12a
0240
(K12a
0240
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 1 4 11 12 5 9 8
Solving Sequence
8,11
9 12
4,10
5 1 7 3 6 2
c
8
c
11
c
9
c
4
c
12
c
7
c
3
c
6
c
2
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h43u
91
− 1146u
90
+ ··· + 32b + 579, 141u
91
− 878u
90
+ ··· + 4a + 144, u
92
− 7u
91
+ ··· + 3u − 1i
I
u
2
= hb
6
+ b
5
− b
4
− 2b
3
+ b + 1, a, u + 1i
* 2 irreducible components of dim
C
= 0, with total 98 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
= h43u
91
− 1146u
90
+ · · · + 32b + 579, 141u
91
− 878u
90
+ · · · + 4a +
144, u
92
− 7u
91
+ · · · + 3u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
12
=
u
−u
3
+ u
a
4
=
−
141
4
u
91
+
439
2
u
90
+ ··· + 80u − 36
−1.34375u
91
+ 35.8125u
90
+ ··· + 26.1875u − 18.0938
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
259
2
u
91
− 723u
90
+ ··· −
407
2
u +
359
4
84.1563u
91
− 414.438u
90
+ ··· − 88.5625u + 26.4063
a
1
=
−u
3
+ 2u
−u
3
+ u
a
7
=
−u
9
+ 4u
7
+ 2u
6
− 5u
5
− 6u
4
+ 4u
2
+ 3u + 2
0.0312500u
91
− 0.187500u
90
+ ··· − 1.06250u + 0.0312500
a
3
=
73.0313u
91
− 436.188u
90
+ ··· − 135.313u + 68.5313
367.031u
91
− 2093.25u
90
+ ··· − 628.625u + 279.844
a
6
=
−0.0312500u
91
+ 0.187500u
90
+ ··· + 3.06250u + 1.96875
0.0312500u
91
− 0.187500u
90
+ ··· − 1.06250u + 0.0312500
a
2
=
2.68750u
91
− 15.9375u
90
+ ··· + 4.81250u + 3.50000
15.4688u
91
− 91.0625u
90
+ ··· − 29.1875u + 13.8438
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3517
8
u
91
+
39449
16
u
90
+ ··· +
11229
16
u −
4837
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
92
+ 42u
91
+ ··· + 6u + 1
c
2
, c
5
u
92
+ 2u
91
+ ··· − 2u + 1
c
3
, c
7
u
92
− 2u
91
+ ··· + 110u + 25
c
4
, c
10
u
92
− u
91
+ ··· + 128u − 64
c
6
u
92
+ 6u
91
+ ··· − 18u − 5
c
8
, c
9
, c
11
u
92
+ 7u
91
+ ··· − 3u − 1
c
12
u
92
− 39u
91
+ ··· − 45056u + 4096
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
92
+ 18y
91
+ ··· − 50y + 1
c
2
, c
5
y
92
− 42y
91
+ ··· − 6y + 1
c
3
, c
7
y
92
− 66y
91
+ ··· + 23850y + 625
c
4
, c
10
y
92
− 39y
91
+ ··· − 45056y + 4096
c
6
y
92
+ 6y
91
+ ··· + 1086y + 25
c
8
, c
9
, c
11
y
92
− 81y
91
+ ··· + y + 1
c
12
y
92
+ 17y
91
+ ··· − 201326592y + 16777216
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.855893 + 0.527606I
a = 0.393530 − 0.308819I
b = 1.238450 − 0.035861I
5.58340 − 0.40937I 0
u = −0.855893 − 0.527606I
a = 0.393530 + 0.308819I
b = 1.238450 + 0.035861I
5.58340 + 0.40937I 0
u = −0.823622 + 0.543416I
a = −0.299409 + 0.297378I
b = −1.200290 + 0.200271I
3.94188 − 5.47839I 0
u = −0.823622 − 0.543416I
a = −0.299409 − 0.297378I
b = −1.200290 − 0.200271I
3.94188 + 5.47839I 0
u = −0.934785 + 0.428419I
a = −0.640037 + 0.141332I
b = −0.946030 − 0.426464I
0.419646 + 0.395087I 0
u = −0.934785 − 0.428419I
a = −0.640037 − 0.141332I
b = −0.946030 + 0.426464I
0.419646 − 0.395087I 0
u = −0.926309 + 0.516475I
