12n
0352
(K12n
0352
)
A knot diagram
1
Linearized knot diagam
3 5 12 10 2 11 3 4 5 9 7 8
Solving Sequence
4,10 5,12
3 2 1 9 8 7 11 6
c
4
c
3
c
2
c
1
c
9
c
8
c
7
c
11
c
6
c
5
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
18
− 24u
17
+ ··· + 2b − 14, −11u
18
− 95u
17
+ ··· + 8a − 48, u
19
+ 9u
18
+ ··· + 44u + 8i
I
u
2
= h2u
15
− 2u
14
+ 9u
13
− 7u
12
+ 19u
11
− 14u
10
+ 23u
9
− 16u
8
+ 16u
7
− 12u
6
+ 7u
5
− 4u
4
+ 4u
3
+ b + 2u + 1,
u
15
+ 2u
14
+ ··· + a + 4,
u
16
− u
15
+ 5u
14
− 4u
13
+ 12u
12
− 9u
11
+ 17u
10
− 12u
9
+ 15u
8
− 11u
7
+ 9u
6
− 6u
5
+ 5u
4
− 2u
3
+ 3u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 35 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
18
− 24u
17
+ · · · + 2b − 14, −11u
18
− 95u
17
+ · · · + 8a −
48, u
19
+ 9u
18
+ · · · + 44u + 8i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−u
2
a
12
=
11
8
u
18
+
95
8
u
17
+ ··· +
67
2
u + 6
3
2
u
18
+ 12u
17
+ ··· +
67
2
u + 7
a
3
=
3
8
u
18
+
21
8
u
17
+ ··· −
73
4
u −
9
2
1
4
u
18
+
5
4
u
17
+ ··· − 9u − 1
a
2
=
1
8
u
18
+
3
8
u
17
+ ··· −
157
4
u −
19
2
1
4
u
18
+
5
4
u
17
+ ··· − 11u − 1
a
1
=
3
8
u
18
+
15
8
u
17
+ ··· −
85
2
u − 10
−u
18
−
19
2
u
17
+ ··· −
131
2
u − 13
a
9
=
−u
u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
1
8
u
18
+
11
8
u
17
+ ··· +
31
4
u + 1
−
1
4
u
18
−
7
4
u
17
+ ··· −
23
2
u − 3
a
11
=
−u
3
u
5
+ u
3
+ u
a
6
=
−
7
8
u
18
−
61
8
u
17
+ ··· −
129
4
u − 7
1
4
u
18
+
7
4
u
17
+ ··· +
57
2
u + 7
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −8u
18
−66u
17
−305u
16
−975u
15
−2367u
14
−4600u
13
−7383u
12
−10049u
11
−11804u
10
−
12134u
9
−11000u
8
−8792u
7
−6229u
6
−3932u
5
−2281u
4
−1230u
3
−584u
2
−212u − 42
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 50u
18
+ ··· + 9u − 1
c
2
, c
5
, c
7
u
19
+ 25u
17
+ ··· − u + 1
c
3
u
19
− 4u
18
+ ··· + 5u − 1
c
4
, c
9
u
19
+ 9u
18
+ ··· + 44u + 8
c
6
, c
11
u
19
+ 3u
18
+ ··· + 28u
2
+ 1
c
8
u
19
− 9u
18
+ ··· − 4116u + 1960
c
10
u
19
+ 9u
18
+ ··· − 112u − 64
c
12
u
19
− u
18
+ ··· − 104u + 193
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− 202y
18
+ ··· + 101y −1
c
2
, c
5
, c
7
y
19
+ 50y
18
+ ··· + 9y −1
c
3
y
19
− 2y
18
+ ··· + y −1
c
4
, c
9
y
19
+ 9y
18
+ ··· − 112y −64
c
6
, c
11
y
19
− 49y
18
+ ··· − 56y −1
c
8
y
19
− 91y
18
+ ··· + 10638096y −3841600
