12n
0308
(K12n
0308
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 12 11 4 7 8 10
Solving Sequence
7,12 4,8
3 11 10 5 1 6 2 9
c
7
c
3
c
11
c
10
c
4
c
12
c
6
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h6u
39
+ 24u
38
+ ··· + 4b − 9, 9u
39
+ 30u
38
+ ··· + 4a − 24, u
40
+ 4u
39
+ ··· − 2u − 1i
I
u
2
= hb + u, a + 1, u
3
− u
2
+ 2u − 1i
I
u
3
= h−au + b, u
2
a + a
2
− au + 3u
2
+ a − u + 5, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 49 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
=
h6u
39
+24u
38
+· · ·+4b−9, 9u
39
+30u
38
+· · ·+4a−24, u
40
+4u
39
+· · ·−2u−1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−
9
4
u
39
−
15
2
u
38
+ ··· −
37
4
u + 6
−
3
2
u
39
− 6u
38
+ ··· −
3
2
u +
9
4
a
8
=
1
u
2
a
3
=
−3.75000u
39
− 13.5000u
38
+ ··· − 10.7500u + 8.25000
−
3
2
u
39
− 6u
38
+ ··· −
3
2
u +
9
4
a
11
=
u
u
3
+ u
a
10
=
u
3
+ 2u
u
3
+ u
a
5
=
−
11
4
u
39
−
17
2
u
38
+ ··· −
43
4
u + 7
−
3
2
u
39
− 5u
38
+ ··· −
5
2
u +
11
4
a
1
=
−u
7
− 4u
5
− 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
6
=
1
2
u
39
+
7
4
u
38
+ ··· −
21
4
u +
5
4
1
4
u
39
+ u
38
+ ··· −
5
4
u −
1
2
a
2
=
−
3
4
u
39
−
11
4
u
38
+ ··· +
13
2
u +
1
4
−
1
4
u
39
− u
38
+ ··· +
9
4
u +
1
2
a
9
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
9
2
u
39
−
37
2
u
38
+ ··· −
39
2
u −
7
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
40
+ 16u
39
+ ··· + 14u + 1
c
2
, c
5
u
40
+ 4u
39
+ ··· − 6u − 1
c
3
u
40
− 4u
39
+ ··· + 7u
2
− 1
c
4
, c
9
u
40
− u
39
+ ··· + 1536u + 512
c
7
, c
8
, c
11
u
40
− 4u
39
+ ··· + 2u − 1
c
10
u
40
+ 4u
39
+ ··· + 7u − 2
c
12
u
40
− 20u
39
+ ··· − 2480u + 20513
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
40
+ 20y
39
+ ··· − 142y + 1
c
2
, c
5
y
40
− 16y
39
+ ··· − 14y + 1
c
3
y
40
− 64y
39
+ ··· − 14y + 1
c
4
, c
9
y
40
− 49y
39
+ ··· − 655360y + 262144
c
7
, c
8
, c
11
y
40
+ 32y
39
+ ··· − 22y + 1
c
10
y
40
− 48y
39
+ ··· − 109y + 4
c
12
y
40
− 76y
39
+ ··· − 22287329974y + 420783169
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.965795
a = −2.34182
b = −2.26172
−14.3054 −18.4760
u = 0.139471 + 0.935681I
a = 0.435322 + 0.697825I
b = 0.592227 − 0.504649I
−0.336971 + 0.164971I −14.2424 − 0.6515I
u = 0.139471 − 0.935681I
a = 0.435322 − 0.697825I
b = 0.592227 + 0.504649I
−0.336971 − 0.164971I −14.2424 + 0.6515I
u = −0.938638 + 0.087954I
a = −2.38096 − 0.31083I
b = −2.26220 − 0.08234I
−9.79280 + 8.10641I −15.5163 − 4.8945I
u = −0.938638 − 0.087954I
a = −2.38096 + 0.31083I
b = −2.26220 + 0.08234I
−9.79280 − 8.10641I −15.5163 + 4.8945I
u = −0.916347 + 0.055169I
a = 2.49555 + 0.21808I
b = 2.29882 + 0.06216I
−7.85713 + 2.28541I −13.64199 − 0.57988I
u = −0.916347 − 0.055169I
a = 2.49555 − 0.21808I
b = 2.29882 − 0.06216I
−7.85713 − 2.28541I −13.64199 + 0.57988I
u = −0.119891 + 1.171300I
a = 1.32469 + 0.51665I
b = 0.76397 − 1.48966I
5.65080 + 4.37709I −8.98534 − 1.57605I
u = −0.119891 − 1.171300I
a = 1.32469 − 0.51665I
b = 0.76397 + 1.48966I
5.65080 − 4.37709I −8.98534 + 1.57605I
u = −0.075508 + 1.198880I
a = −1.212150 − 0.270191I
b = −0.41545 + 1.43282I
6.05510 − 1.44432I −7.22921 + 3.78916I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.075508 − 1.198880I
a = −1.212150 + 0.270191I
b = −0.41545 − 1.43282I
