12a
0409
(K12a
0409
)
A knot diagram
1
Linearized knot diagam
3 6 9 12 2 10 11 5 1 7 8 4
Solving Sequence
6,10
7 11 8
3,12
2 1 5 4 9
c
6
c
10
c
7
c
11
c
2
c
1
c
5
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h1.36563 × 10
102
u
76
− 6.07669 × 10
102
u
75
+ ··· + 1.56636 × 10
101
b − 2.53112 × 10
102
,
1.74520 × 10
103
u
76
− 7.65702 × 10
103
u
75
+ ··· + 7.83180 × 10
100
a − 2.71806 × 10
103
, u
77
− 5u
76
+ ··· − 21u + 1i
I
u
2
= h−a
2
+ 2b, a
4
− 2a
3
+ 2a
2
− 4a + 4, u − 1i
* 2 irreducible components of dim
C
= 0, with total 81 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
= h1.37 × 10
102
u
76
− 6.08 × 10
102
u
75
+ · · · + 1.57 × 10
101
b − 2.53 ×
10
102
, 1.75 × 10
103
u
76
− 7.66 × 10
103
u
75
+ · · · + 7.83 × 10
100
a − 2.72 ×
10
103
, u
77
− 5u
76
+ · · · − 21u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
u
2
a
11
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−222.835u
76
+ 977.683u
75
+ ··· − 6806.53u + 347.055
−8.71850u
76
+ 38.7950u
75
+ ··· − 311.143u + 16.1592
a
12
=
u
3
− 2u
u
5
− 3u
3
+ u
a
2
=
−231.553u
76
+ 1016.48u
75
+ ··· − 7117.67u + 363.214
−8.71850u
76
+ 38.7950u
75
+ ··· − 311.143u + 16.1592
a
1
=
−347.436u
76
+ 1528.95u
75
+ ··· − 11419.1u + 612.993
8.56092u
76
− 38.6252u
75
+ ··· + 321.987u − 18.8227
a
5
=
86.7977u
76
− 385.491u
75
+ ··· + 3342.73u − 188.896
15.5783u
76
− 67.5759u
75
+ ··· + 440.398u − 21.5051
a
4
=
67.6379u
76
− 301.817u
75
+ ··· + 2732.19u − 156.914
16.0006u
76
− 69.5975u
75
+ ··· + 461.286u − 22.7965
a
9
=
−1492.68u
76
+ 6577.28u
75
+ ··· − 48928.0u + 2566.44
−157.740u
76
+ 692.289u
75
+ ··· − 4990.09u + 254.857
(ii) Obstruction class = −1
(iii) Cusp Shapes = 213.329u
76
− 938.680u
75
+ ··· + 6844.56u − 366.282
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
77
+ 35u
76
+ ··· + 129u + 4
c
2
, c
5
u
77
+ 5u
76
+ ··· + 23u + 2
c
3
u
77
− u
76
+ ··· + 11u − 1
c
4
, c
12
u
77
+ 5u
76
+ ··· + 13u + 1
c
6
, c
7
, c
10
c
11
u
77
+ 5u
76
+ ··· − 21u − 1
c
8
u
77
+ 19u
76
+ ··· − 1017075u − 2694247
c
9
u
77
+ 11u
76
+ ··· − 34199u − 5203
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
77
+ 17y
76
+ ··· + 16241y −16
c
2
, c
5
y
77
− 35y
76
+ ··· + 129y −4
c
3
y
77
− y
76
+ ··· + 113y −1
c
4
, c
12
y
77
+ 63y
76
+ ··· + y −1
c
6
, c
7
, c
10
c
11
y
77
− 93y
76
+ ··· + 161y −1
c
8
y
77
− 315y
76
+ ··· + 223681910731807y −7258966897009
c
9
y
77
+ 241y
76
+ ··· + 1048539415y −27071209
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.871635 + 0.521957I
a = 0.806284 + 0.772806I
b = −0.423802 − 0.855380I
−3.37937 + 7.74744I 0