a = 0.581393 − 0.353795I
b = 1.344830 + 0.308771I
5.25124 + 2.21296I 0
u = −0.926309 − 0.516475I
a = 0.581393 + 0.353795I
b = 1.344830 − 0.308771I
5.25124 − 2.21296I 0
u = −0.951668 + 0.520377I
a = −0.650526 + 0.382532I
b = −1.40538 − 0.44067I
3.32536 + 7.33336I 0
u = −0.951668 − 0.520377I
a = −0.650526 − 0.382532I
b = −1.40538 + 0.44067I
3.32536 − 7.33336I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.257095 + 0.850924I
a = −1.05101 + 1.49174I
b = −1.50653 + 0.51583I
1.19521 − 12.17710I 0
u = −0.257095 − 0.850924I
a = −1.05101 − 1.49174I
b = −1.50653 − 0.51583I
1.19521 + 12.17710I 0
u = −0.266178 + 0.838902I
a = 1.01110 − 1.43448I
b = 1.41908 − 0.41643I
3.21993 − 7.00055I 0
u = −0.266178 − 0.838902I
a = 1.01110 + 1.43448I
b = 1.41908 + 0.41643I
3.21993 + 7.00055I 0
u = −0.298969 + 0.807596I
a = 0.91610 − 1.25406I
b = 1.201580 − 0.120872I
3.86636 − 4.28759I 0
u = −0.298969 − 0.807596I
a = 0.91610 + 1.25406I
b = 1.201580 + 0.120872I
3.86636 + 4.28759I 0
u = −0.320326 + 0.790880I
a = −0.86648 + 1.14685I
b = −1.082220 − 0.039888I
2.40332 + 0.79509I 0
u = −0.320326 − 0.790880I
a = −0.86648 − 1.14685I
b = −1.082220 + 0.039888I
2.40332 − 0.79509I 0
u = −0.238388 + 0.811798I
a = −0.85620 + 1.51375I
b = −1.149650 + 0.585085I
−1.74435 − 4.87706I 0
u = −0.238388 − 0.811798I
a = −0.85620 − 1.51375I
b = −1.149650 − 0.585085I
−1.74435 + 4.87706I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.126370 + 0.322522I
a = 0.984386 + 0.298675I
b = 0.160829 + 0.964145I
−1.59588 + 2.31274I 0
u = −1.126370 − 0.322522I
a = 0.984386 − 0.298675I
b = 0.160829 − 0.964145I
−1.59588 − 2.31274I 0
u = −1.158340 + 0.231833I
a = −0.817877 − 0.574536I
b = 0.147057 − 0.642703I
1.15869 − 1.03212I 0
u = −1.158340 − 0.231833I
a = −0.817877 + 0.574536I
b = 0.147057 + 0.642703I
1.15869 + 1.03212I 0
u = −0.119442 + 0.754619I
a = 0.44100 − 1.86219I
b = 0.374544 − 1.118990I
−4.64187 − 6.24622I 0
u = −0.119442 − 0.754619I
a = 0.44100 + 1.86219I
b = 0.374544 + 1.118990I
−4.64187 + 6.24622I 0
u = −1.207270 + 0.288848I
a = 1.100770 + 0.600111I
b = −0.328351 + 0.984733I
−1.41302 − 4.72361I 0
u = −1.207270 − 0.288848I
a = 1.100770 − 0.600111I
b = −0.328351 − 0.984733I
−1.41302 + 4.72361I 0
u = −1.258690 + 0.052256I
a = −0.241511 − 1.169820I
b = 0.200788 + 0.093222I
2.44595 − 1.87550I 0
u = −1.258690 − 0.052256I
a = −0.241511 + 1.169820I
b = 0.200788 − 0.093222I
2.44595 + 1.87550I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.265130 + 0.134678I
a = 0.298029 − 0.315517I
b = 1.20646 − 0.83292I
2.67098 − 4.72600I 0
u = 1.265130 − 0.134678I
a = 0.298029 + 0.315517I
b = 1.20646 + 0.83292I
2.67098 + 4.72600I 0
u = −0.060720 + 0.707583I
a = 0.18373 − 1.95736I
b = −0.097751 − 1.124380I
−4.91657 + 1.10042I 0
u = −0.060720 − 0.707583I
a = 0.18373 + 1.95736I
b = −0.097751 + 1.124380I
−4.91657 − 1.10042I 0
u = −0.128623 + 0.697439I
a = −0.27048 + 1.73365I
