55956 + 1.92579I −8.00122 − 3.82169I
u = −0.083038 − 1.167020I
a = −0.182812 − 0.329365I
b = −0.28373 − 1.50091I
1.55956 − 1.92579I −8.00122 + 3.82169I
u = −0.083038 − 1.167020I
a = −1.82248 + 0.57547I
b = 0.180439 + 0.168950I
1.55956 − 1.92579I −8.00122 + 3.82169I
u = 0.179330 + 1.269600I
a = −0.753270 − 0.865498I
b = −0.137976 − 0.536137I
7.63579 − 4.78537I −4.65540 + 3.59229I
u = 0.179330 + 1.269600I
a = 1.148590 + 0.147882I
b = −0.404842 − 0.185728I
7.63579 − 4.78537I −4.65540 + 3.59229I
u = 0.179330 − 1.269600I
a = −0.753270 + 0.865498I
b = −0.137976 + 0.536137I
7.63579 + 4.78537I −4.65540 − 3.59229I
u = 0.179330 − 1.269600I
a = 1.148590 − 0.147882I
b = −0.404842 + 0.185728I
7.63579 + 4.78537I −4.65540 − 3.59229I
u = 0.379427 + 0.590112I
a = −0.841955 − 0.244681I
b = −1.020470 + 0.268374I
1.59236 − 2.83275I −7.00318 + 5.17990I
u = 0.379427 + 0.590112I
a = 0.330450 − 0.407996I
b = 0.615126 + 0.383997I
1.59236 − 2.83275I −7.00318 + 5.17990I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.379427 − 0.590112I
a = −0.841955 + 0.244681I
b = −1.020470 − 0.268374I
1.59236 + 2.83275I −7.00318 − 5.17990I
u = 0.379427 − 0.590112I
a = 0.330450 + 0.407996I
b = 0.615126 − 0.383997I
1.59236 + 2.83275I −7.00318 − 5.17990I
u = 0.485085
a = −0.500820 + 1.088090I
b = 0.327558 + 0.106231I
−0.173769 −11.9170
u = 0.485085
a = −0.500820 − 1.088090I
b = 0.327558 − 0.106231I
−0.173769 −11.9170
u = −0.245118 + 0.346982I
a = 0.833919 + 0.205281I
b = 1.26442 + 0.90129I
−3.34890 + 0.88691I −13.3039 − 7.8258I
u = −0.245118 + 0.346982I
a = 0.69156 + 3.05215I
b = −0.067874 + 0.176233I
−3.34890 + 0.88691I −13.3039 − 7.8258I
u = −0.245118 − 0.346982I
a = 0.833919 − 0.205281I
b = 1.26442 − 0.90129I
−3.34890 − 0.88691I −13.3039 + 7.8258I
u = −0.245118 − 0.346982I
a = 0.69156 − 3.05215I
b = −0.067874 − 0.176233I
−3.34890 − 0.88691I −13.3039 + 7.8258I
u = −0.01838 + 1.78025I
a = 0.031468 − 1.234190I
b = −0.01017 − 1.63025I
12.36340 + 2.35177I −7.64300 − 2.76650I
u = −0.01838 + 1.78025I
a = −3.09176 − 0.79285I
b = −6.07832 − 1.71326I
12.36340 + 2.35177I −7.64300 − 2.76650I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.01838 − 1.78025I
a = 0.031468 + 1.234190I
b = −0.01017 + 1.63025I
12.36340 − 2.35177I −7.64300 + 2.76650I
u = −0.01838 − 1.78025I
a = −3.09176 + 0.79285I
b = −6.07832 + 1.71326I
12.36340 − 2.35177I −7.64300 + 2.76650I
u = 0.04523 + 1.80316I
a = −1.62778 − 0.56147I
b = −3.25509 − 1.42971I
18.9406 − 5.8171I −4.43476 + 2.75393I
u = 0.04523 + 1.80316I
a = 2.78489 + 0.06061I
b = 5.37092 − 0.01991I
18.9406 − 5.8171I −4.43476 + 2.75393I
u = 0.04523 − 1.80316I
a = −1.62778 + 0.56147I
b = −3.25509 + 1.42971I
18.9406 + 5.8171I −4.43476 − 2.75393I
u = 0.04523 − 1.80316I
a = 2.78489 − 0.06061I
b = 5.37092 + 0.01991I
18.9406 + 5.8171I −4.43476 − 2.75393I
12
III. I
u
3
= hb + u − 2, 3a + 2u − 3, u
2
+ 3i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
3
a
1
=
−
2
3
u + 1
−u + 2
a
11
=
−u
−2u
a
12
=
−
5
3
u + 1
−3u + 2
a
3
=
−2
−3
a
2
=
−
2
3
u − 1
−u − 1
a
9
=
u
u
a
6
=
−2
−3
a
8
=
−u
−2u
a
7
=
2
3
u − 1
u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 3
c
6
, c
12
(u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 3)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205I
a = 1.00000 − 1.15470I
b = 2.00000 − 1.73205I
9.86960 −12.0000
u = − 1.73205I
a = 1.00000 + 1.15470I
b = 2.00000 + 1.73205I
9.86960 −12.0000
16
IV. I
u
4
= hb − u, a − 1, u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
1
a
1
=
1
u
a
11
=
−u
0
a
12
=
−u + 1
u
a
3
=
0
1
a
2
=
1
u + 1
a
9
=
u
u
a
6
=
0
−1
a
8
=
u
0
a
7
=
1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
2
c
2
, c
7
(u + 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.00000
b = 1.000000I
0 −12.0000
u = − 1.000000I
a = 1.00000
b = − 1.000000I
0 −12.0000
20
V. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
−1
0
a
4
=
1
0
a
1
=
0
−1
a
11
=
−1
0
a
12
=
−1
−1
a
3
=
1
0
a
2
=
1
−1
a
9
=
−1
0
a
6
=
1
0
a
8
=
−1
0
a
7
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
12
u + 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
((u − 1)
5
)(u
19
+ 7u
18
+ ··· + 13u + 1)(u
26
+ 13u
25
+ ··· + 385u + 64)
c
2
, c
7
((u − 1)
3
)(u + 1)
2
(u
19
− u
18
+ ··· + u + 1)(u
26
− u
25
+ ··· − 7u + 8)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u(u
2
+ 1)(u
2
+ 3)(u
13
+ u
12
+ ··· + u + 1)
2
(u
19
− 3u
18
+ ··· − 6u + 2)
c
6
, c
12
((u − 1)
2
)(u + 1)
3
(u
19
− u
18
+ ··· + u + 1)(u
26
− u
25
+ ··· − 7u + 8)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
((y − 1)
5
)(y
19
+ 17y
18
+ ··· + 37y − 1)
· (y
26
− y
25
+ ··· + 33151y + 4096)
c
2
, c
6
, c
7
c
12
((y − 1)
5
)(y
19
− 7y
18
+ ··· + 13y − 1)(y
26
− 13y
25
+ ··· − 385y + 64)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y(y + 1)
2
(y + 3)
2
(y
13
+ 19y
12
+ ··· − 5y − 1)
2
· (y
19
+ 27y
18
+ ··· + 24y − 4)
26
