12n
0822
(K12n
0822
)
A knot diagram
1
Linearized knot diagam
4 6 10 8 2 11 12 5 6 4 7 8
Solving Sequence
7,11
12
4,8
5 1 6 10 3 2 9
c
11
c
7
c
4
c
12
c
6
c
10
c
3
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
15
+ 14u
14
+ ··· + 2b + 10, −21u
15
+ 100u
14
+ ··· + 4a + 68, u
16
− 6u
15
+ ··· + 2u + 4i
I
u
2
= h475u
4
a
3
− 65u
4
a
2
+ ··· + 311a + 1939, −u
4
a
3
+ 2u
4
a
2
+ ··· − 2a + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
I
u
3
= hu
4
− 3u
2
+ b + 1, u
7
− 6u
5
+ u
4
+ 11u
3
− 3u
2
+ a − 6u + 1,
u
9
+ u
8
− 6u
7
− 5u
6
+ 12u
5
+ 7u
4
− 9u
3
− 2u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 45 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−3u
15
+ 14u
14
+ · · · + 2b + 10, −21u
15
+ 100u
14
+ · · · + 4a +
68, u
16
− 6u
15
+ · · · + 2u + 4i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
4
=
21
4
u
15
− 25u
14
+ ··· −
99
4
u − 17
3
2
u
15
− 7u
14
+ ··· −
13
2
u − 5
a
8
=
−u
−u
3
+ u
a
5
=
13
4
u
15
− 16u
14
+ ··· −
67
4
u − 11
−
3
2
u
15
+ 5u
14
+ ··· −
1
2
u + 1
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
u
a
10
=
3
2
u
15
−
11
2
u
14
+ ··· −
3
2
u −
3
2
7
2
u
15
− 15u
14
+ ··· −
15
2
u − 8
a
3
=
25
2
u
15
−
113
2
u
14
+ ··· −
85
2
u −
67
2
19
2
u
15
− 43u
14
+ ··· −
61
2
u − 26
a
2
=
11
2
u
15
−
51
2
u
14
+ ··· −
43
2
u −
31
2
5
2
u
15
− 12u
14
+ ··· −
19
2
u − 8
a
9
=
9
2
u
15
−
41
2
u
14
+ ··· −
29
2
u −
23
2
13
2
u
15
− 30u
14
+ ··· −
41
2
u − 18
(ii) Obstruction class = −1
(iii) Cusp Shapes = −11u
15
+ 53u
14
− 38u
13
− 147u
12
+ 140u
11
+ 198u
10
+ 121u
9
−
664u
8
− 111u
7
+ 438u
6
+ 477u
5
− 133u
4
− 460u
3
+ 26u
2
+ 46u + 30
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 2u
15
+ ··· − 6u + 1
c
2
, c
5
u
16
+ 13u
15
+ ··· + 208u + 32
c
3
, c
4
, c
8
c
10
u
16
− u
15
+ ··· + u + 1
c
6
, c
7
, c
11
c
12
u
16
− 6u
15
+ ··· + 2u + 4
c
9
u
16
− u
15
+ ··· − 15u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 22y
15
+ ··· − 52y + 1
c
2
, c
5
y
16
− 5y
15
+ ··· − 5888y + 1024
c
3
, c
4
, c
8
c
10
y
16
− 19y
15
+ ··· − 7y + 1
c
6
, c
7
, c
11
c
12
y
16
− 18y
15
+ ··· − 60y + 16
c
9
y
16
+ 27y
15
+ ··· − 229y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.526647 + 0.921315I
a = −0.158275 + 0.852282I
b = 1.61938 + 0.26122I
12.5756 + 7.7658I −0.37746 − 4.72182I
u = −0.526647 − 0.921315I
a = −0.158275 − 0.852282I
b = 1.61938 − 0.26122I
12.5756 − 7.7658I −0.37746 + 4.72182I
u = −0.641134 + 0.907508I
a = −0.131631 − 0.748203I
b = −1.57975 + 0.10158I
12.24930 − 1.78364I −0.280416 + 0.020773I
u = −0.641134 − 0.907508I
a = −0.131631 + 0.748203I
b = −1.57975 − 0.10158I
