12a
1285
(K12a
1285
)
A knot diagram
1
Linearized knot diagam
5 10 8 9 12 1 11 4 2 3 7 6
Solving Sequence
5,12
6 1 2
7,9
10 4 8 3 11
c
5
c
12
c
1
c
6
c
9
c
4
c
8
c
3
c
11
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
23
+ 2u
22
+ ··· + b − 1, −7u
23
+ 13u
22
+ ··· + 2a − 12, u
24
− 3u
23
+ ··· − 6u − 2i
I
u
2
= h−10u
13
a + 21u
13
+ ··· − 17a + 30, −2u
13
a + 2u
13
+ ··· − 2a + 2,
u
14
+ u
13
− 5u
12
− 4u
11
+ 10u
10
+ 5u
9
− 7u
8
+ 2u
7
− 4u
6
− 8u
5
+ 8u
4
+ 2u
3
− 2u
2
+ 3u − 1i
I
u
3
= hb + 1, 2u
3
− 3u
2
+ 3a − 3u + 3, u
4
− 3u
2
+ 3i
I
u
4
= hb − 1, −u
2
+ a + u + 1, u
4
− u
2
− 1i
I
v
1
= ha, b + 1, v + 1i
* 5 irreducible components of dim
C
= 0, with total 61 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
=
h−u
23
+2u
22
+· · ·+b−1, −7u
23
+13u
22
+· · ·+2a−12, u
24
−3u
23
+· · ·−6u−2i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
7
2
u
23
−
13
2
u
22
+ ··· +
37
2
u + 6
u
23
− 2u
22
+ ··· + 5u + 1
a
10
=
5
2
u
23
−
9
2
u
22
+ ··· +
25
2
u + 4
u
23
− 2u
22
+ ··· + 4u + 1
a
4
=
−
1
2
u
23
+
1
2
u
22
+ ··· −
17
2
u
2
−
7
2
u
u
23
− u
22
+ ··· + 4u + 1
a
8
=
−u
8
+ 3u
6
− 3u
4
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
3
=
−
1
2
u
23
+
1
2
u
22
+ ··· −
13
2
u − 1
2u
23
− 3u
22
+ ··· + 12u + 3
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
21
− 16u
19
− 6u
18
+ 54u
17
+ 42u
16
− 82u
15
− 120u
14
+ 4u
13
+ 148u
12
+ 172u
11
+
4u
10
− 204u
9
− 204u
8
− 20u
7
+ 146u
6
+ 168u
5
+ 66u
4
− 40u
3
− 76u
2
− 56u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u
24
+ 9u
23
+ ··· − 194u − 22
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
24
+ u
23
+ ··· − u + 1
c
5
, c
6
, c
12
u
24
− 3u
23
+ ··· − 6u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
y
24
+ 25y
23
+ ··· + 1744y + 484
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
24
− 33y
23
+ ··· − 3y + 1
c
5
, c
6
, c
12
y
24
− 19y
23
+ ··· + 32y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.067210 + 0.918173I
a = 2.61385 − 0.53592I
b = −1.64955 − 0.33368I
−18.9666 + 8.2466I 9.52974 − 3.63812I
u = −0.067210 − 0.918173I
a = 2.61385 + 0.53592I
b = −1.64955 + 0.33368I
−18.9666 − 8.2466I 9.52974 + 3.63812I
u = −0.810234 + 0.401414I
a = 0.839601 − 0.631574I
b = −1.60712 + 0.05636I
10.53670 − 0.39581I 6.64151 − 1.21019I
u = −0.810234 − 0.401414I
a = 0.839601 + 0.631574I
b = −1.60712 − 0.05636I
10.53670 + 0.39581I 6.64151 + 1.21019I
u = −0.005460 + 0.831838I
a = −1.165050 − 0.241677I
b = 0.554556 + 0.402781I
5.61187 + 1.45036I 4.65438 − 4.77575I
u = −0.005460 − 0.831838I
a = −1.165050 + 0.241677I
b = 0.554556 − 0.402781I
5.61187 − 1.45036I 4.65438 + 4.77575I
u = −1.21253
a = −0.256764
b = 0.409349
−2.69200 −0.296190
u = −0.310605 + 0.687272I
a = −1.79744 + 1.27189I
b = 1.58804 + 0.14110I
12.05890 + 4.45584I 8.65503 − 4.30738I
u = −0.310605 − 0.687272I
a = −1.79744 − 1.27189I
b = 1.58804 − 0.14110I
12.05890 − 4.45584I 8.65503 + 4.30738I
u = 1.267450 + 0.119306I
a = −0.465945 − 0.973738I
b = 0.187464 − 0.543825I
−4.27340 − 2.36049I −6.43774 + 5.77001I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.267450 − 0.119306I
