12a
0035
(K12a
0035
)
A knot diagram
1
Linearized knot diagam
3 5 6 10 2 11 12 1 4 9 8 7
Solving Sequence
7,12
8
1,3
2 9 11 6 4 5 10
c
7
c
12
c
1
c
8
c
11
c
6
c
3
c
5
c
10
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−6u
93
19u
92
+ ··· + 2b + 4, 3u
93
+ 15u
92
+ ··· + 2a 8, u
94
+ 3u
93
+ ··· 4u 1i
I
u
2
= h−u
2
a + b, u
2
a + a
2
+ u
2
+ a u + 2, u
3
u
2
+ 2u 1i
* 2 irreducible components of dim
C
= 0, with total 100 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−6u
93
19u
92
+· · ·+2b+4, 3u
93
+15u
92
+· · ·+2a8, u
94
+3u
93
+· · ·4u1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
u
u
a
3
=
3
2
u
93
15
2
u
92
+ ··· +
29
2
u + 4
3u
93
+
19
2
u
92
+ ···
17
2
u 2
a
2
=
1
2
u
93
3
2
u
92
+ ···
47
2
u
2
+
17
2
u
1
2
u
92
+ u
91
+ ··· 7u
2
+
3
2
u
a
9
=
u
4
u
2
+ 1
u
4
+ 2u
2
a
11
=
u
u
3
+ u
a
6
=
u
4
u
2
+ 1
u
6
2u
4
u
2
a
4
=
u
93
9u
92
+ ··· + 18u +
9
2
9
2
u
93
+ 15u
92
+ ··· 17u
9
2
a
5
=
u
93
+ 3u
92
+ ··· 4u
1
2
3
2
u
93
3u
92
+ ··· 2u
3
2
a
10
=
u
11
+ 4u
9
+ 4u
7
2u
5
3u
3
+ 2u
u
11
5u
9
8u
7
3u
5
+ 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
93
+
15
2
u
92
+ ···
33
2
u
11
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
94
+ 44u
93
+ ··· 9u + 1
c
2
, c
5
u
94
+ 4u
93
+ ··· + 9u + 1
c
3
u
94
4u
93
+ ··· 3441u + 306
c
4
, c
9
u
94
+ u
93
+ ··· 160u 64
c
6
, c
8
u
94
+ 3u
93
+ ··· 33u 34
c
7
, c
11
, c
12
u
94
3u
93
+ ··· + 4u 1
c
10
u
94
+ 35u
93
+ ··· + 66560u + 4096
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
94
+ 16y
93
+ ··· 369y + 1
c
2
, c
5
y
94
+ 44y
93
+ ··· 9y + 1
c
3
y
94
12y
93
+ ··· 6463449y + 93636
c
4
, c
9
y
94
35y
93
+ ··· 66560y + 4096
c
6
, c
8
y
94
53y
93
+ ··· 12309y + 1156
c
7
, c
11
, c
12
y
94
+ 79y
93
+ ··· 16y + 1
c
10
y
94
+ 37y
93
+ ··· 336592896y + 16777216
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.351586 + 1.054580I
a = 1.15398 + 1.86992I
b = 0.313058 0.701062I
1.40013 8.27926I 0
u = 0.351586 1.054580I
a = 1.15398 1.86992I
b = 0.313058 + 0.701062I
1.40013 + 8.27926I 0
u = 0.319387 + 1.064870I
a = 0.60014 1.46592I
b = 0.269640 + 0.910868I
0.91044 3.19835I 0
u = 0.319387 1.064870I
a = 0.60014 + 1.46592I
b = 0.269640 0.910868I
0.91044 + 3.19835I 0
u = 0.272022 + 1.136290I
a = 1.42508 + 2.49568I
b = 0.66890 1.54095I
0.75736 + 2.48496I 0
u = 0.272022 1.136290I
a = 1.42508 2.49568I
b = 0.66890 + 1.54095I
0.75736 2.48496I 0
u = 0.341633 + 1.119760I
a = 0.36104 + 1.97232I
b = 0.355067 1.058820I
3.79950 0.44484I 0
u = 0.341633 1.119760I
a = 0.36104 1.97232I
b = 0.355067 + 1.058820I
3.79950 + 0.44484I 0
u = 0.805155 + 0.159834I
a = 0.683311 0.812685I
b = 2.10771 0.56090I
4.13551 + 12.51930I 12.4410 9.1338I
u = 0.805155 0.159834I
a = 0.683311 + 0.812685I