12.24930 + 1.78364I −0.280416 − 0.020773I
u = −0.860541
a = 0.313373
b = 0.501732
−1.66742 −3.52050
u = 1.384200 + 0.067843I
a = −0.23731 − 1.57085I
b = 0.063351 − 0.856630I
−5.42049 − 2.20486I −9.58545 + 3.34239I
u = 1.384200 − 0.067843I
a = −0.23731 + 1.57085I
b = 0.063351 + 0.856630I
−5.42049 + 2.20486I −9.58545 − 3.34239I
u = 0.536874
a = −1.68270
b = 0.418387
−2.43643 6.61560
u = 1.54541 + 0.35416I
a = −0.78815 + 1.39880I
b = −1.60127 + 0.41996I
5.91450 − 12.45070I −3.55733 + 5.83875I
u = 1.54541 − 0.35416I
a = −0.78815 − 1.39880I
b = −1.60127 − 0.41996I
5.91450 + 12.45070I −3.55733 − 5.83875I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.252945 + 0.318927I
a = 0.736930 − 0.632401I
b = −0.145126 − 0.490024I
−0.252639 + 0.904244I −5.12440 − 7.64172I
u = −0.252945 − 0.318927I
a = 0.736930 + 0.632401I
b = −0.145126 + 0.490024I
−0.252639 − 0.904244I −5.12440 + 7.64172I
u = −1.65011
a = −0.204743
b = −0.895334
−10.2599 −6.17920
u = 1.62313 + 0.35288I
a = 0.898424 − 0.854793I
b = 1.47438 − 0.04596I
4.87833 − 2.98005I −2.44851 + 1.44008I
u = 1.62313 − 0.35288I
a = 0.898424 + 0.854793I
b = 1.47438 + 0.04596I
4.87833 + 2.98005I −2.44851 − 1.44008I
u = 1.70974
a = −0.565907
b = −0.686707
−10.9818 0.831250
6
II. I
u
2
= h475u
4
a
3
− 65u
4
a
2
+ · · · + 311a + 1939, −u
4
a
3
+ 2u
4
a
2
+ · · · − 2a +
1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
4
=
a
−0.304292a
3
u
4
+ 0.0416400a
2
u
4
+ ··· − 0.199231a − 1.24215
a
8
=
−u
−u
3
+ u
a
5
=
−0.136451a
3
u
4
+ 0.502883a
2
u
4
+ ··· + 0.516976a + 0.152466
−0.431134a
3
u
4
+ 0.227418a
2
u
4
+ ··· + 0.450352a − 1.86099
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
u
a
10
=
−0.0743113a
3
u
4
+ 0.452274a
2
u
4
+ ··· − 0.225496a + 1.03139
0.394619a
3
u
4
+ 0.977578a
2
u
4
+ ··· − 0.354260a + 2.59193
a
3
=
−0.274183a
3
u
4
+ 1.04805a
2
u
4
+ ··· + 0.616272a − 0.125561
0.0204997a
3
u
4
+ 1.32351a
2
u
4
+ ··· + 0.682896a + 0.887892
a
2
=
−0.0999359a
3
u
4
+ 0.297886a
2
u
4
+ ··· + 0.420884a + 0.421525
0.194747a
3
u
4
+ 0.573350a
2
u
4
+ ··· + 0.487508a + 1.43498
a
9
=
0.281230a
3
u
4
− 0.280589a
2
u
4
+ ··· − 0.319026a + 0.493274
0.750160a
3
u
4
+ 0.244715a
2
u
4
+ ··· − 0.447790a + 2.05381
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1108
1561
u
4
a
3
−
900
1561
u
4
a
2
+ ··· +
944
1561
a −
1002
223
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
− 7u
19
+ ··· − 454u + 73
c
2
, c
5
(u
2
− u + 1)
10
c
3
, c
4
, c
8
c
10
u
20
+ u
19
+ ··· − 40u + 7
c
6
, c
7
, c
11
c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
4
c
9
u
20
+ 3u
19
+ ··· + 60u + 7