b = −0.165115 + 0.841230I
−1.87128 − 2.38560I 6.00000 + 4.07768I
u = −0.128623 − 0.697439I
a = −0.27048 − 1.73365I
b = −0.165115 − 0.841230I
−1.87128 + 2.38560I 6.00000 − 4.07768I
u = 1.283140 + 0.186796I
a = 0.373253 − 0.484907I
b = 0.780681 − 1.002100I
0.67630 + 2.73155I 0
u = 1.283140 − 0.186796I
a = 0.373253 + 0.484907I
b = 0.780681 + 1.002100I
0.67630 − 2.73155I 0
u = 1.294300 + 0.139031I
a = −0.232339 + 0.384153I
b = −1.028070 + 0.689952I
4.93199 + 0.09462I 0
u = 1.294300 − 0.139031I
a = −0.232339 − 0.384153I
b = −1.028070 − 0.689952I
4.93199 − 0.09462I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.370573 + 0.549861I
a = −0.036226 + 0.815695I
b = 0.015187 − 0.145705I
0.29821 − 3.60185I 8.77802 + 7.91967I
u = −0.370573 − 0.549861I
a = −0.036226 − 0.815695I
b = 0.015187 + 0.145705I
0.29821 + 3.60185I 8.77802 − 7.91967I
u = −1.320460 + 0.224539I
a = −1.13840 − 1.21130I
b = 1.099530 − 0.487703I
1.26119 − 2.89599I 0
u = −1.320460 − 0.224539I
a = −1.13840 + 1.21130I
b = 1.099530 + 0.487703I
1.26119 + 2.89599I 0
u = 1.311480 + 0.275755I
a = −0.585969 + 0.814922I
b = 0.075543 + 1.278460I
−0.61934 + 2.43878I 0
u = 1.311480 − 0.275755I
a = −0.585969 − 0.814922I
b = 0.075543 − 1.278460I
−0.61934 − 2.43878I 0
u = −0.560744 + 0.338028I
a = −0.401947 − 0.291968I
b = −0.417009 + 0.124386I
0.930041 + 0.097702I 11.39299 + 0.73864I
u = −0.560744 − 0.338028I
a = −0.401947 + 0.291968I
b = −0.417009 − 0.124386I
0.930041 − 0.097702I 11.39299 − 0.73864I
u = −1.356550 + 0.186660I
a = 1.03317 + 1.48094I
b = −1.238570 + 0.053586I
6.71288 − 2.16341I 0
u = −1.356550 − 0.186660I
a = 1.03317 − 1.48094I
b = −1.238570 − 0.053586I
6.71288 + 2.16341I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.359340 + 0.167370I
a = −0.93577 − 1.53603I
b = 1.157740 + 0.103088I
5.16406 + 2.93183I 0
u = −1.359340 − 0.167370I
a = −0.93577 + 1.53603I
b = 1.157740 − 0.103088I
5.16406 − 2.93183I 0
u = 1.338010 + 0.306526I
a = −0.634728 + 1.012130I
b = 0.535704 + 1.230630I
−0.05500 + 10.07060I 0
u = 1.338010 − 0.306526I
a = −0.634728 − 1.012130I
b = 0.535704 − 1.230630I
−0.05500 − 10.07060I 0
u = 1.344200 + 0.280833I
a = 0.508661 − 0.958326I
b = −0.321400 − 1.006570I
2.77854 + 5.93136I 0
u = 1.344200 − 0.280833I
a = 0.508661 + 0.958326I
b = −0.321400 + 1.006570I
2.77854 − 5.93136I 0
u = −1.356800 + 0.223131I
a = 1.22453 + 1.39437I
b = −1.40250 + 0.35712I
6.21974 − 4.87713I 0
u = −1.356800 − 0.223131I
a = 1.22453 − 1.39437I
b = −1.40250 − 0.35712I
6.21974 + 4.87713I 0
u = −1.359380 + 0.235892I
a = −1.29717 − 1.37298I
b = 1.47387 − 0.46581I
4.24684 − 10.04410I 0
u = −1.359380 − 0.235892I
a = −1.29717 + 1.37298I
b = 1.47387 + 0.46581I
4.24684 + 10.04410I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.373680 + 0.165838I
a = −0.030998 + 0.649563I
b = −0.565392 + 0.278503I
6.41119 + 1.66341I 0
u = 1.373680 − 0.165838I