b = 2.10771 + 0.56090I
4.13551 12.51930I 12.4410 + 9.1338I
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.819292 + 0.023297I
a = 0.557966 0.104325I
b = 2.01706 1.04963I
9.72584 + 4.11008I 17.4683 3.8689I
u = 0.819292 0.023297I
a = 0.557966 + 0.104325I
b = 2.01706 + 1.04963I
9.72584 4.11008I 17.4683 + 3.8689I
u = 0.223727 + 1.159260I
a = 0.038821 0.996676I
b = 0.793555 + 1.158040I
2.56110 0.96081I 0
u = 0.223727 1.159260I
a = 0.038821 + 0.996676I
b = 0.793555 1.158040I
2.56110 + 0.96081I 0
u = 0.800713 + 0.124544I
a = 0.056147 0.232819I
b = 1.45417 + 0.99662I
6.82446 + 4.61345I 16.0257 3.7320I
u = 0.800713 0.124544I
a = 0.056147 + 0.232819I
b = 1.45417 0.99662I
6.82446 4.61345I 16.0257 + 3.7320I
u = 0.791562 + 0.153666I
a = 0.673326 + 0.346910I
b = 1.43596 + 0.24106I
1.85813 + 7.30815I 9.64546 5.45633I
u = 0.791562 0.153666I
a = 0.673326 0.346910I
b = 1.43596 0.24106I
1.85813 7.30815I 9.64546 + 5.45633I
u = 0.795310
a = 0.653225
b = 1.34153
6.33214 13.9450
u = 0.207009 + 1.187620I
a = 0.54176 1.63085I
b = 0.28783 + 1.45505I
2.77826 1.83941I 0
6
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.207009 1.187620I
a = 0.54176 + 1.63085I
b = 0.28783 1.45505I
2.77826 + 1.83941I 0
u = 0.763490 + 0.126859I
a = 0.996501 0.724536I
b = 2.36823 0.61181I
2.24813 6.31785I 11.54818 + 5.93405I
u = 0.763490 0.126859I
a = 0.996501 + 0.724536I
b = 2.36823 + 0.61181I
2.24813 + 6.31785I 11.54818 5.93405I
u = 0.361057 + 0.671513I
a = 0.862756 + 0.799832I
b = 1.113130 0.534733I
0.51313 8.14873I 8.97384 + 8.84166I
u = 0.361057 0.671513I
a = 0.862756 0.799832I
b = 1.113130 + 0.534733I
0.51313 + 8.14873I 8.97384 8.84166I
u = 0.246747 + 1.213070I
a = 0.344077 + 0.628748I
b = 1.32614 0.49237I
1.93636 + 4.08463I 0
u = 0.246747 1.213070I
a = 0.344077 0.628748I
b = 1.32614 + 0.49237I
1.93636 4.08463I 0
u = 0.742646 + 0.133805I
a = 0.305512 0.849898I
b = 0.247171 0.370551I
0.40083 + 4.55861I 9.48261 6.77535I
u = 0.742646 0.133805I
a = 0.305512 + 0.849898I
b = 0.247171 + 0.370551I
0.40083 4.55861I 9.48261 + 6.77535I
u = 0.745684 + 0.058825I
a = 0.284002 0.610144I
b = 1.55723 + 1.53202I
4.01862 + 0.63484I 14.7380 0.8437I
7
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.745684 0.058825I
a = 0.284002 + 0.610144I
b = 1.55723 1.53202I
4.01862 0.63484I 14.7380 + 0.8437I
u = 0.297640 + 1.220630I
a = 0.91602 + 3.00986I
b = 0.17095 2.37493I
0.46784 4.41267I 0
u = 0.297640 1.220630I
a = 0.91602 3.00986I
b = 0.17095 + 2.37493I
0.46784 + 4.41267I 0
u = 0.730077 + 0.133883I
a = 0.875274 + 0.280144I
b = 1.370320 + 0.318322I
0.21470 1.60724I 7.97056 + 1.57129I
u = 0.730077 0.133883I
a = 0.875274 0.280144I
b = 1.370320 0.318322I
0.21470 + 1.60724I 7.97056 1.57129I
u = 0.717097 + 0.108393I