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 19y
19
+ ··· + 54932y + 5329
c
2
, c
5
(y
2
+ y + 1)
10
c
3
, c
4
, c
8
c
10
y
20
− 21y
19
+ ··· − 1404y + 49
c
6
, c
7
, c
11
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
4
c
9
y
20
+ 27y
19
+ ··· + 2672y + 49
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = 0.182571 + 1.126310I
b = −1.267850 − 0.008241I
2.53372 + 2.02988I −1.48114 − 3.46410I
u = 1.21774
a = 0.182571 − 1.126310I
b = −1.267850 + 0.008241I
2.53372 − 2.02988I −1.48114 + 3.46410I
u = 1.21774
a = 1.26348 + 1.37832I
b = 1.65127 + 0.67233I
2.53372 + 2.02988I −1.48114 − 3.46410I
u = 1.21774
a = 1.26348 − 1.37832I
b = 1.65127 − 0.67233I
2.53372 − 2.02988I −1.48114 + 3.46410I
u = 0.309916 + 0.549911I
a = −0.208082 + 0.883906I
b = −0.825780 + 0.914155I
4.60570 − 3.56046I −0.51511 + 7.89475I
u = 0.309916 + 0.549911I
a = 0.423531 + 0.774423I
b = −1.51045 − 0.06114I
4.60570 + 0.49930I −0.515115 + 0.966547I
u = 0.309916 + 0.549911I
a = 0.96461 − 1.56415I
b = 0.628697 + 0.178647I
4.60570 + 0.49930I −0.515115 + 0.966547I
u = 0.309916 + 0.549911I
a = −1.16991 − 1.69121I
b = 1.368420 − 0.209289I
4.60570 − 3.56046I −0.51511 + 7.89475I
u = 0.309916 − 0.549911I
a = −0.208082 − 0.883906I
b = −0.825780 − 0.914155I
4.60570 + 3.56046I −0.51511 − 7.89475I
u = 0.309916 − 0.549911I
a = 0.423531 − 0.774423I
b = −1.51045 + 0.06114I
4.60570 − 0.49930I −0.515115 − 0.966547I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.309916 − 0.549911I
a = 0.96461 + 1.56415I
b = 0.628697 − 0.178647I
4.60570 − 0.49930I −0.515115 − 0.966547I
u = 0.309916 − 0.549911I
a = −1.16991 + 1.69121I
b = 1.368420 + 0.209289I
4.60570 + 3.56046I −0.51511 − 7.89475I
u = −1.41878 + 0.21917I
a = −0.639121 − 0.914346I
b = −0.205274 − 0.110634I
−0.93776 + 2.37095I −4.74431 − 0.03448I
u = −1.41878 + 0.21917I
a = 0.97762 + 1.05920I
b = 1.47248 + 0.31606I
−0.93776 + 2.37095I −4.74431 − 0.03448I
u = −1.41878 + 0.21917I
a = −0.53015 − 1.69434I
b = −1.341620 − 0.334073I
−0.93776 + 6.43072I −4.74431 − 6.96269I
u = −1.41878 + 0.21917I
a = 0.23545 + 1.91506I
b = 0.53011 + 1.32879I
−0.93776 + 6.43072I −4.74431 − 6.96269I
u = −1.41878 − 0.21917I
a = −0.639121 + 0.914346I
b = −0.205274 + 0.110634I
−0.93776 − 2.37095I −4.74431 + 0.03448I
u = −1.41878 − 0.21917I
a = 0.97762 − 1.05920I
b = 1.47248 − 0.31606I
−0.93776 − 2.37095I −4.74431 + 0.03448I
u = −1.41878 − 0.21917I
a = −0.53015 + 1.69434I
b = −1.341620 + 0.334073I
−0.93776 − 6.43072I −4.74431 + 6.96269I
u = −1.41878 − 0.21917I
a = 0.23545 − 1.91506I
b = 0.53011 − 1.32879I
−0.93776 − 6.43072I −4.74431 + 6.96269I
11
III.
