12a
0052
(K12a
0052
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 11 12 1 6 8 9
Solving Sequence
8,11
12 9 1
4,7
3 2 6 10 5
c
11
c
8
c
12
c
7
c
3
c
1
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h1.62429 × 10
21
u
62
+ 5.48543 × 10
21
u
61
+ ··· + 1.76821 × 10
20
b − 1.09318 × 10
21
,
1.33327 × 10
20
u
62
+ 3.96132 × 10
20
u
61
+ ··· + 1.76821 × 10
20
a + 5.67190 × 10
20
, u
63
+ 5u
62
+ ··· − 8u − 1i
I
u
2
= h7a
2
u − 4a
2
− 9au + 61b − 21a + 46u − 35, a
3
+ a
2
u + a
2
− au + 6a + 5u + 2, u
2
− u − 1i
I
u
3
= hu
2
+ b + u − 2, u
2
+ a + u − 2, u
3
+ u
2
− 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 72 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
= h1.62 × 10
21
u
62
+ 5.49 × 10
21
u
61
+ · · · + 1.77 × 10
20
b − 1.09 × 10
21
, 1.33 ×
10
20
u
62
+3.96×10
20
u
61
+· · ·+1.77×10
20
a+5.67×10
20
, u
63
+5u
62
+· · ·−8u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−0.754022u
62
− 2.24030u
61
+ ··· − 21.4495u − 3.20771
−9.18606u
62
− 31.0226u
61
+ ··· + 46.9211u + 6.18243
a
7
=
u
u
a
3
=
−23.4953u
62
− 77.2375u
61
+ ··· + 77.1418u + 10.1702
−31.9273u
62
− 106.020u
61
+ ··· + 145.512u + 19.5604
a
2
=
−10.7523u
62
− 35.7397u
61
+ ··· + 57.9187u + 7.96787
−2.43432u
62
− 7.52201u
61
+ ··· + 4.03934u + 0.358011
a
6
=
24.0607u
62
+ 78.4765u
61
+ ··· − 116.822u − 16.4195
20.0955u
62
+ 65.6712u
61
+ ··· − 84.1906u − 11.8348
a
10
=
u
3
− 2u
u
5
− 3u
3
+ u
a
5
=
−13.0012u
62
− 42.0114u
61
+ ··· + 38.8447u + 4.43096
−21.3192u
62
− 70.2291u
61
+ ··· + 92.7241u + 12.0408
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
910404764759386405011
29470106428947974938
u
62
−
3960180972357089673875
29470106428947974938
u
61
+ ··· +
9246140476465749326853
29470106428947974938
u +
1190469186010506308511
29470106428947974938
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
63
+ 32u
62
+ ··· + 328u + 1
c
2
, c
4
u
63
− 6u
62
+ ··· + 12u + 1
c
3
, c
6
u
63
− 3u
62
+ ··· − 20u + 8
c
5
, c
10
u
63
− 2u
62
+ ··· − 224u − 64
c
7
, c
8
, c
9
c
11
, c
12
u
63
+ 5u
62
+ ··· − 8u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
63
+ 4y
62
+ ··· + 101996y −1
c
2
, c
4
y
63
− 32y
62
+ ··· + 328y −1
c
3
, c
6
y
63
+ 27y
62
+ ··· + 1872y −64
c
5
, c
10
y
63
− 40y
62
+ ··· + 160768y −4096
c
7
, c
8
, c
9
c
11
, c
12
y
63
− 85y
62
+ ··· − 52y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.028890 + 0.152703I
a = 0.905018 − 0.380951I
b = 0.0268490 + 0.0773734I
−4.77176 − 0.89078I 0
u = 1.028890 − 0.152703I
a = 0.905018 + 0.380951I
b = 0.0268490 − 0.0773734I
−4.77176 + 0.89078I 0
u = 0.999217 + 0.355880I
a = −0.49337 + 1.91103I
b = −1.42854 + 1.09253I
−2.32837 − 6.62955I 0
u = 0.999217 − 0.355880I
a = −0.49337 − 1.91103I
b = −1.42854 − 1.09253I
−2.32837 + 6.62955I 0
u = −0.896327 + 0.253744I
a = −0.749447 − 0.960775I
b = −1.295060 − 0.092213I
0.036006 + 1.082630I 0
u = −0.896327 − 0.253744I
a = −0.749447 + 0.960775I
b = −1.295060 + 0.092213I
0.036006 − 1.082630I 0
u = −1.055640 + 0.212243I
a = 0.668837 + 0.903910I
b = 1.341030 + 0.052198I
−1.79480 + 5.60016I 0
u = −1.055640 − 0.212243I