6.05510 + 1.44432I −7.22921 − 3.78916I
u = 0.182743 + 1.209970I
a = −0.361246 − 0.222448I
b = −0.203140 + 0.477747I
2.74721 − 2.07700I −5.89369 + 3.03885I
u = 0.182743 − 1.209970I
a = −0.361246 + 0.222448I
b = −0.203140 − 0.477747I
2.74721 + 2.07700I −5.89369 − 3.03885I
u = 0.359488 + 1.180450I
a = 0.169664 + 0.421096I
b = 0.436088 − 0.351658I
1.06997 − 5.73235I −12.00000 + 6.50930I
u = 0.359488 − 1.180450I
a = 0.169664 − 0.421096I
b = 0.436088 + 0.351658I
1.06997 + 5.73235I −12.00000 − 6.50930I
u = 0.758575 + 0.094110I
a = 0.049336 − 0.391801I
b = −0.074297 + 0.292567I
−2.22421 + 1.66592I −15.6530 − 3.4641I
u = 0.758575 − 0.094110I
a = 0.049336 + 0.391801I
b = −0.074297 − 0.292567I
−2.22421 − 1.66592I −15.6530 + 3.4641I
u = 0.633093 + 0.358144I
a = 0.135758 − 0.528918I
b = −0.275376 + 0.286233I
−1.63533 − 3.53225I −15.3501 + 6.0239I
u = 0.633093 − 0.358144I
a = 0.135758 + 0.528918I
b = −0.275376 − 0.286233I
−1.63533 + 3.53225I −15.3501 − 6.0239I
u = −0.494907 + 1.218760I
a = −0.89625 − 1.31642I
b = −2.04796 + 0.44080I
−6.31623 − 3.01599I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.494907 − 1.218760I
a = −0.89625 + 1.31642I
b = −2.04796 − 0.44080I
−6.31623 + 3.01599I 0
u = −0.457832 + 1.243030I
a = 0.88954 + 1.38304I
b = 2.12642 − 0.47253I
−4.18941 + 2.61474I 0
u = −0.457832 − 1.243030I
a = 0.88954 − 1.38304I
b = 2.12642 + 0.47253I
−4.18941 − 2.61474I 0
u = 0.285992 + 1.310300I
a = −0.053918 − 0.293250I
b = −0.368826 + 0.154516I
2.18000 − 2.07454I 0
u = 0.285992 − 1.310300I
a = −0.053918 + 0.293250I
b = −0.368826 − 0.154516I
2.18000 + 2.07454I 0
u = −0.480023 + 1.304310I
a = −0.77326 − 1.37509I
b = −2.16472 + 0.34850I
−10.25510 + 5.14488I 0
u = −0.480023 − 1.304310I
a = −0.77326 + 1.37509I
b = −2.16472 − 0.34850I
−10.25510 − 5.14488I 0
u = 0.146448 + 1.384710I
a = −0.301378 + 0.307270I
b = 0.469617 + 0.372324I
4.81791 − 1.66817I 0
u = 0.146448 − 1.384710I
a = −0.301378 − 0.307270I
b = 0.469617 − 0.372324I
4.81791 + 1.66817I 0
u = −0.427217 + 1.328920I
a = 0.72481 + 1.49260I
b = 2.29320 − 0.32556I
−3.52880 + 7.09218I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.427217 − 1.328920I
a = 0.72481 − 1.49260I
b = 2.29320 + 0.32556I
−3.52880 − 7.09218I 0
u = −0.43256 + 1.35367I
a = −0.66257 − 1.47304I
b = −2.28062 + 0.25972I
−5.27211 + 13.00920I 0
u = −0.43256 − 1.35367I
a = −0.66257 + 1.47304I
b = −2.28062 − 0.25972I
−5.27211 − 13.00920I 0
u = 0.19436 + 1.41468I
a = 0.190363 − 0.356157I
b = −0.540845 − 0.200080I
4.09613 − 6.42410I 0
u = 0.19436 − 1.41468I
a = 0.190363 + 0.356157I
b = −0.540845 + 0.200080I
4.09613 + 6.42410I 0
u = 0.269571 + 0.358650I
a = −0.373006 + 0.916266I
b = 0.429170 − 0.113220I
−0.650305 + 0.185166I −12.44941 + 0.70433I
u = 0.269571 − 0.358650I
a = −0.373006 − 0.916266I
b = 0.429170 + 0.113220I
−0.650305 − 0.185166I −12.44941 − 0.70433I
u = 0.334754
a = −0.815567
b = 0.273014
−0.669605 −14.6520
u = −0.311291 + 0.046384I
a = 0.17839 + 3.50865I
b = 0.218277 + 1.083940I
2.49750 − 2.73965I −2.44237 + 2.28441I
u = −0.311291 − 0.046384I
a = 0.17839 − 3.50865I
b = 0.218277 − 1.083940I
2.49750 + 2.73965I −2.44237 − 2.28441I
8
II. I
u
2
= hb + u, a + 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−1
−u
a
8
=
1
u
2
a
3
=
−u − 1
−u