I
u
3
= hu
4
−3u
2
+b+1, u
7
−6u
5
+u
4
+11u
3
−3u
2
+a−6u+1, u
9
+u
8
+· · ·+u−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
4
=
−u
7
+ 6u
5
− u
4
− 11u
3
+ 3u
2
+ 6u − 1
−u
4
+ 3u
2
− 1
a
8
=
−u
−u
3
+ u
a
5
=
−u
7
+ 5u
5
− u
4
− 8u
3
+ 3u
2
+ 5u − 1
−u
7
+ 4u
5
− u
4
− 4u
3
+ 3u
2
+ u − 1
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
u
a
10
=
2u
8
+ u
7
− 11u
6
− 5u
5
+ 18u
4
+ 8u
3
− 8u
2
− 5u
u
8
− 6u
6
+ 11u
4
− 6u
2
+ 1
a
3
=
u
7
− 5u
5
+ u
4
+ 7u
3
− 2u
2
− 3u − 1
u
7
− 5u
5
+ u
4
+ 7u
3
− 3u
2
− 3u + 1
a
2
=
u
7
− 5u
5
+ 7u
3
− 3u − 1
u
7
− 5u
5
+ 7u
3
− u
2
− 3u + 1
a
9
=
u
8
+ u
7
− 6u
6
− 6u
5
+ 11u
4
+ 11u
3
− 6u
2
− 6u
−u
6
− u
5
+ 4u
4
+ 3u
3
− 4u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
8
− u
7
− 10u
6
+ 6u
5
+ 29u
4
− 17u
3
− 27u
2
+ 18u − 3
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 2u
8
+ 3u
7
+ 2u
6
− u
5
− 4u
4
− 4u
3
− u
2
+ 2u + 1
c
2
u
9
+ 2u
8
− u
7
− 4u
6
− 4u
5
− u
4
+ 2u
3
+ 3u
2
+ 2u + 1
c
3
, c
8
u
9
+ u
8
− 5u
7
− 5u
6
+ 10u
5
+ 10u
4
− 9u
3
− 8u
2
+ 3u + 1
c
4
, c
10
u
9
− u
8
− 5u
7
+ 5u
6
+ 10u
5
− 10u
4
− 9u
3
+ 8u
2
+ 3u − 1
c
5
u
9
− 2u
8
− u
7
+ 4u
6
− 4u
5
+ u
4
+ 2u
3
− 3u
2
+ 2u − 1
c
6
, c
7
u
9
− u
8
− 6u
7
+ 5u
6
+ 12u
5
− 7u
4
− 9u
3
+ 2u
2
+ u + 1
c
9
u
9
+ u
8
+ 2u
7
− u
4
− 7u
3
− 5u
2
− 3u − 1
c
11
, c
12
u
9
+ u
8
− 6u
7
− 5u
6
+ 12u
5
+ 7u
4
− 9u
3
− 2u
2
+ u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 2y
8
− y
7
− 2y
6
+ y
5
+ 4y
4
− 9y
2
+ 6y −1
c
2
, c
5
y
9
− 6y
8
+ 9y
7
− 4y
5
− y
4
+ 2y
3
+ y
2
− 2y −1
c
3
, c
4
, c
8
c
10
y
9
− 11y
8
+ ··· + 25y −1
c
6
, c
7
, c
11
c
12
y
9
− 13y
8
+ 70y
7
− 201y
6
+ 328y
5
− 295y
4
+ 123y
3
− 8y
2
− 3y −1
c
9
y
9
+ 3y
8
+ 4y
7
− 12y
6
− 24y
5
− 11y
4
+ 39y
3
+ 15y
2
− y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.166850 + 0.186778I
a = 0.546502 − 0.075866I
b = 1.40995 + 0.15111I
1.80577 + 0.24484I −3.61013 + 0.69147I
u = 1.166850 − 0.186778I
a = 0.546502 + 0.075866I
b = 1.40995 − 0.15111I
1.80577 − 0.24484I −3.61013 − 0.69147I
u = −0.701278
a = −1.11470
b = 0.233515
−2.77702 −18.0900
u = −1.45070 + 0.17281I
a = 1.08872 + 1.49683I
b = 1.171140 + 0.576247I
−0.36210 + 4.32575I −2.99049 − 3.69672I
u = −1.45070 − 0.17281I
a = 1.08872 − 1.49683I