c
10
y
19
+ y
18
+ ··· − 35584y −4096
c
12
y
19
+ 57y
18
+ ··· + 85314y −37249
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.346161 + 0.956386I
a = 1.46736 + 0.28036I
b = 0.418807 − 0.392483I
−0.63169 + 2.23536I 1.18472 − 3.58566I
u = 0.346161 − 0.956386I
a = 1.46736 − 0.28036I
b = 0.418807 + 0.392483I
−0.63169 − 2.23536I 1.18472 + 3.58566I
u = −0.302273 + 1.061440I
a = −1.21082 + 0.86313I
b = −1.020080 + 0.439454I
−3.68086 − 0.50062I −7.63466 + 1.65999I
u = −0.302273 − 1.061440I
a = −1.21082 − 0.86313I
b = −1.020080 − 0.439454I
−3.68086 + 0.50062I −7.63466 − 1.65999I
u = −0.800161
a = 1.21345
b = 0.717995
−3.42695 −3.98880
u = −0.536472 + 1.088430I
a = −1.78211 + 0.61588I
b = −0.874784 − 0.944864I
−2.09426 − 6.57381I −0.65064 + 5.14701I
u = −0.536472 − 1.088430I
a = −1.78211 − 0.61588I
b = −0.874784 + 0.944864I
−2.09426 + 6.57381I −0.65064 − 5.14701I
u = −0.628002 + 0.338042I
a = 0.039501 − 0.508734I
b = −0.649313 + 0.779497I
0.03067 + 1.98876I 2.15859 − 2.97799I
u = −0.628002 − 0.338042I
a = 0.039501 + 0.508734I
b = −0.649313 − 0.779497I
0.03067 − 1.98876I 2.15859 + 2.97799I
u = −0.455443 + 1.212920I
a = 1.89609 − 0.58745I
b = 0.758353 + 0.039269I
−6.99596 − 4.49122I −9.18387 − 0.01438I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.455443 − 1.212920I
a = 1.89609 + 0.58745I
b = 0.758353 − 0.039269I
−6.99596 + 4.49122I −9.18387 + 0.01438I
u = 0.413988 + 0.520782I
a = 0.288663 − 0.426959I
b = −0.084685 + 0.588329I
0.657337 + 1.088170I 4.45180 − 5.21852I
u = 0.413988 − 0.520782I
a = 0.288663 + 0.426959I
b = −0.084685 − 0.588329I
0.657337 − 1.088170I 4.45180 + 5.21852I
u = −1.41615 + 0.03594I
a = 0.802995 − 0.190440I
b = 1.04189 + 1.04688I
18.3876 − 3.8412I −1.79058 + 1.95309I
u = −1.41615 − 0.03594I
a = 0.802995 + 0.190440I
b = 1.04189 − 1.04688I
18.3876 + 3.8412I −1.79058 − 1.95309I
u = −0.78204 + 1.42833I
a = 0.573067 − 0.779029I
b = 0.97983 − 1.07686I
14.2330 − 3.7497I −3.12835 + 0.88970I
u = −0.78204 − 1.42833I
a = 0.573067 + 0.779029I
b = 0.97983 + 1.07686I
14.2330 + 3.7497I −3.12835 − 0.88970I
u = −0.73969 + 1.46029I
a = 1.81853 − 0.33078I
b = 1.07098 + 0.99229I
13.8839 − 11.3207I −3.41262 + 4.66030I
u = −0.73969 − 1.46029I
a = 1.81853 + 0.33078I
b = 1.07098 − 0.99229I
13.8839 + 11.3207I −3.41262 − 4.66030I
6
II.