a = 0.308251 + 1.174220I
b = 0.130133 + 0.598547I
1.35579 0.62536I 12.06986 1.95150I
u = 0.717097 0.108393I
a = 0.308251 1.174220I
b = 0.130133 0.598547I
1.35579 + 0.62536I 12.06986 + 1.95150I
u = 0.293385 + 0.658021I
a = 0.266291 0.707620I
b = 0.598578 + 0.548638I
1.60349 3.29234I 5.23174 + 4.77747I
u = 0.293385 0.658021I
a = 0.266291 + 0.707620I
b = 0.598578 0.548638I
1.60349 + 3.29234I 5.23174 4.77747I
u = 0.365972 + 1.235490I
a = 2.15333 + 1.40823I
b = 1.72605 0.32067I
5.98368 + 0.15479I 0
8
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.365972 1.235490I
a = 2.15333 1.40823I
b = 1.72605 + 0.32067I
5.98368 0.15479I 0
u = 0.658370 + 0.267531I
a = 0.199386 + 1.344550I
b = 0.308441 + 0.617622I
1.88023 + 4.47482I 11.92967 3.68864I
u = 0.658370 0.267531I
a = 0.199386 1.344550I
b = 0.308441 0.617622I
1.88023 4.47482I 11.92967 + 3.68864I
u = 0.146043 + 1.287240I
a = 0.519747 0.895621I
b = 0.651519 + 1.152850I
3.09448 2.01504I 0
u = 0.146043 1.287240I
a = 0.519747 + 0.895621I
b = 0.651519 1.152850I
3.09448 + 2.01504I 0
u = 0.346129 + 1.258550I
a = 0.76971 1.71108I
b = 0.93264 + 1.31896I
2.43413 + 4.11256I 0
u = 0.346129 1.258550I
a = 0.76971 + 1.71108I
b = 0.93264 1.31896I
2.43413 4.11256I 0
u = 0.655988 + 0.207772I
a = 0.303076 0.671050I
b = 0.339247 0.122021I
0.0223914 0.0925396I 8.17235 + 0.51763I
u = 0.655988 0.207772I
a = 0.303076 + 0.671050I
b = 0.339247 + 0.122021I
0.0223914 + 0.0925396I 8.17235 0.51763I
u = 0.365714 + 1.275590I
a = 0.53429 + 2.97606I
b = 1.51702 2.35674I
5.68982 + 8.37124I 0
9
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.365714 1.275590I
a = 0.53429 2.97606I
b = 1.51702 + 2.35674I
5.68982 8.37124I 0
u = 0.313278 + 1.307810I
a = 2.86579 + 0.42092I
b = 2.66846 + 0.41995I
0.25992 3.19129I 0
u = 0.313278 1.307810I
a = 2.86579 0.42092I
b = 2.66846 0.41995I
0.25992 + 3.19129I 0
u = 0.367469 + 0.521123I
a = 0.06904 + 1.46105I
b = 0.348565 + 0.361785I
2.59885 0.99613I 12.66422 + 3.51334I
u = 0.367469 0.521123I
a = 0.06904 1.46105I
b = 0.348565 0.361785I
2.59885 + 0.99613I 12.66422 3.51334I
u = 0.070031 + 1.366780I
a = 0.123188 0.592014I
b = 0.264084 + 1.332300I
3.22572 2.28737I 0
u = 0.070031 1.366780I
a = 0.123188 + 0.592014I
b = 0.264084 1.332300I
3.22572 + 2.28737I 0
u = 0.305834 + 1.335140I
a = 0.074584 0.836640I
b = 0.89820 + 1.42609I
3.19308 + 3.09736I 0
u = 0.305834 1.335140I
a = 0.074584 + 0.836640I
b = 0.89820 1.42609I
3.19308 3.09736I 0
u = 0.538245 + 0.320223I
a = 1.031060 0.094532I
b = 0.872555 0.075771I
3.30326 2.28034I 15.1915 + 4.6781I
10
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.538245 0.320223I
a = 1.031060 + 0.094532I
b = 0.872555 + 0.075771I
3.30326 + 2.28034I 15.1915 4.6781I