u = −0.871635 − 0.521957I
a = 0.806284 − 0.772806I
b = −0.423802 + 0.855380I
−3.37937 − 7.74744I 0
u = 0.849846 + 0.434640I
a = −0.359051 + 1.128330I
b = −0.074629 − 0.367441I
−3.72246 − 0.44807I 0
u = 0.849846 − 0.434640I
a = −0.359051 − 1.128330I
b = −0.074629 + 0.367441I
−3.72246 + 0.44807I 0
u = −1.008120 + 0.363672I
a = 0.329054 + 0.212618I
b = 1.196760 + 0.069492I
−9.11045 + 5.28746I 0
u = −1.008120 − 0.363672I
a = 0.329054 − 0.212618I
b = 1.196760 − 0.069492I
−9.11045 − 5.28746I 0
u = 0.788462 + 0.754022I
a = −0.04992 − 1.69456I
b = 1.045250 + 0.397646I
−6.31504 − 3.48175I 0
u = 0.788462 − 0.754022I
a = −0.04992 + 1.69456I
b = 1.045250 − 0.397646I
−6.31504 + 3.48175I 0
u = 0.889138 + 0.185542I
a = 0.481935 − 0.278927I
b = 0.910187 − 0.481008I
−1.73035 + 1.95114I 0
u = 0.889138 − 0.185542I
a = 0.481935 + 0.278927I
b = 0.910187 + 0.481008I
−1.73035 − 1.95114I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.924156 + 0.594199I
a = −0.33784 − 1.89395I
b = −1.124630 + 0.628045I
−5.4921 + 13.2430I 0
u = −0.924156 − 0.594199I
a = −0.33784 + 1.89395I
b = −1.124630 − 0.628045I
−5.4921 − 13.2430I 0
u = −0.788058 + 0.431200I
a = 0.57941 + 1.85354I
b = 1.125230 − 0.619677I
−0.72268 + 8.28581I 0
u = −0.788058 − 0.431200I
a = 0.57941 − 1.85354I
b = 1.125230 + 0.619677I
−0.72268 − 8.28581I 0
u = −0.003925 + 0.882924I
a = 0.80702 + 1.44156I
b = −1.073580 − 0.572372I
−2.67980 − 8.37636I 0
u = −0.003925 − 0.882924I
a = 0.80702 − 1.44156I
b = −1.073580 + 0.572372I
−2.67980 + 8.37636I 0
u = 0.297530 + 0.762269I
a = −0.783831 + 0.708395I
b = 1.009260 − 0.219949I
−4.99404 − 1.67489I 0
u = 0.297530 − 0.762269I
a = −0.783831 − 0.708395I
b = 1.009260 + 0.219949I
−4.99404 + 1.67489I 0
u = −0.817383
a = −0.614795
b = −1.27028
−4.46353 0
u = 1.143130 + 0.308998I
a = −0.393897 + 1.339210I
b = −0.667233 − 0.206047I
−4.00145 − 0.28027I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.143130 − 0.308998I
a = −0.393897 − 1.339210I
b = −0.667233 + 0.206047I
−4.00145 + 0.28027I 0
u = 1.226660 + 0.068946I
a = 0.424240 − 0.795406I
b = 0.793103 + 0.587761I
−1.57293 − 2.28596I 0
u = 1.226660 − 0.068946I
a = 0.424240 + 0.795406I
b = 0.793103 − 0.587761I
−1.57293 + 2.28596I 0
u = −0.765979 + 0.060037I
a = −0.505222 − 1.217440I
b = −1.165500 + 0.677513I
−4.10595 + 2.48021I 0
u = −0.765979 − 0.060037I
a = −0.505222 + 1.217440I
b = −1.165500 − 0.677513I
−4.10595 − 2.48021I 0
u = 1.065830 + 0.645154I
a = 0.282851 − 0.390245I
b = −1.048050 + 0.473536I
−5.82204 + 3.18193I 0
u = 1.065830 − 0.645154I
a = 0.282851 + 0.390245I