I
u
2
= h2u
15
−2u
14
+· · ·+b+1, u
15
+2u
14
+· · ·+a+4, u
16
−u
15
+· · ·+3u
2
+1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−u
2
a
12
=
−u
15
− 2u
14
+ ··· + u − 4
−2u
15
+ 2u
14
+ ··· − 2u − 1
a
3
=
−u
15
− 2u
13
+ ··· + u − 3
−2u
15
+ 3u
14
+ ··· − 2u − 1
a
2
=
−u
15
− u
13
+ ··· + 2u − 3
−3u
15
+ 4u
14
+ ··· − 2u − 1
a
1
=
−u
15
− u
14
+ ··· + 2u − 3
−3u
15
+ 3u
14
+ ··· − 2u − 1
a
9
=
−u
u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
u
14
− 2u
13
+ ··· − 3u + 1
u
15
− u
14
+ ··· + u + 1
a
11
=
−u
3
u
5
+ u
3
+ u
a
6
=
2u
14
− 3u
13
+ ··· − 3u + 2
u
15
+ 3u
13
+ ··· + 2u
2
+ 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
15
− 8u
14
+ 36u
13
− 27u
12
+ 74u
11
− 55u
10
+ 90u
9
− 67u
8
+
63u
7
− 54u
6
+ 32u
5
− 18u
4
+ 16u
3
+ 3u
2
+ 10u − 3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 11u
15
+ ··· − 9u + 1
c
2
, c
7
u
16
− u
15
+ ··· − u + 1
c
3
u
16
+ 7u
15
+ ··· + 5u + 1
c
4
u
16
− u
15
+ ··· + 3u
2
+ 1
c
5
u
16
+ u
15
+ ··· + u + 1
c
6
u
16
+ 2u
15
+ ··· + 2u + 1
c
8
u
16
− u
15
+ ··· + 4u
2
+ 1
c
9
u
16
+ u
15
+ ··· + 3u
2
+ 1
c
10
u
16
+ 9u
15
+ ··· + 6u + 1
c
11
u
16
− 2u
15
+ ··· − 2u + 1
c
12
u
16
+ 7u
14
+ ··· + 4u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 5y
15
+ ··· + 5y + 1
c
2
, c
5
, c
7
y
16
+ 11y
15
+ ··· + 9y + 1
c
3
y
16
− y
15
+ ··· + y + 1
c
4
, c
9
y
16
+ 9y
15
+ ··· + 6y + 1
c
6
, c
11
y
16
− 12y
15
+ ··· − 6y + 1
c
8
y
16
− 7y
15
+ ··· + 8y + 1
c
10
y
16
+ y
15
+ ··· + 2y + 1
c
12
y
16
+ 14y
15
+ ··· + 8y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.335104 + 0.911069I
a = 2.45662 − 0.85727I
b = 0.921522 + 0.158810I
−7.02394 + 1.39379I −9.47279 + 0.73629I
u = 0.335104 − 0.911069I
a = 2.45662 + 0.85727I
b = 0.921522 − 0.158810I
−7.02394 − 1.39379I −9.47279 − 0.73629I
u = −0.379248 + 1.028620I
a = −0.646901 + 1.136830I
b = −1.14774 + 0.83032I
−3.77746 + 0.63307I −8.74418 − 3.43739I
u = −0.379248 − 1.028620I
a = −0.646901 − 1.136830I
b = −1.14774 − 0.83032I
−3.77746 − 0.63307I −8.74418 + 3.43739I
u = 0.814712 + 0.313052I
a = −0.637439 + 0.085695I
b = −0.219047 + 0.658555I
−2.04483 + 1.12270I 0.01080 − 2.10787I
u = 0.814712 − 0.313052I
a = −0.637439 − 0.085695I
b = −0.219047 − 0.658555I
−2.04483 − 1.12270I 0.01080 + 2.10787I
u = −0.532348 + 1.055360I
a = −1.92201 + 0.51905I
b = −0.99317 − 1.17536I
−2.65168 − 7.12816I −8.0544 + 12.2171I
u = −0.532348 − 1.055360I
a = −1.92201 − 0.51905I
