12n
0245
(K12n
0245
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 11 10 12 3 5 6 8 7
Solving Sequence
5,11 3,6
2 1 4 10 7 9 8 12
c
5
c
2
c
1
c
4
c
10
c
6
c
9
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
− u
8
+ 5u
7
− 7u
6
+ 9u
5
− 16u
4
+ 7u
3
− 10u
2
+ 4b + 3u + 3,
u
9
+ 3u
8
+ 5u
7
+ 5u
6
+ 5u
5
− 8u
4
− 5u
3
− 22u
2
+ 8a − u − 5,
u
10
+ 4u
8
− 2u
7
+ 6u
6
− 7u
5
+ 3u
4
− 7u
3
+ u
2
− 2u − 1i
I
u
2
= h22u
15
+ 71u
14
+ ··· + 125b + 158, 121u
15
+ 203u
14
+ ··· + 125a − 6, u
16
+ 2u
15
+ ··· + 2u + 1i
I
u
3
= hb + 1, u
2
+ 2a + u + 3, u
3
+ 2u − 1i
I
u
4
= hb + 1, u
3
+ u
2
+ a + u + 2, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 4 irreducible components of dim
C
= 0, with total 33 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
= hu
9
− u
8
+ · · · + 4b + 3, u
9
+ 3u
8
+ · · · + 8a − 5, u
10
+ 4u
8
+ · · · − 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−
1
8
u
9
−
3
8
u
8
+ ··· +
1
8
u +
5
8
−
1
4
u
9
+
1
4
u
8
+ ··· −
3
4
u −
3
4
a
6
=
1
u
2
a
2
=
−
3
8
u
9
−
1
8
u
8
+ ··· −
5
8
u −
1
8
−
1
4
u
9
+
1
4
u
8
+ ··· −
3
4
u −
3
4
a
1
=
−u
3
− 2u
u
9
+ 3u
7
− 2u
6
+ 2u
5
− 5u
4
− 3u
3
− 2u
2
a
4
=
−
13
8
u
9
+
1
8
u
8
+ ··· −
3
8
u +
1
8
−
5
4
u
9
+
1
4
u
8
+ ··· −
3
4
u −
3
4
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
−1
u
8
+ 3u
6
− 2u
5
+ 3u
4
− 5u
3
− u
2
− 2u − 1
a
12
=
−u
u
9
+ 3u
7
− 2u
6
+ 3u
5
− 5u
4
− u
3
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
15
16
u
9
−
51
16
u
8
+
51
16
u
7
−
197
16
u
6
+
171
16
u
5
−
35
2
u
4
+
293
16
u
3
−
53
8
u
2
+
145
16
u −
219
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 14u
9
+ ··· + 625u + 16
c
2
, c
4
u
10
− 2u
9
− 5u
8
+ 7u
7
+ 14u
6
− 36u
4
− 8u
3
+ 42u
2
− 17u − 4
c
3
, c
8
u
10
+ 3u
9
+ ··· + 88u + 32
c
5
, c
6
, c
7
c
10
, c
11
, c
12
u
10
+ 4u
8
− 2u
7
+ 6u
6
− 7u
5
+ 3u
4
− 7u
3
+ u
2
− 2u − 1
c
9
u
10
− 6u
9
+ 9u
8
+ 2u
7
+ 7u
6
− 51u
5
+ 50u
4
− 20u
3
+ 7u
2
− 4u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 34y
9
+ ··· − 333665y + 256
c
2
, c
4
y
10
− 14y
9
+ ··· − 625y + 16
c
3
, c
8
y
10
− 15y
9
+ ··· − 3392y + 1024
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
10
+ 8y
9
+ ··· − 6y + 1
c
9
y
10
− 18y
9
+ ··· − 72y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.408860 + 1.019830I
a = −0.623231 − 0.763538I
b = −0.84308 + 1.25661I
0.76752 + 5.23818I −10.88397 − 7.30305I
u = −0.408860 − 1.019830I
a = −0.623231 + 0.763538I
b = −0.84308 − 1.25661I
0.76752 − 5.23818I −10.88397 + 7.30305I
u = 1.10481
a = 1.89244
b = 1.98176
−16.4276 −16.3110
u = 0.331850 + 0.653227I
a = −0.698518 + 0.937685I
b = −1.56733 − 0.09555I
−1.76206 − 1.44138I −14.1408 + 4.6887I
u = 0.331850 − 0.653227I
a = −0.698518 − 0.937685I
b = −1.56733 + 0.09555I
−1.76206 + 1.44138I −14.1408 − 4.6887I
u = 0.24366 + 1.40906I
a = 0.325341 − 0.085189I
b = 0.683278 − 0.384377I
8.38588 − 4.49014I −3.00164 + 0.77612I
u = 0.24366 − 1.40906I
a = 0.325341 + 0.085189I
b = 0.683278 + 0.384377I
8.38588 + 4.49014I −3.00164 − 0.77612I
u = −0.56211 + 1.36382I
a = 0.87591 + 1.14263I
b = 1.81943 − 0.43136I
−7.9485 + 11.8019I −10.95990 − 5.74637I
u = −0.56211 − 1.36382I