u = 0.025452 + 0.620139I
a = 0.742007 0.914591I
b = 0.346461 + 0.713419I
2.38027 1.44221I 2.82236 + 3.43033I
u = 0.025452 0.620139I
a = 0.742007 + 0.914591I
b = 0.346461 0.713419I
2.38027 + 1.44221I 2.82236 3.43033I
u = 0.310881 + 1.345020I
a = 1.08999 2.22156I
b = 1.52900 + 2.06908I
4.44484 5.39215I 0
u = 0.310881 1.345020I
a = 1.08999 + 2.22156I
b = 1.52900 2.06908I
4.44484 + 5.39215I 0
u = 0.016489 + 1.381180I
a = 2.02420 + 0.83379I
b = 2.94086 0.32282I
6.88593 + 3.56319I 0
u = 0.016489 1.381180I
a = 2.02420 0.83379I
b = 2.94086 + 0.32282I
6.88593 3.56319I 0
u = 0.316216 + 1.345920I
a = 0.0516053 + 0.0153543I
b = 0.676154 0.292686I
4.26127 + 8.40222I 0
u = 0.316216 1.345920I
a = 0.0516053 0.0153543I
b = 0.676154 + 0.292686I
4.26127 8.40222I 0
u = 0.325860 + 1.343950I
a = 1.91588 + 3.33804I
b = 2.93290 2.93126I
2.38301 10.26160I 0
11
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.325860 1.343950I
a = 1.91588 3.33804I
b = 2.93290 + 2.93126I
2.38301 + 10.26160I 0
u = 0.200914 + 1.368680I
a = 1.313830 + 0.182615I
b = 1.93679 + 0.32568I
1.96196 4.91979I 0
u = 0.200914 1.368680I
a = 1.313830 0.182615I
b = 1.93679 0.32568I
1.96196 + 4.91979I 0
u = 0.344918 + 1.344910I
a = 2.19325 + 0.39504I
b = 2.16703 + 0.59757I
2.20181 + 8.74640I 0
u = 0.344918 1.344910I
a = 2.19325 0.39504I
b = 2.16703 0.59757I
2.20181 8.74640I 0
u = 0.003577 + 1.389610I
a = 1.11891 1.18307I
b = 1.65509 + 0.99510I
8.44452 1.51231I 0
u = 0.003577 1.389610I
a = 1.11891 + 1.18307I
b = 1.65509 0.99510I
8.44452 + 1.51231I 0
u = 0.275455 + 1.363070I
a = 0.102702 0.238819I
b = 0.517274 + 0.037833I
4.96173 3.51095I 0
u = 0.275455 1.363070I
a = 0.102702 + 0.238819I
b = 0.517274 0.037833I
4.96173 + 3.51095I 0
u = 0.337562 + 1.359830I
a = 1.23843 1.96495I
b = 1.75415 + 1.70909I
2.91527 + 11.38800I 0
12
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.337562 1.359830I
a = 1.23843 + 1.96495I
b = 1.75415 1.70909I
2.91527 11.38800I 0
u = 0.261096 + 1.380600I
a = 0.092849 0.803720I
b = 0.79183 + 1.51038I
3.31639 + 1.14144I 0
u = 0.261096 1.380600I
a = 0.092849 + 0.803720I
b = 0.79183 1.51038I
3.31639 1.14144I 0
u = 0.343711 + 1.364600I
a = 1.90058 + 2.78698I
b = 2.99172 2.39077I
0.6743 + 16.6665I 0
u = 0.343711 1.364600I
a = 1.90058 2.78698I
b = 2.99172 + 2.39077I
0.6743 16.6665I 0
u = 0.039853 + 1.411050I
a = 0.965916 0.939378I
b = 1.42920 + 0.68536I
8.01316 4.10190I 0
u = 0.039853 1.411050I
a = 0.965916 + 0.939378I
b = 1.42920 0.68536I
8.01316 + 4.10190I 0
u = 0.05124 + 1.42228I
a = 1.89938 + 0.53953I
b = 2.82345 + 0.01727I
6.06023 9.18761I 0
u = 0.05124 1.42228I
a = 1.89938 0.53953I
b = 2.82345 0.01727I
6.06023 + 9.18761I 0
u = 0.125946 + 0.536133I
a = 1.34468 + 1.38361I
b = 0.933554 0.595599I