b = −1.048050 − 0.473536I
−5.82204 − 3.18193I 0
u = −0.654081 + 0.366173I
a = −0.800157 − 0.489010I
b = 0.384980 + 0.843441I
1.46897 + 2.86772I 0
u = −0.654081 − 0.366173I
a = −0.800157 + 0.489010I
b = 0.384980 − 0.843441I
1.46897 − 2.86772I 0
u = −0.028223 + 0.746166I
a = 0.40028 − 1.68284I
b = −0.431130 + 0.682340I
−0.80801 − 3.51202I −8.00000 + 0.I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.028223 − 0.746166I
a = 0.40028 + 1.68284I
b = −0.431130 − 0.682340I
−0.80801 + 3.51202I −8.00000 + 0.I
u = −0.701002 + 0.185836I
a = 0.127831 − 0.967471I
b = −0.614469 + 0.908672I
−2.32786 + 3.73049I −13.7491 − 12.8450I
u = −0.701002 − 0.185836I
a = 0.127831 + 0.967471I
b = −0.614469 − 0.908672I
−2.32786 − 3.73049I −13.7491 + 12.8450I
u = 0.618969 + 0.344051I
a = −0.51111 + 2.87161I
b = −0.944612 − 0.463574I
−1.67002 − 3.05655I −8.00000 + 6.69690I
u = 0.618969 − 0.344051I
a = −0.51111 − 2.87161I
b = −0.944612 + 0.463574I
−1.67002 + 3.05655I −8.00000 − 6.69690I
u = −0.083013 + 0.598652I
a = −1.12653 − 1.23667I
b = 1.022510 + 0.594189I
1.39951 − 4.78176I −4.95226 + 4.64563I
u = −0.083013 − 0.598652I
a = −1.12653 + 1.23667I
b = 1.022510 − 0.594189I
1.39951 + 4.78176I −4.95226 − 4.64563I
u = 0.601180 + 0.002263I
a = −14.1881 + 11.7877I
b = 0.851370 + 0.493083I
−2.60411 − 2.03169I −136.2940 − 14.6490I
u = 0.601180 − 0.002263I
a = −14.1881 − 11.7877I
b = 0.851370 − 0.493083I
−2.60411 + 2.03169I −136.2940 + 14.6490I
u = −0.212039 + 0.506396I
a = −0.10650 + 1.80456I
b = 0.564116 − 0.692422I
2.77376 + 0.18726I −1.78780 − 2.14448I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.212039 − 0.506396I
a = −0.10650 − 1.80456I
b = 0.564116 + 0.692422I
2.77376 − 0.18726I −1.78780 + 2.14448I
u = 0.348984 + 0.348224I
a = 1.94963 − 0.62772I
b = −0.814043 + 0.325199I
−0.965555 + 0.405754I −7.83651 + 1.83330I
u = 0.348984 − 0.348224I
a = 1.94963 + 0.62772I
b = −0.814043 − 0.325199I
−0.965555 − 0.405754I −7.83651 − 1.83330I
u = 0.479840
a = 0.890816
b = −0.179116
−0.737088 −13.2280
u = −1.57486 + 0.04270I
a = 0.597648 + 0.370331I
b = −0.420518 − 0.504083I
−7.73145 + 0.41162I 0
u = −1.57486 − 0.04270I
a = 0.597648 − 0.370331I
b = −0.420518 + 0.504083I
−7.73145 − 0.41162I 0
u = −1.60958 + 0.01197I
a = −2.62680 − 0.11024I
b = 0.753976 − 0.522222I
−10.36160 + 2.12144I 0
u = −1.60958 − 0.01197I
a = −2.62680 + 0.11024I
b = 0.753976 + 0.522222I
−10.36160 − 2.12144I 0
u = 1.61089 + 0.07581I
a = −0.267958 + 0.344213I
b = 0.304142 − 1.035400I
−6.33159 − 4.36200I 0
u = 1.61089 − 0.07581I
a = −0.267958 − 0.344213I
b = 0.304142 + 1.035400I