a = −0.030998 − 0.649563I
b = −0.565392 − 0.278503I
6.41119 − 1.66341I 0
u = 0.157895 + 0.574898I
a = 0.47890 + 2.17258I
b = 1.33051 + 0.56157I
−0.57833 + 7.04233I 2.90061 − 5.37699I
u = 0.157895 − 0.574898I
a = 0.47890 − 2.17258I
b = 1.33051 − 0.56157I
−0.57833 − 7.04233I 2.90061 + 5.37699I
u = 1.393190 + 0.205065I
a = 0.062760 − 0.858438I
b = 0.195021 − 0.215669I
5.80644 + 6.29039I 0
u = 1.393190 − 0.205065I
a = 0.062760 + 0.858438I
b = 0.195021 + 0.215669I
5.80644 − 6.29039I 0
u = 0.060811 + 0.581784I
a = 0.32226 + 2.03401I
b = 0.869516 + 0.680199I
−3.09730 − 0.04455I −0.715184 + 0.480345I
u = 0.060811 − 0.581784I
a = 0.32226 − 2.03401I
b = 0.869516 − 0.680199I
−3.09730 + 0.04455I −0.715184 − 0.480345I
u = 0.147924 + 0.538492I
a = −0.50740 − 2.10909I
b = −1.224020 − 0.407211I
1.42723 + 2.03586I 6.04382 − 1.04481I
u = 0.147924 − 0.538492I
a = −0.50740 + 2.10909I
b = −1.224020 + 0.407211I
1.42723 − 2.03586I 6.04382 + 1.04481I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.40611 + 0.33247I
a = 0.52147 − 1.41501I
b = −1.29248 − 0.62592I
3.48271 + 9.00992I 0
u = 1.40611 − 0.33247I
a = 0.52147 + 1.41501I
b = −1.29248 + 0.62592I
3.48271 − 9.00992I 0
u = 1.42239 + 0.34002I
a = −0.49530 + 1.52622I
b = 1.50359 + 0.45510I
8.5960 + 11.2525I 0
u = 1.42239 − 0.34002I
a = −0.49530 − 1.52622I
b = 1.50359 − 0.45510I
8.5960 − 11.2525I 0
u = 1.42069 + 0.34734I
a = 0.53981 − 1.54675I
b = −1.59516 − 0.52951I
6.5335 + 16.4961I 0
u = 1.42069 − 0.34734I
a = 0.53981 + 1.54675I
b = −1.59516 + 0.52951I
6.5335 − 16.4961I 0
u = 1.42947 + 0.31906I
a = −0.36213 + 1.47494I
b = 1.253540 + 0.228818I
9.38007 + 8.35415I 0
u = 1.42947 − 0.31906I
a = −0.36213 − 1.47494I
b = 1.253540 − 0.228818I
9.38007 − 8.35415I 0
u = 1.43299 + 0.30718I
a = 0.29061 − 1.44097I
b = −1.107560 − 0.118760I
8.00030 + 3.16463I 0
u = 1.43299 − 0.30718I
a = 0.29061 + 1.44097I
b = −1.107560 + 0.118760I
8.00030 − 3.16463I 0
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.527801
a = −0.723957
b = −0.374114
0.870536 11.8250
u = 1.47354
a = 0.647269
b = −1.32717
8.16717 0
u = 1.50558 + 0.01546I
a = −0.830211 − 0.093239I
b = 1.47970 + 0.09284I
13.64150 + 1.52842I 0
u = 1.50558 − 0.01546I
a = −0.830211 + 0.093239I
b = 1.47970 − 0.09284I
13.64150 − 1.52842I 0
u = 1.50649 + 0.02862I
a = 0.830893 + 0.173036I
b = −1.46630 − 0.17312I
11.88820 + 6.80502I 0
u = 1.50649 − 0.02862I
a = 0.830893 − 0.173036I
b = −1.46630 + 0.17312I
11.88820 − 6.80502I 0
u = 0.163934 + 0.407741I
a = −0.64977 − 2.02768I
b = −1.030710 + 0.089763I
1.88700 − 0.17058I 6.42139 − 0.17260I
u = 0.163934 − 0.407741I
a = −0.64977 + 2.02768I
b = −1.030710 − 0.089763I
1.88700 + 0.17058I 6.42139 + 0.17260I
u = 0.218461 + 0.348098I
a = 0.67430 + 2.05517I
b = 1.021640 − 0.377057I
0.22019 − 4.99067I 2.83178 + 4.98707I
u = 0.218461 − 0.348098I
a = 0.67430 − 2.05517I
b = 1.021640 + 0.377057I
0.22019 + 4.99067I 2.83178 − 4.98707I