a = −0.465945 + 0.973738I
b = 0.187464 + 0.543825I
−4.27340 + 2.36049I −6.43774 − 5.77001I
u = −1.227700 + 0.469098I
a = −1.142420 + 0.522004I
b = 1.66590 − 0.30646I
16.9366 − 3.3077I 6.61436 + 0.33127I
u = −1.227700 − 0.469098I
a = −1.142420 − 0.522004I
b = 1.66590 + 0.30646I
16.9366 + 3.3077I 6.61436 − 0.33127I
u = −1.264550 + 0.372340I
a = 0.346462 − 0.320746I
b = −0.563190 + 0.341593I
1.70636 + 2.87549I 0.74677 + 1.38068I
u = −1.264550 − 0.372340I
a = 0.346462 + 0.320746I
b = −0.563190 − 0.341593I
1.70636 − 2.87549I 0.74677 − 1.38068I
u = 1.274190 + 0.379172I
a = 0.955758 + 0.604843I
b = −0.541567 + 0.462644I
1.63700 − 5.80273I 0.52316 + 7.89295I
u = 1.274190 − 0.379172I
a = 0.955758 − 0.604843I
b = −0.541567 − 0.462644I
1.63700 + 5.80273I 0.52316 − 7.89295I
u = 1.378330 + 0.236400I
a = −0.10842 + 1.66407I
b = −1.52672 + 0.17912I
6.69086 − 7.69527I 3.83110 + 5.36935I
u = 1.378330 − 0.236400I
a = −0.10842 − 1.66407I
b = −1.52672 − 0.17912I
6.69086 + 7.69527I 3.83110 − 5.36935I
u = 1.333410 + 0.425756I
a = −1.31381 − 1.82736I
b = 1.62816 − 0.35030I
16.1301 − 13.0526I 5.84267 + 6.20915I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.333410 − 0.425756I
a = −1.31381 + 1.82736I
b = 1.62816 + 0.35030I
16.1301 + 13.0526I 5.84267 − 6.20915I
u = 1.40803
a = 1.10728
b = 1.49513
3.58125 2.27540
u = −0.165371 + 0.320976I
a = 0.812144 − 0.253890I
b = −0.188218 − 0.317776I
0.012470 + 0.742718I 0.40941 − 9.38538I
u = −0.165371 − 0.320976I
a = 0.812144 + 0.253890I
b = −0.188218 + 0.317776I
0.012470 − 0.742718I 0.40941 + 9.38538I
7
II. I
u
2
= h−10u
13
a + 21u
13
+ · · · − 17a + 30, −2u
13
a + 2u
13
+ · · · − 2a +
2, u
14
+ u
13
+ · · · + 3u − 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
a
0.526316au
13
− 1.10526u
13
+ ··· + 0.894737a − 1.57895
a
10
=
0.526316au
13
− 0.105263u
13
+ ··· + 1.89474a − 0.578947
0.368421au
13
− 1.47368u
13
+ ··· + 0.526316a − 1.10526
a
4
=
0.105263au
13
− 0.421053u
13
+ ··· + 1.57895a + 0.684211
−0.631579au
13
+ 0.526316u
13
+ ··· − 0.473684a + 1.89474
a
8
=
−u
8
+ 3u
6
− 3u
4
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
3
=
0.736842au
13
− 0.947368u
13
+ ··· + 2.05263a − 1.21053
0.210526au
13
+ 0.157895u
13
+ ··· + 0.157895a + 1.36842
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
12
+ 20u
10
− 4u
9
− 36u
8
+ 16u
7
+ 12u
6
− 20u
5
+ 36u
4
− 4u
3
− 28u
2
+ 20u − 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
(u
14
− 3u
13
+ ··· − 5u + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
28
+ u
27
+ ··· − 38u + 7
c
5
, c
6
, c
12
(u
14
+ u
13
+ ··· + 3u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
(y
14
+ 17y
13
+ ··· − y + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
28
− 25y
27
+ ··· − 3040y + 49
c
5
, c
6
, c
12
(y
14
− 11y
13
+ ··· − 5y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.021800 + 0.901952I
a = −1.047000 + 0.562730I
b = 0.653487 − 0.990965I
12.94110 − 3.26499I 8.09314 + 2.49004I
u = 0.021800 + 0.901952I
a = 3.03667 + 0.27891I
b = −1.57846 + 0.10746I