b = 1.171140 − 0.576247I
−0.36210 − 4.32575I −2.99049 + 3.69672I
u = 0.169241 + 0.365052I
a = 0.44926 + 2.73952I
b = −1.309540 + 0.396544I
5.17385 − 2.30230I 3.82886 + 2.71981I
u = 0.169241 − 0.365052I
a = 0.44926 − 2.73952I
b = −1.309540 − 0.396544I
5.17385 + 2.30230I 3.82886 − 2.71981I
u = 1.68460
a = −0.119390
b = −0.539942
−11.4397 −17.8090
u = −1.75412
a = −0.934890
b = −1.23668
−8.88794 −1.55820
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
9
+ 2u
8
+ 3u
7
+ 2u
6
− u
5
− 4u
4
− 4u
3
− u
2
+ 2u + 1)
· (u
16
+ 2u
15
+ ··· − 6u + 1)(u
20
− 7u
19
+ ··· − 454u + 73)
c
2
(u
2
− u + 1)
10
(u
9
+ 2u
8
− u
7
− 4u
6
− 4u
5
− u
4
+ 2u
3
+ 3u
2
+ 2u + 1)
· (u
16
+ 13u
15
+ ··· + 208u + 32)
c
3
, c
8
(u
9
+ u
8
− 5u
7
− 5u
6
+ 10u
5
+ 10u
4
− 9u
3
− 8u
2
+ 3u + 1)
· (u
16
− u
15
+ ··· + u + 1)(u
20
+ u
19
+ ··· − 40u + 7)
c
4
, c
10
(u
9
− u
8
− 5u
7
+ 5u
6
+ 10u
5
− 10u
4
− 9u
3
+ 8u
2
+ 3u − 1)
· (u
16
− u
15
+ ··· + u + 1)(u
20
+ u
19
+ ··· − 40u + 7)
c
5
(u
2
− u + 1)
10
(u
9
− 2u
8
− u
7
+ 4u
6
− 4u
5
+ u
4
+ 2u
3
− 3u
2
+ 2u − 1)
· (u
16
+ 13u
15
+ ··· + 208u + 32)
c
6
, c
7
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
4
· (u
9
− u
8
− 6u
7
+ 5u
6
+ 12u
5
− 7u
4
− 9u
3
+ 2u
2
+ u + 1)
· (u
16
− 6u
15
+ ··· + 2u + 4)
c
9
(u
9
+ u
8
+ ··· − 3u − 1)(u
16
− u
15
+ ··· − 15u + 1)
· (u
20
+ 3u
19
+ ··· + 60u + 7)
c
11
, c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
4
· (u
9
+ u
8
− 6u
7
− 5u
6
+ 12u
5
+ 7u
4
− 9u
3
− 2u
2
+ u − 1)
· (u
16
− 6u
15
+ ··· + 2u + 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
9
+ 2y
8
− y
7
− 2y
6
+ y
5
+ 4y
4
− 9y
2
+ 6y −1)
· (y
16
+ 22y
15
+ ··· − 52y + 1)(y
20
+ 19y
19
+ ··· + 54932y + 5329)
c
2
, c
5
(y
2
+ y + 1)
10
(y
9
− 6y
8
+ 9y
7
− 4y
5
− y
4
+ 2y
3
+ y
2
− 2y −1)
· (y
16
− 5y
15
+ ··· − 5888y + 1024)
c
3
, c
4
, c
8
c
10
(y
9
− 11y
8
+ ··· + 25y −1)(y
16
− 19y
15
+ ··· − 7y + 1)
· (y
20
− 21y
19
+ ··· − 1404y + 49)
c
6
, c
7
, c
11
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
4
· (y
9
− 13y
8
+ 70y
7
− 201y
6
+ 328y
5
− 295y
4
+ 123y
3
− 8y
2
− 3y −1)
· (y
16
− 18y
15
+ ··· − 60y + 16)
c
9
(y
9
+ 3y
8
+ 4y
7
− 12y
6
− 24y
5
− 11y
4
+ 39y
3
+ 15y
2
− y −1)
· (y
16
+ 27y
15
+ ··· − 229y + 1)(y
20
+ 27y
19
+ ··· + 2672y + 49)
17