a = 0.668837 − 0.903910I
b = 1.341030 − 0.052198I
−1.79480 − 5.60016I 0
u = 1.044900 + 0.298813I
a = −0.829234 + 0.625125I
b = −0.059797 − 0.170377I
−6.88408 − 5.57625I 0
u = 1.044900 − 0.298813I
a = −0.829234 − 0.625125I
b = −0.059797 + 0.170377I
−6.88408 + 5.57625I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.063910 + 0.242269I
a = 0.77322 − 2.30918I
b = 1.43939 − 1.55305I
−7.53862 − 2.77757I 0
u = 1.063910 − 0.242269I
a = 0.77322 + 2.30918I
b = 1.43939 + 1.55305I
−7.53862 + 2.77757I 0
u = −0.886505 + 0.065475I
a = 0.866237 − 0.584689I
b = 1.55540 + 0.00724I
−3.50514 + 0.98775I 0
u = −0.886505 − 0.065475I
a = 0.866237 + 0.584689I
b = 1.55540 − 0.00724I
−3.50514 − 0.98775I 0
u = 1.048840 + 0.413504I
a = 0.60788 − 1.71015I
b = 1.60656 − 1.01476I
−5.14579 − 12.09490I 0
u = 1.048840 − 0.413504I
a = 0.60788 + 1.71015I
b = 1.60656 + 1.01476I
−5.14579 + 12.09490I 0
u = −0.559821 + 0.591306I
a = 0.608474 + 1.194700I
b = 0.988829 − 0.154158I
−2.11826 − 4.21827I −12.00000 + 0.I
u = −0.559821 − 0.591306I
a = 0.608474 − 1.194700I
b = 0.988829 + 0.154158I
−2.11826 + 4.21827I −12.00000 + 0.I
u = 0.801660 + 0.061483I
a = 0.08712 + 2.74634I
b = −0.240999 + 0.989685I
1.61353 − 3.07418I −21.7450 + 6.3725I
u = 0.801660 − 0.061483I
a = 0.08712 − 2.74634I
b = −0.240999 − 0.989685I
1.61353 + 3.07418I −21.7450 − 6.3725I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.626636 + 0.416140I
a = −0.724621 − 1.147740I
b = −1.018610 − 0.016299I
0.0003302 + 0.0283867I −12.00000 − 1.58557I
u = −0.626636 − 0.416140I
a = −0.724621 + 1.147740I
b = −1.018610 + 0.016299I
0.0003302 − 0.0283867I −12.00000 + 1.58557I
u = −0.240977 + 0.680886I
a = 0.303437 − 0.160247I
b = −1.41332 − 0.53392I
−1.15974 + 8.39259I −13.8921 − 7.9238I
u = −0.240977 − 0.680886I
a = 0.303437 + 0.160247I
b = −1.41332 + 0.53392I
−1.15974 − 8.39259I −13.8921 + 7.9238I
u = 1.309330 + 0.213784I
a = −0.511554 + 0.626098I
b = −0.244470 − 0.005554I
−8.22487 + 1.35085I 0
u = 1.309330 − 0.213784I
a = −0.511554 − 0.626098I
b = −0.244470 + 0.005554I
−8.22487 − 1.35085I 0
u = −0.181089 + 0.593484I
a = −0.476070 + 0.258477I
b = 1.33722 + 0.54379I
1.31779 + 3.40957I −9.77047 − 4.49596I
u = −0.181089 − 0.593484I
a = −0.476070 − 0.258477I
b = 1.33722 − 0.54379I
1.31779 − 3.40957I −9.77047 + 4.49596I
u = −0.275760 + 0.524784I
a = 0.59469 + 1.49476I
b = 0.752601 − 0.186307I
−2.78020 + 2.76904I −15.9302 − 5.0017I
u = −0.275760 − 0.524784I
a = 0.59469 − 1.49476I
b = 0.752601 + 0.186307I
−2.78020 − 2.76904I −15.9302 + 5.0017I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.362579 + 0.450678I
a = 0.062756 − 0.676284I
b = −1.43437 − 0.73425I
−3.12199 + 0.43767I −16.9567 − 5.0899I
u = −0.362579 − 0.450678I
a = 0.062756 + 0.676284I
b = −1.43437 + 0.73425I
−3.12199 − 0.43767I −16.9567 + 5.0899I
u = −0.546898
a = −3.12245
b = −3.56875
−2.45024 −97.9560
u = 0.127965 + 0.446646I
a = −1.274990 − 0.311863I
b = 1.060070 + 0.298515I
3.14642 + 1.35383I −6.03722 − 2.02193I
u = 0.127965 − 0.446646I
a = −1.274990 + 0.311863I
b = 1.060070 − 0.298515I