13
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.167778 + 0.119576I
a = 0.59959 + 2.50203I
b = 0.335757 − 0.546609I
−1.65717 + 1.14756I −1.67384 − 1.02015I
u = 0.167778 − 0.119576I
a = 0.59959 − 2.50203I
b = 0.335757 + 0.546609I
−1.65717 − 1.14756I −1.67384 + 1.02015I
14
II. I
u
2
= hb
6
+ b
5
− b
4
− 2b
3
+ b + 1, a, u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
−1
a
9
=
1
−1
a
12
=
−1
0
a
4
=
0
b
a
10
=
0
−1
a
5
=
0
b
a
1
=
−1
0
a
7
=
1
b
2
a
3
=
−b
−b
3
+ b
a
6
=
−b
2
+ 1
b
2
a
2
=
−b
4
+ b
2
− 1
b
5
+ b
4
− 2b
3
− b
2
+ b + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −b
5
− 4b
4
+ 2b
3
+ 4b
2
+ 2b + 5
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
2
, c
7
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
3
, c
5
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
4
, c
10
, c
12
u
6
c
8
, c
9
(u + 1)
6
c
11
(u − 1)
6
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
2
, c
3
, c
5
c
7
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
4
, c
10
, c
12
y
6
c
8
, c
9
, c
11
(y −1)
6
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.002190 + 0.295542I
3.53554 + 0.92430I 10.03026 − 0.88960I
u = −1.00000
a = 0
b = 1.002190 − 0.295542I
3.53554 − 0.92430I 10.03026 + 0.88960I
u = −1.00000
a = 0
b = −0.428243 + 0.664531I
−0.245672 + 0.924305I 5.20252 − 1.68215I
u = −1.00000
a = 0
b = −0.428243 − 0.664531I
−0.245672 − 0.924305I 5.20252 + 1.68215I
u = −1.00000
a = 0
b = −1.073950 + 0.558752I
1.64493 − 5.69302I 6.76721 + 6.15196I
u = −1.00000
a = 0
b = −1.073950 − 0.558752I
1.64493 + 5.69302I 6.76721 − 6.15196I
18
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
92
+ 42u
91
+ ··· + 6u + 1)
c
2
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
92
+ 2u
91
+ ··· − 2u + 1)
c
3
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
92
− 2u
91
+ ··· + 110u + 25)
c
4
, c
10
u
6
(u
92
− u
91
+ ··· + 128u − 64)
c
5
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
92
+ 2u
91
+ ··· − 2u + 1)
c
6
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
92
+ 6u
91
+ ··· − 18u − 5)
c
7
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
92
− 2u
91
+ ··· + 110u + 25)
c
8
, c
9
((u + 1)
6
)(u
92
+ 7u
91
+ ··· − 3u − 1)
c
11
((u − 1)
6
)(u
92
+ 7u
91
+ ··· − 3u − 1)
c
12
u
6
(u
92
− 39u
91
+ ··· − 45056u + 4096)
19
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
92
+ 18y
91
+ ··· − 50y + 1)
c
2
, c
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
92
− 42y
91
+ ··· − 6y + 1)
c
3
, c
7
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
92
− 66y
91
+ ··· + 23850y + 625)
c
4
, c
10
y
6
(y
92
− 39y
91
+ ··· − 45056y + 4096)
c
6
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
92
+ 6y
91
+ ··· + 1086y + 25)
c
8
, c
9
, c
11
((y −1)
6
)(y
92
− 81y
91
+ ··· + y + 1)
c
12
y
6
(y
92
+ 17y
91
+ ··· − 2.01327 × 10
8
y + 1.67772 × 10
7
)
20