a
11
=
u
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
a
5
=
−1
−u
a
1
=
−1
0
a
6
=
−u + 1
−u
2
a
2
=
−u
2
− u − 1
−u
2
a
9
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
2
+ 8u − 20
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
9
u
3
c
5
, c
10
, c
12
u
3
− u
2
+ 1
c
6
, c
11
u
3
+ u
2
+ 2u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
y
3
+ 3y
2
+ 2y − 1
c
2
, c
5
, c
10
c
12
y
3
− y
2
+ 2y − 1
c
4
, c
9
y
3
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.00000
b = −0.215080 − 1.307140I
6.04826 − 5.65624I −6.64285 + 6.52117I
u = 0.215080 − 1.307140I
a = −1.00000
b = −0.215080 + 1.307140I
6.04826 + 5.65624I −6.64285 − 6.52117I
u = 0.569840
a = −1.00000
b = −0.569840
−2.22691 −17.7140
12
III. I
u
3
= h−au + b, u
2
a + a
2
− au + 3u
2
+ a − u + 5, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
au
a
8
=
1
u
2
a
3
=
au + a
au
a
11
=
u
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
a
5
=
a
au
a
1
=
−1
0
a
6
=
−au + 2u
2
+ a − u + 4
−u
2
a + au + u
2
− u + 2
a
2
=
−u
2
a + 3u
2
+ a − 2u + 4
−u
2
a + au + u
2
− u + 2
a
9
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
2
a − 3au − 2u
2
+ a + 4u − 19
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
, c
10
, c
12
(u
3
− u
2
+ 1)
2
c
6
, c
11
(u
3
+ u
2
+ 2u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y − 1)
2
c
2
, c
5
, c
10
c
12
(y
3
− y
2
+ 2y − 1)
2
c
4
, c
9
y
6
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.947279 + 0.320410I
b = −0.215080 + 1.307140I
6.04826 −7.95781 + 0.50299I
u = 0.215080 + 1.307140I
a = −0.069840 + 0.424452I
b = −0.569840
1.91067 − 2.82812I −12.8076 + 6.7630I
u = 0.215080 − 1.307140I
a = 0.947279 − 0.320410I
b = −0.215080 − 1.307140I
6.04826 −7.95781 − 0.50299I
u = 0.215080 − 1.307140I
a = −0.069840 − 0.424452I
b = −0.569840
1.91067 + 2.82812I −12.8076 − 6.7630I
u = 0.569840
a = −0.37744 + 2.29387I
b = −0.215080 + 1.307140I
1.91067 + 2.82812I −16.7346 − 3.8621I
u = 0.569840
a = −0.37744 − 2.29387I
b = −0.215080 − 1.307140I
1.91067 − 2.82812I −16.7346 + 3.8621I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
40
+ 16u
39
+ ··· + 14u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
40
+ 4u
39
+ ··· − 6u − 1)
c
3
((u
3
− u
2
+ 2u − 1)
3
)(u
40
− 4u
39
+ ··· + 7u
2
− 1)
c
4
, c
9
u
9
(u
40
− u
39
+ ··· + 1536u + 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
40
+ 4u
39
+ ··· − 6u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
40
+ 16u
39
+ ··· + 14u + 1)
c
7
, c
8
((u
3
− u
2
+ 2u − 1)
3
)(u
40
− 4u
39
+ ··· + 2u − 1)
c
10
((u
3
− u
2
+ 1)
3
)(u
40
+ 4u
39
+ ··· + 7u − 2)
c
11
((u
3
+ u
2
+ 2u + 1)
3
)(u
40
− 4u
39
+ ··· + 2u − 1)
c
12
((u
3
− u
2
+ 1)
3
)(u
40
− 20u
39
+ ··· − 2480u + 20513)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
40
+ 20y
39
+ ··· − 142y + 1)
c
2
, c
5
((y
3
− y
2
+ 2y − 1)
3
)(y
40
− 16y
39
+ ··· − 14y + 1)
c
3
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
40
− 64y
39
+ ··· − 14y + 1)
c
4
, c
9
y
9
(y
40
− 49y
39
+ ··· − 655360y + 262144)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
40
+ 32y
39
+ ··· − 22y + 1)
c
10
((y
3
− y
2
+ 2y − 1)
3
)(y
40
− 48y
39
+ ··· − 109y + 4)
c
12
(y
3
− y
2
+ 2y − 1)
3
· (y
40
− 76y
39
+ ··· − 22287329974y + 420783169)
18