3.14642 − 1.35383I −6.03722 + 2.02193I
u = 0.300748 + 0.343620I
a = 1.44873 + 1.07671I
b = −0.909905 − 0.082810I
2.43021 − 3.64228I −6.61693 + 6.48458I
u = 0.300748 − 0.343620I
a = 1.44873 − 1.07671I
b = −0.909905 + 0.082810I
2.43021 + 3.64228I −6.61693 − 6.48458I
u = 1.55975 + 0.08766I
a = 0.524965 − 1.191820I
b = 0.600823 − 0.640135I
−7.43465 − 1.76582I 0
u = 1.55975 − 0.08766I
a = 0.524965 + 1.191820I
b = 0.600823 + 0.640135I
−7.43465 + 1.76582I 0
u = 1.61037
a = 3.62252
b = 3.92432
−10.1147 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.381909
a = −1.00049
b = −0.382079
−0.657964 −14.9870
u = −1.68795 + 0.01367I
a = 0.14081 + 2.60287I
b = 0.23802 + 1.66652I
−7.35150 + 3.34536I 0
u = −1.68795 − 0.01367I
a = 0.14081 − 2.60287I
b = 0.23802 − 1.66652I
−7.35150 − 3.34536I 0
u = 1.69104 + 0.05015I
a = 1.24766 − 1.01147I
b = 1.58417 − 0.51110I
−9.10013 − 2.17742I 0
u = 1.69104 − 0.05015I
a = 1.24766 + 1.01147I
b = 1.58417 + 0.51110I
−9.10013 + 2.17742I 0
u = 1.70113 + 0.01268I
a = −1.57689 − 0.79292I
b = −2.02027 − 0.39058I
−12.78410 − 1.26409I 0
u = 1.70113 − 0.01268I
a = −1.57689 + 0.79292I
b = −2.02027 + 0.39058I
−12.78410 + 1.26409I 0
u = −1.72001 + 0.09428I
a = 1.03143 + 2.22415I
b = 1.49512 + 1.52284I
−11.9480 + 8.4481I 0
u = −1.72001 − 0.09428I
a = 1.03143 − 2.22415I
b = 1.49512 − 1.52284I
−11.9480 − 8.4481I 0
u = −1.72905 + 0.04304I
a = −0.288124 − 0.148815I
b = 0.389884 + 0.095224I
−14.6550 + 1.7190I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.72905 − 0.04304I
a = −0.288124 + 0.148815I
b = 0.389884 − 0.095224I
−14.6550 − 1.7190I 0
u = −1.73268 + 0.07792I
a = 0.239402 + 0.232586I
b = −0.419372 − 0.168398I
−16.7785 + 7.1236I 0
u = −1.73268 − 0.07792I
a = 0.239402 − 0.232586I
b = −0.419372 + 0.168398I
−16.7785 − 7.1236I 0
u = −1.73373 + 0.11357I
a = −1.20066 − 2.06097I
b = −1.72265 − 1.40659I
−14.9724 + 14.2765I 0
u = −1.73373 − 0.11357I
a = −1.20066 + 2.06097I
b = −1.72265 + 1.40659I
−14.9724 − 14.2765I 0
u = −1.73663 + 0.06289I
a = −1.09099 − 2.68531I
b = −1.42955 − 2.04344I
−17.5514 + 4.0414I 0
u = −1.73663 − 0.06289I
a = −1.09099 + 2.68531I
b = −1.42955 + 2.04344I
−17.5514 − 4.0414I 0
u = 1.73782 + 0.05723I
a = −1.19559 + 0.80808I
b = −1.60415 + 0.24819I
−11.81270 − 6.72860I 0
u = 1.73782 − 0.05723I
a = −1.19559 − 0.80808I
b = −1.60415 − 0.24819I
−11.81270 + 6.72860I 0
u = −1.80049 + 0.02971I
a = 0.095999 + 0.111650I
b = −0.178594 − 0.166882I
−19.6506 − 0.4097I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.80049 − 0.02971I
a = 0.095999 − 0.111650I
b = −0.178594 + 0.166882I
−19.6506 + 0.4097I 0
u = −0.030116 + 0.163245I
a = −1.54490 − 3.56904I
b = −0.483050 + 0.114818I
−0.977525 − 0.103718I −10.13328 − 1.14919I
u = −0.030116 − 0.163245I
a = −1.54490 + 3.56904I
b = −0.483050 − 0.114818I
−0.977525 + 0.103718I −10.13328 + 1.14919I
11
II. I
u
2
=
h7a
2
u−4a
2
−9au+61b−21a+46u−35, a
3
+a
2
u+a
2
−au+6a+5u+2, u
2
−u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u + 1
a
9
=
−u
−u − 1
a
1
=
−u
−u
a
4
=
a
−0.114754a
2
u + 0.147541au + ··· + 0.344262a + 0.573770
a
7
=
u
u
a
3
=