−6.33159 + 4.36200I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61710 + 0.08976I
a = −1.02416 − 1.56228I
b = −1.037440 + 0.506606I
−9.47283 + 4.60615I 0
u = −1.61710 − 0.08976I
a = −1.02416 + 1.56228I
b = −1.037440 − 0.506606I
−9.47283 − 4.60615I 0
u = 1.63407 + 0.03802I
a = 0.003139 + 0.666872I
b = −0.669565 − 1.103280I
−10.52050 − 4.48505I 0
u = 1.63407 − 0.03802I
a = 0.003139 − 0.666872I
b = −0.669565 + 1.103280I
−10.52050 + 4.48505I 0
u = −1.64816 + 0.01239I
a = 1.51628 + 0.07749I
b = 0.960623 + 0.388315I
−10.42720 − 1.45912I 0
u = −1.64816 − 0.01239I
a = 1.51628 − 0.07749I
b = 0.960623 − 0.388315I
−10.42720 + 1.45912I 0
u = 1.64827 + 0.01455I
a = −0.559053 + 0.775844I
b = −1.30014 − 0.75889I
−12.59660 − 2.75325I 0
u = 1.64827 − 0.01455I
a = −0.559053 − 0.775844I
b = −1.30014 + 0.75889I
−12.59660 + 2.75325I 0
u = 1.64464 + 0.11365I
a = 0.85856 − 1.15813I
b = 1.210340 + 0.641926I
−9.11064 − 10.32360I 0
u = 1.64464 − 0.11365I
a = 0.85856 + 1.15813I
b = 1.210340 − 0.641926I
−9.11064 + 10.32360I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.095003 + 0.332687I
a = 2.10515 − 0.52416I
b = −0.435064 − 0.531419I
−0.66814 − 1.87825I −4.74322 + 2.54988I
u = −0.095003 − 0.332687I
a = 2.10515 + 0.52416I
b = −0.435064 + 0.531419I
−0.66814 + 1.87825I −4.74322 − 2.54988I
u = 1.65568
a = −0.782927
b = −1.47869
−13.1360 0
u = −1.65750 + 0.14801I
a = −0.335612 − 0.692340I
b = 0.230603 + 0.666767I
−12.26620 + 2.82177I 0
u = −1.65750 − 0.14801I
a = −0.335612 + 0.692340I
b = 0.230603 − 0.666767I
−12.26620 − 2.82177I 0
u = 1.67002 + 0.14858I
a = 0.473642 − 0.470437I
b = −0.414424 + 0.964985I
−12.1217 − 10.3513I 0
u = 1.67002 − 0.14858I
a = 0.473642 + 0.470437I
b = −0.414424 − 0.964985I
−12.1217 + 10.3513I 0
u = 1.68810 + 0.17252I
a = −0.73378 + 1.40207I
b = −1.170620 − 0.661665I
−14.4500 − 16.2652I 0
u = 1.68810 − 0.17252I
a = −0.73378 − 1.40207I
b = −1.170620 + 0.661665I
−14.4500 + 16.2652I 0
u = −1.68571 + 0.22627I
a = 0.53610 + 1.31327I
b = 1.111390 − 0.508532I
−14.7554 + 7.3067I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.68571 − 0.22627I
a = 0.53610 − 1.31327I
b = 1.111390 + 0.508532I
−14.7554 − 7.3067I 0
u = 1.70007 + 0.09625I
a = 0.737615 − 0.162232I
b = 1.337770 − 0.063513I
−18.5664 − 7.1193I 0
u = 1.70007 − 0.09625I
a = 0.737615 + 0.162232I
b = 1.337770 + 0.063513I
−18.5664 + 7.1193I 0
u = −1.75832 + 0.10945I
a = −0.528353 + 0.351806I
b = −1.099490 − 0.317079I
−16.0277 − 0.1764I 0
u = −1.75832 − 0.10945I
a = −0.528353 − 0.351806I
b = −1.099490 + 0.317079I
−16.0277 + 0.1764I 0
u = 0.1016190 + 0.0298609I