12.94110 − 3.26499I 8.09314 + 2.49004I
u = 0.021800 − 0.901952I
a = −1.047000 − 0.562730I
b = 0.653487 + 0.990965I
12.94110 + 3.26499I 8.09314 − 2.49004I
u = 0.021800 − 0.901952I
a = 3.03667 − 0.27891I
b = −1.57846 − 0.10746I
12.94110 + 3.26499I 8.09314 − 2.49004I
u = 1.126450 + 0.176078I
a = 0.285171 + 0.774418I
b = 0.882087 + 0.470065I
1.87700 − 0.85224I 4.40198 + 0.38712I
u = 1.126450 + 0.176078I
a = 1.55549 + 1.18730I
b = −1.287820 + 0.132216I
1.87700 − 0.85224I 4.40198 + 0.38712I
u = 1.126450 − 0.176078I
a = 0.285171 − 0.774418I
b = 0.882087 − 0.470065I
1.87700 + 0.85224I 4.40198 − 0.38712I
u = 1.126450 − 0.176078I
a = 1.55549 − 1.18730I
b = −1.287820 − 0.132216I
1.87700 + 0.85224I 4.40198 − 0.38712I
u = −1.28972
a = 0.697582
b = 1.06109
−2.27008 −4.70520
u = −1.28972
a = −1.30932
b = −0.136131
−2.27008 −4.70520
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.279790 + 0.223785I
a = 0.007778 + 1.298290I
b = 0.368198 + 0.626753I
0.31026 + 4.88256I 0.31401 − 6.44337I
u = −1.279790 + 0.223785I
a = 0.57163 − 1.69896I
b = −1.287520 − 0.156522I
0.31026 + 4.88256I 0.31401 − 6.44337I
u = −1.279790 − 0.223785I
a = 0.007778 − 1.298290I
b = 0.368198 − 0.626753I
0.31026 − 4.88256I 0.31401 + 6.44337I
u = −1.279790 − 0.223785I
a = 0.57163 + 1.69896I
b = −1.287520 + 0.156522I
0.31026 − 4.88256I 0.31401 + 6.44337I
u = 1.264560 + 0.437504I
a = −0.212178 + 0.241475I
b = −0.697903 − 0.968584I
9.09089 − 1.51934I 4.87778 + 0.64840I
u = 1.264560 + 0.437504I
a = −1.68041 − 0.87564I
b = 1.57724 + 0.07154I
9.09089 − 1.51934I 4.87778 + 0.64840I
u = 1.264560 − 0.437504I
a = −0.212178 − 0.241475I
b = −0.697903 + 0.968584I
9.09089 + 1.51934I 4.87778 − 0.64840I
u = 1.264560 − 0.437504I
a = −1.68041 + 0.87564I
b = 1.57724 − 0.07154I
9.09089 + 1.51934I 4.87778 − 0.64840I
u = −1.299190 + 0.426336I
a = 1.125940 − 0.741448I
b = −0.608008 − 1.000040I
8.82756 + 8.01486I 4.36796 − 5.37427I
u = −1.299190 + 0.426336I
a = −1.69482 + 1.50999I
b = 1.56993 + 0.13979I
8.82756 + 8.01486I 4.36796 − 5.37427I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.299190 − 0.426336I
a = 1.125940 + 0.741448I
b = −0.608008 + 1.000040I
8.82756 − 8.01486I 4.36796 + 5.37427I
u = −1.299190 − 0.426336I
a = −1.69482 − 1.50999I
b = 1.56993 − 0.13979I
8.82756 − 8.01486I 4.36796 + 5.37427I
u = 0.129663 + 0.583715I
a = 0.574640 + 0.645353I
b = −0.582162 + 0.557704I
4.64212 − 1.98638I 7.34408 + 5.08636I
u = 0.129663 + 0.583715I
a = −2.71168 − 0.96145I
b = 1.309070 − 0.039650I
4.64212 − 1.98638I 7.34408 + 5.08636I
u = 0.129663 − 0.583715I
a = 0.574640 − 0.645353I
b = −0.582162 − 0.557704I
4.64212 + 1.98638I 7.34408 − 5.08636I
u = 0.129663 − 0.583715I
a = −2.71168 + 0.96145I
b = 1.309070 + 0.039650I
4.64212 + 1.98638I 7.34408 − 5.08636I
u = 0.362713
a = 0.600452
b = −1.15455
2.55923 −2.09270
u = 0.362713
a = 2.38882
b = 0.593309
2.55923 −2.09270
13
III. I
u
3
= hb + 1, 2u
3
− 3u
2
+ 3a − 3u + 3, u
4
− 3u
2
+ 3i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
2
+ 3
a
9
=
−
2
3
u
3
+ u
2
+ u − 1
−1
a
10
=
1
3
u
3
+ u
2
− u − 1
−u
3
+ u − 1
a
4
=
2
3
u
3