a = 0.87591 − 1.14263I
b = 1.81943 + 0.43136I
−7.9485 − 11.8019I −10.95990 + 5.74637I
u = −0.313895
a = 0.848562
b = −0.166359
−0.552314 −17.9670
5
II. I
u
2
= h22u
15
+ 71u
14
+ · · · + 125b + 158, 121u
15
+ 203u
14
+ · · · + 125a −
6, u
16
+ 2u
15
+ · · · + 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−0.968000u
15
− 1.62400u
14
+ ··· − 0.976000u + 0.0480000
−0.176000u
15
− 0.568000u
14
+ ··· − 0.632000u − 1.26400
a
6
=
1
u
2
a
2
=
−1.14400u
15
− 2.19200u
14
+ ··· − 1.60800u − 1.21600
−0.176000u
15
− 0.568000u
14
+ ··· − 0.632000u − 1.26400
a
1
=
−1.93600u
15
− 3.24800u
14
+ ··· − 1.95200u − 2.90400
−0.496000u
15
− 0.328000u
14
+ ··· + 1.12800u − 0.744000
a
4
=
−2.03200u
15
− 2.37600u
14
+ ··· − 3.02400u − 1.04800
−1.03200u
15
− 1.37600u
14
+ ··· − 2.02400u − 2.04800
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
0.784000u
15
+ 0.712000u
14
+ ··· + 3.08800u + 1.17600
0.904000u
15
+ 0.872000u
14
+ ··· + 0.928000u + 1.85600
a
12
=
−0.144000u
15
− 1.19200u
14
+ ··· − 0.608000u − 1.21600
1.79200u
15
+ 2.05600u
14
+ ··· + 3.34400u + 1.68800
(ii) Obstruction class = −1
(iii) Cusp Shapes =
121
125
u
15
−
47
125
u
14
+ ··· +
622
125
u −
1506
125
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
8
+ 13u
7
+ 68u
6
+ 185u
5
+ 287u
4
+ 249u
3
+ 77u
2
+ 3u + 1)
2
c
2
, c
4
(u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1)
2
c
3
, c
8
(u
8
− u
7
− 7u
6
+ 4u
5
+ 16u
4
+ 3u
3
− 9u
2
+ 8u − 4)
2
c
5
, c
6
, c
7
c
10
, c
11
, c
12
u
16
+ 2u
15
+ ··· + 2u + 1
c
9
(u
8
+ 2u
7
− 7u
6
− 12u
5
+ 5u
4
− 3u
3
− 2u
2
− 2u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
8
− 33y
7
+ ··· + 145y + 1)
2
c
2
, c
4
(y
8
− 13y
7
+ 68y
6
− 185y
5
+ 287y
4
− 249y
3
+ 77y
2
− 3y + 1)
2
c
3
, c
8
(y
8
− 15y
7
+ 89y
6
− 252y
5
+ 366y
4
− 305y
3
− 95y
2
+ 8y + 16)
2
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
16
+ 10y
15
+ ··· + 12y
2
+ 1
c
9
(y
8
− 18y
7
+ 107y
6
− 206y
5
− 9y
4
− 91y
3
+ 2y
2
− 8y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.152816 + 1.034440I
a = 1.33690 − 2.28052I
b = −0.736738
2.18625 −12.78715 + 0.I
u = −0.152816 − 1.034440I
a = 1.33690 + 2.28052I
b = −0.736738
2.18625 −12.78715 + 0.I
u = 0.316903 + 0.894740I
a = −0.695071 + 1.182330I
b = −1.178780 − 0.606721I
−1.14222 − 1.62541I −14.5850 + 1.4256I
u = 0.316903 − 0.894740I
a = −0.695071 − 1.182330I
b = −1.178780 + 0.606721I
−1.14222 + 1.62541I −14.5850 − 1.4256I
u = −1.103920 + 0.013257I
a = 1.85395 + 0.11352I
b = 1.89776 + 0.22684I
−12.14610 − 5.90409I −13.72541 + 2.82977I
u = −1.103920 − 0.013257I
a = 1.85395 − 0.11352I
b = 1.89776 − 0.22684I
−12.14610 + 5.90409I −13.72541 − 2.82977I
u = −0.125010 + 1.233150I
a = 0.441765 − 0.140806I
b = 0.238510 + 0.243220I
2.92647 + 1.66195I −6.61632 − 3.48117I
u = −0.125010 − 1.233150I
a = 0.441765 + 0.140806I
b = 0.238510 − 0.243220I
2.92647 − 1.66195I −6.61632 + 3.48117I
u = 0.506035 + 0.355900I
a = 1.129350 − 0.256604I
b = 0.238510 − 0.243220I
2.92647 − 1.66195I −6.61632 + 3.48117I
u = 0.506035 − 0.355900I
a = 1.129350 + 0.256604I
b = 0.238510 + 0.243220I