b = −0.99317 + 1.17536I
−2.65168 + 7.12816I −8.0544 − 12.2171I
u = −0.569437 + 0.482937I
a = 0.475047 + 0.043194I
b = −0.782251 + 1.053490I
−0.93348 + 2.66812I −2.81990 − 6.15304I
u = −0.569437 − 0.482937I
a = 0.475047 − 0.043194I
b = −0.782251 − 1.053490I
−0.93348 − 2.66812I −2.81990 + 6.15304I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.182107 + 0.721236I
a = −2.18930 + 1.30740I
b = −0.958246 − 0.624440I
−2.38549 − 3.31503I −6.02251 + 3.66103I
u = −0.182107 − 0.721236I
a = −2.18930 − 1.30740I
b = −0.958246 + 0.624440I
−2.38549 + 3.31503I −6.02251 − 3.66103I
u = 0.614974 + 1.102350I
a = 0.301915 + 0.157095I
b = 0.241401 − 0.693802I
−4.28291 + 4.20394I −2.40515 − 3.76480I
u = 0.614974 − 1.102350I
a = 0.301915 − 0.157095I
b = 0.241401 + 0.693802I
−4.28291 − 4.20394I −2.40515 + 3.76480I
u = 0.398350 + 1.236570I
a = −1.83794 + 0.04383I
b = −0.562464 + 0.307777I
−6.50903 + 5.01414I −1.49188 − 7.53690I
u = 0.398350 − 1.236570I
a = −1.83794 − 0.04383I
b = −0.562464 − 0.307777I
−6.50903 − 5.01414I −1.49188 + 7.53690I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
16
− 11u
15
+ ··· − 9u + 1)(u
19
+ 50u
18
+ ··· + 9u − 1)
c
2
, c
7
(u
16
− u
15
+ ··· − u + 1)(u
19
+ 25u
17
+ ··· − u + 1)
c
3
(u
16
+ 7u
15
+ ··· + 5u + 1)(u
19
− 4u
18
+ ··· + 5u − 1)
c
4
(u
16
− u
15
+ ··· + 3u
2
+ 1)(u
19
+ 9u
18
+ ··· + 44u + 8)
c
5
(u
16
+ u
15
+ ··· + u + 1)(u
19
+ 25u
17
+ ··· − u + 1)
c
6
(u
16
+ 2u
15
+ ··· + 2u + 1)(u
19
+ 3u
18
+ ··· + 28u
2
+ 1)
c
8
(u
16
− u
15
+ ··· + 4u
2
+ 1)(u
19
− 9u
18
+ ··· − 4116u + 1960)
c
9
(u
16
+ u
15
+ ··· + 3u
2
+ 1)(u
19
+ 9u
18
+ ··· + 44u + 8)
c
10
(u
16
+ 9u
15
+ ··· + 6u + 1)(u
19
+ 9u
18
+ ··· − 112u − 64)
c
11
(u
16
− 2u
15
+ ··· − 2u + 1)(u
19
+ 3u
18
+ ··· + 28u
2
+ 1)
c
12
(u
16
+ 7u
14
+ ··· + 4u
2
+ 1)(u
19
− u
18
+ ··· − 104u + 193)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
16
− 5y
15
+ ··· + 5y + 1)(y
19
− 202y
18
+ ··· + 101y −1)
c
2
, c
5
, c
7
(y
16
+ 11y
15
+ ··· + 9y + 1)(y
19
+ 50y
18
+ ··· + 9y −1)
c
3
(y
16
− y
15
+ ··· + y + 1)(y
19
− 2y
18
+ ··· + y −1)
c
4
, c
9
(y
16
+ 9y
15
+ ··· + 6y + 1)(y
19
+ 9y
18
+ ··· − 112y −64)
c
6
, c
11
(y
16
− 12y
15
+ ··· − 6y + 1)(y
19
− 49y
18
+ ··· − 56y −1)
c
8
(y
16
− 7y
15
+ ··· + 8y + 1)
· (y
19
− 91y
18
+ ··· + 10638096y −3841600)
c
10
(y
16
+ y
15
+ ··· + 2y + 1)(y
19
+ y
18
+ ··· − 35584y −4096)
c
12
(y
16
+ 14y
15
+ ··· + 8y + 1)(y
19
+ 57y
18
+ ··· + 85314y −37249)
13