1.01261 + 3.21488I 4.87205 3.43985I
13
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.125946 0.536133I
a = 1.34468 1.38361I
b = 0.933554 + 0.595599I
1.01261 3.21488I 4.87205 + 3.43985I
u = 0.408365
a = 0.235762
b = 0.359010
0.736099 13.3290
u = 0.192609 + 0.116954I
a = 0.12284 + 3.63556I
b = 0.351760 + 0.622816I
0.31015 1.80079I 2.04373 + 2.87212I
u = 0.192609 0.116954I
a = 0.12284 3.63556I
b = 0.351760 0.622816I
0.31015 + 1.80079I 2.04373 2.87212I
14
II. I
u
2
= h−u
2
a + b, u
2
a + a
2
+ u
2
+ a u + 2, u
3
u
2
+ 2u 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
u
u
a
3
=
a
u
2
a
a
2
=
u
2
+ a u + 1
u
2
a + 1
a
9
=
u
u
2
u + 1
a
11
=
u
u
2
u + 1
a
6
=
u
u
a
4
=
au
u
2
a au + a
a
5
=
au
u
2
a au + a
a
10
=
u
u
2
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
2
a + 3au 5u
2
4a + 5u 16
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
u + 1)
3
c
2
(u
2
+ u + 1)
3
c
4
, c
9
, c
10
u
6
c
6
, c
8
(u
3
+ u
2
1)
2
c
7
(u
3
u
2
+ 2u 1)
2
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
3
c
4
, c
9
, c
10
y
6
c
6
, c
8
(y
3
y
2
+ 2y 1)
2
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.818128 + 0.292480I
b = 1.52448 0.02619I
3.02413 0.79824I 6.43615 0.68567I
u = 0.215080 + 1.307140I
a = 0.155769 0.854759I
b = 0.73956 + 1.33333I
3.02413 4.85801I 2.88198 + 6.08229I
u = 0.215080 1.307140I
a = 0.818128 0.292480I
b = 1.52448 + 0.02619I
3.02413 + 0.79824I 6.43615 + 0.68567I
u = 0.215080 1.307140I
a = 0.155769 + 0.854759I
b = 0.73956 1.33333I
3.02413 + 4.85801I 2.88198 6.08229I
u = 0.569840
a = 0.662359 + 1.147240I
b = 0.215080 + 0.372529I
1.11345 + 2.02988I 12.18187 4.49037I
u = 0.569840
a = 0.662359 1.147240I
b = 0.215080 0.372529I
1.11345 2.02988I 12.18187 + 4.49037I
18
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
u + 1)
3
)(u
94
+ 44u
93
+ ··· 9u + 1)
c
2
((u
2
+ u + 1)
3
)(u
94
+ 4u
93
+ ··· + 9u + 1)
c
3
((u
2
u + 1)
3
)(u
94
4u
93
+ ··· 3441u + 306)
c
4
, c
9
u
6
(u
94
+ u
93
+ ··· 160u 64)
c
5
((u
2
u + 1)
3
)(u
94
+ 4u
93
+ ··· + 9u + 1)
c
6
, c
8
((u
3
+ u
2
1)
2
)(u
94
+ 3u
93
+ ··· 33u 34)
c
7
((u
3
u
2
+ 2u 1)
2
)(u
94
3u
93
+ ··· + 4u 1)
c
10
u
6
(u
94
+ 35u
93
+ ··· + 66560u + 4096)
c
11
, c
12
((u
3
+ u
2
+ 2u + 1)
2
)(u
94
3u
93
+ ··· + 4u 1)
19
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
3
)(y
94
+ 16y
93
+ ··· 369y + 1)
c
2
, c
5
((y
2
+ y + 1)
3
)(y
94
+ 44y
93
+ ··· 9y + 1)
c
3
((y
2
+ y + 1)
3
)(y
94
12y
93
+ ··· 6463449y + 93636)
c
4
, c
9
y
6
(y
94
35y
93
+ ··· 66560y + 4096)
c
6
, c
8
((y
3
y
2
+ 2y 1)
2
)(y
94
53y
93
+ ··· 12309y + 1156)
c
7
, c
11
, c
12
((y
3
+ 3y
2
+ 2y 1)
2
)(y
94
+ 79y
93
+ ··· 16y + 1)
c
10
y
6
(y
94
+ 37y
93
+ ··· 3.36593 × 10
8
y + 1.67772 × 10
7
)
20