− u
2
− u + 2
1
a
8
=
1
0
a
3
=
2
3
u
3
− u
2
− u + 1
1
a
11
=
u
3
− 2u
−u
3
+ 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
7
, c
11
u
4
+ 3u
2
+ 3
c
2
, c
8
(u − 1)
4
c
3
, c
4
, c
9
c
10
(u + 1)
4
c
5
, c
6
, c
12
u
4
− 3u
2
+ 3
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
(y
2
+ 3y + 3)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y −1)
4
c
5
, c
6
, c
12
(y
2
− 3y + 3)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.271230 + 0.340625I
a = 0.696660 + 0.132080I
b = −1.00000
3.28987 − 4.05977I 6.00000 + 3.46410I
u = 1.271230 − 0.340625I
a = 0.696660 − 0.132080I
b = −1.00000
3.28987 + 4.05977I 6.00000 − 3.46410I
u = −1.271230 + 0.340625I
a = 0.30334 − 1.59997I
b = −1.00000
3.28987 + 4.05977I 6.00000 − 3.46410I
u = −1.271230 − 0.340625I
a = 0.30334 + 1.59997I
b = −1.00000
3.28987 − 4.05977I 6.00000 + 3.46410I
17
IV. I
u
4
= hb − 1, −u
2
+ a + u + 1, u
4
− u
2
− 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
− 1
a
9
=
u
2
− u − 1
1
a
10
=
−u
3
+ u
2
+ u − 1
u
3
− u + 1
a
4
=
u
2
− u
1
a
8
=
−1
0
a
3
=
u
2
− u − 1
1
a
11
=
−u
3
+ 2u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 8
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u
4
+ u
2
− 1
c
2
, c
8
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u − 1)
4
c
5
, c
6
, c
12
u
4
− u
2
− 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
(y
2
+ y −1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y −1)
4
c
5
, c
6
, c
12
(y
2
− y −1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151I
a = −1.61803 − 0.78615I
b = 1.00000
7.23771 10.4720
u = − 0.786151I
a = −1.61803 + 0.78615I
b = 1.00000
7.23771 10.4720
u = 1.27202
a = −0.653986
b = 1.00000
−0.657974 1.52790
u = −1.27202
a = 1.89005
b = 1.00000
−0.657974 1.52790
21
V. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
−1
0
a
6
=
1
0
a
1
=
−1
0
a
2
=
−1
0
a
7
=
1
0
a
9
=
0
−1
a
10
=
−1
−1
a
4
=
1
1
a
8
=
1
0
a
3
=
0
1
a
11
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u
c
2
, c
8
u − 1
c
3
, c
4
, c
9
c
10
u + 1
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
y
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y − 1
24
(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
25
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
14
− 3u
13
+ ··· − 5u + 1)
2
· (u
24
+ 9u
23
+ ··· − 194u − 22)
c
2
, c
8
((u − 1)
5
)(u + 1)
4
(u
24
+ u
23
+ ··· − u + 1)(u
28
+ u
27
+ ··· − 38u + 7)
c
3
, c
4
, c
9
c
10
((u − 1)
4
)(u + 1)
5
(u
24
+ u
23
+ ··· − u + 1)(u
28
+ u
27
+ ··· − 38u + 7)
c
5
, c
6
, c
12
u(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)(u
14
+ u
13
+ ··· + 3u − 1)
2
· (u
24
− 3u
23
+ ··· − 6u − 2)
26
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
y(y
2
+ y − 1)
2
(y
2
+ 3y + 3)
2
(y
14
+ 17y
13
+ ··· − y + 1)
2
· (y
24
+ 25y
23
+ ··· + 1744y + 484)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
((y − 1)
9
)(y
24
− 33y
23
+ ··· − 3y + 1)(y
28
− 25y
27
+ ··· − 3040y + 49)
c
5
, c
6
, c
12
y(y
2
− 3y + 3)
2
(y
2
− y − 1)
2
(y
14
− 11y
13
+ ··· − 5y + 1)
2
· (y
24
− 19y
23
+ ··· + 32y + 4)
27