−0.163934a
2
u − 0.360656au + ··· + 0.491803a − 0.180328
−0.278689a
2
u − 0.213115au + ··· − 0.163934a + 0.393443
a
2
=
0.0163934a
2
u − 0.163934au + ··· − 0.0491803a − 0.0819672
−0.278689a
2
u − 0.213115au + ··· − 0.163934a + 0.393443
a
6
=
0.295082a
2
u + 0.0491803au + ··· + 0.114754a − 0.475410
0
a
10
=
1
0
a
5
=
0.295082a
2
u + 0.0491803au + ··· + 0.114754a − 0.475410
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =
93
61
a
2
u −
27
61
a
2
+
46
61
au +
26
61
a +
341
61
u −
831
61
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
10
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
9
(u
2
+ u − 1)
3
c
11
, c
12
(u
2
− u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
10
y
6
c
7
, c
8
, c
9
c
11
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.162553
b = 1.08457
−2.10041 −17.1210
u = −0.618034
a = −0.27226 + 2.57535I
b = 0.075747 + 0.460350I
2.03717 + 2.82812I −7.98462 + 1.83947I
u = −0.618034
a = −0.27226 − 2.57535I
b = 0.075747 − 0.460350I
2.03717 − 2.82812I −7.98462 − 1.83947I
u = 1.61803
a = −0.06538 + 2.01307I
b = −0.198308 + 1.205210I
−5.85852 − 2.82812I −12.87990 + 2.78145I
u = 1.61803
a = −0.06538 − 2.01307I
b = −0.198308 − 1.205210I
−5.85852 + 2.82812I −12.87990 − 2.78145I
u = 1.61803
a = −2.48727
b = −2.83945
−9.99610 3.85000
15
III. I
u
3
= hu
2
+ b + u − 2, u
2
+ a + u − 2, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
u
2
− u − 1
a
1
=
−u
2
+ 1
−u
2
+ u + 1
a
4
=
−u
2
− u + 2
−u
2
− u + 2
a
7
=
u
u
a
3
=
−u
2
− u + 2
−u
2
− u + 2
a
2
=
−2u
2
− u + 3
−2u
2
+ 3
a
6
=
u
u
a
10
=
−u
2
+ 1
−u
2
a
5
=
u
2
− 1
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
− 7u + 2
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
c
9
u
3
− u
2
− 2u + 1
c
10
, c
11
, c
12
u
3
+ u
2
− 2u − 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −0.801938
b = −0.801938
−7.98968 −19.1690
u = −0.445042
a = 2.24698
b = 2.24698
−2.34991 3.53080
u = −1.80194
a = 0.554958
b = 0.554958
−19.2692 −11.3620
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
3
− u
2
+ 2u − 1)
2
(u
63
+ 32u
62
+ ··· + 328u + 1)
c
2
((u − 1)
3
)(u
3
+ u
2
− 1)
2
(u
63
− 6u
62
+ ··· + 12u + 1)
c
3
u
3
(u
3
− u
2
+ 2u − 1)
2
(u
63
− 3u
62
+ ··· − 20u + 8)
c
4
((u + 1)
3
)(u
3
− u
2
+ 1)
2
(u
63
− 6u
62
+ ··· + 12u + 1)
c
5
u
6
(u
3
− u
2
− 2u + 1)(u
63
− 2u
62
+ ··· − 224u − 64)
c
6
u
3
(u
3
+ u
2
+ 2u + 1)
2
(u
63
− 3u
62
+ ··· − 20u + 8)
c
7
, c
8
, c
9
((u
2
+ u − 1)
3
)(u
3
− u
2
− 2u + 1)(u
63
+ 5u
62
+ ··· − 8u − 1)
c
10
u
6
(u
3
+ u
2
− 2u − 1)(u
63
− 2u
62
+ ··· − 224u − 64)
c
11
, c
12
((u
2
− u − 1)
3
)(u
3
+ u
2
− 2u − 1)(u
63
+ 5u
62
+ ··· − 8u − 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
3
+ 3y
2
+ 2y −1)
2
(y
63
+ 4y
62
+ ··· + 101996y −1)
c
2
, c
4
((y −1)
3
)(y
3
− y
2
+ 2y −1)
2
(y
63
− 32y
62
+ ··· + 328y −1)
c
3
, c
6
y
3
(y
3
+ 3y
2
+ 2y −1)
2
(y
63
+ 27y
62
+ ··· + 1872y −64)
c
5
, c
10
y
6
(y
3
− 5y
2
+ 6y −1)(y
63
− 40y
62
+ ··· + 160768y −4096)
c
7
, c
8
, c
9
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
3
− 5y
2
+ 6y −1)(y
63
− 85y
62
+ ··· − 52y −1)
21