a = 9.47466 + 7.93380I
b = −0.918640 − 0.537767I
−1.80001 − 2.05748I −8.74387 + 2.56472I
u = 0.1016190 − 0.0298609I
a = 9.47466 − 7.93380I
b = −0.918640 + 0.537767I
−1.80001 + 2.05748I −8.74387 − 2.56472I
12
II. I
u
2
= h−a
2
+ 2b, a
4
− 2a
3
+ 2a
2
− 4a + 4, u − 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
1
a
7
=
1
1
a
11
=
−1
0
a
8
=
0
1
a
3
=
a
1
2
a
2
a
12
=
−1
−1
a
2
=
1
2
a
2
+ a
1
2
a
2
a
1
=
−
1
2
a
3
+
1
2
a
2
+ 2
−
1
2
a
3
+
1
2
a
2
+ 1
a
5
=
−a
3
+
1
2
a
2
− a + 2
−
1
2
a
3
+
1
2
a
2
− a + 1
a
4
=
−
3
2
a
3
+
1
2
a
2
− a + 3
−a
3
+
1
2
a
2
− a + 2
a
9
=
−
3
2
a
3
+
3
2
a
2
− a + 5
−a
3
+ a
2
− a + 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2a
3
− 2a
2
+ 4a − 20
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
2
c
2
, c
5
u
4
− u
2
+ 1
c
3
, c
4
, c
12
(u
2
+ 1)
2
c
6
, c
7
(u − 1)
4
c
8
u
4
− 2u
3
+ 5u
2
− 4u + 1
c
9
u
4
+ 4u
3
+ 5u
2
+ 2u + 1
c
10
, c
11
(u + 1)
4
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
2
c
2
, c
5
(y
2
− y + 1)
2
c
3
, c
4
, c
12
(y + 1)
4
c
6
, c
7
, c
10
c
11
(y −1)
4
c
8
y
4
+ 6y
3
+ 11y
2
− 6y + 1
c
9
y
4
− 6y
3
+ 11y
2
+ 6y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.36603 + 0.36603I
b = 0.866025 + 0.500000I
−3.28987 − 2.02988I −14.0000 + 3.4641I
u = 1.00000
a = 1.36603 − 0.36603I
b = 0.866025 − 0.500000I
−3.28987 + 2.02988I −14.0000 − 3.4641I
u = 1.00000
a = −0.36603 + 1.36603I
b = −0.866025 − 0.500000I
−3.28987 − 2.02988I −14.0000 + 3.4641I
u = 1.00000
a = −0.36603 − 1.36603I
b = −0.866025 + 0.500000I
−3.28987 + 2.02988I −14.0000 − 3.4641I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
77
+ 35u
76
+ ··· + 129u + 4)
c
2
, c
5
(u
4
− u
2
+ 1)(u
77
+ 5u
76
+ ··· + 23u + 2)
c
3
((u
2
+ 1)
2
)(u
77
− u
76
+ ··· + 11u − 1)
c
4
, c
12
((u
2
+ 1)
2
)(u
77
+ 5u
76
+ ··· + 13u + 1)
c
6
, c
7
((u − 1)
4
)(u
77
+ 5u
76
+ ··· − 21u − 1)
c
8
(u
4
− 2u
3
+ 5u
2
− 4u + 1)(u
77
+ 19u
76
+ ··· − 1017075u − 2694247)
c
9
(u
4
+ 4u
3
+ 5u
2
+ 2u + 1)(u
77
+ 11u
76
+ ··· − 34199u − 5203)
c
10
, c
11
((u + 1)
4
)(u
77
+ 5u
76
+ ··· − 21u − 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
2
)(y
77
+ 17y
76
+ ··· + 16241y −16)
c
2
, c
5
((y
2
− y + 1)
2
)(y
77
− 35y
76
+ ··· + 129y −4)
c
3
((y + 1)
4
)(y
77
− y
76
+ ··· + 113y −1)
c
4
, c
12
((y + 1)
4
)(y
77
+ 63y
76
+ ··· + y −1)
c
6
, c
7
, c
10
c
11
((y −1)
4
)(y
77
− 93y
76
+ ··· + 161y −1)
c
8
(y
4
+ 6y
3
+ 11y
2
− 6y + 1)
· (y
77
− 315y
76
+ ··· + 223681910731807y −7258966897009)
c
9
(y
4
− 6y
3
+ 11y
2
+ 6y + 1)
· (y
77
+ 241y
76
+ ··· + 1048539415y −27071209)
18