2.92647 + 1.66195I −6.61632 − 3.48117I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.443597 + 0.298423I
a = 0.117192 − 0.758722I
b = −1.178780 − 0.606721I
−1.14222 − 1.62541I −14.5850 + 1.4256I
u = −0.443597 − 0.298423I
a = 0.117192 + 0.758722I
b = −1.178780 + 0.606721I
−1.14222 + 1.62541I −14.5850 − 1.4256I
u = 0.55989 + 1.37681I
a = 0.721990 − 1.125960I
b = 1.89776 + 0.22684I
−12.14610 − 5.90409I −13.72541 + 2.82977I
u = 0.55989 − 1.37681I
a = 0.721990 + 1.125960I
b = 1.89776 − 0.22684I
−12.14610 + 5.90409I −13.72541 − 2.82977I
u = −0.55749 + 1.39010I
a = 0.593934 + 1.012530I
b = 1.82176
−7.78143 −11.35940 + 0.I
u = −0.55749 − 1.39010I
a = 0.593934 − 1.012530I
b = 1.82176
−7.78143 −11.35940 + 0.I
10
III. I
u
3
= hb + 1, u
2
+ 2a + u + 3, u
3
+ 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−
1
2
u
2
−
1
2
u −
3
2
−1
a
6
=
1
u
2
a
2
=
−
1
2
u
2
−
1
2
u −
5
2
−1
a
1
=
−1
0
a
4
=
−
1
2
u
2
−
1
2
u −
3
2
−1
a
10
=
u
−u + 1
a
7
=
u
2
+ 1
u
a
9
=
1
−u + 1
a
8
=
1
−u + 1
a
12
=
−u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
7
4
u
2
−
21
4
u −
57
4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
u
3
+ 2u − 1
c
9
u
3
+ 3u
2
+ 5u + 2
c
10
, c
11
, c
12
u
3
+ 2u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
9
y
3
+ y
2
+ 13y −4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = −0.335258 − 0.401127I
b = −1.00000
7.79580 + 5.13794I −9.37996 − 6.54094I
u = −0.22670 − 1.46771I
a = −0.335258 + 0.401127I
b = −1.00000
7.79580 − 5.13794I −9.37996 + 6.54094I
u = 0.453398
a = −1.82948
b = −1.00000
−2.43213 −16.9900
14
IV. I
u
4
= hb + 1, u
3
+ u
2
+ a + u + 2, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−u
3
− u
2
− u − 2
−1
a
6
=
1
u
2
a
2
=
−u
3
− u
2
− u − 3
−1
a
1
=
−1
0
a
4
=
−u
3
− u
2
− u − 2
−1
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
−u
3
− 2u − 1
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
u
3
+ 2u
u
3
+ u
a
12
=
−2u
3
− u
2
− 3u − 3
−u
3
− u
2
− u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
3
− 4u − 15
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
8
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
7
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
9
(u
2
− u + 1)
2
c
10
, c
11
, c
12
u
4
− u
3
+ 2u
2
− 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
9
(y
2
+ y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −1.69244 − 0.31815I
b = −1.00000
1.64493 + 2.02988I −13.00000 − 3.46410I
u = −0.621744 − 0.440597I
a = −1.69244 + 0.31815I
b = −1.00000
1.64493 − 2.02988I −13.00000 + 3.46410I
u = 0.121744 + 1.306620I
a = 0.192440 + 0.547877I
b = −1.00000
1.64493 − 2.02988I −13.00000 + 3.46410I
u = 0.121744 − 1.306620I
a = 0.192440 − 0.547877I
b = −1.00000
1.64493 + 2.02988I −13.00000 − 3.46410I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
7
· (u
8
+ 13u
7
+ 68u
6
+ 185u
5
+ 287u
4
+ 249u
3
+ 77u
2
+ 3u + 1)
2
· (u
10
+ 14u
9
+ ··· + 625u + 16)
c
2
(u − 1)
7
(u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1)
2
· (u
10
− 2u
9
− 5u
8
+ 7u
7
+ 14u
6
− 36u
4
− 8u
3
+ 42u
2
− 17u − 4)
c
3
, c
8
u
7
(u
8
− u
7
− 7u
6
+ 4u
5
+ 16u
4
+ 3u
3
− 9u
2
+ 8u − 4)
2
· (u
10
+ 3u
9
+ ··· + 88u + 32)
c
4
(u + 1)
7
(u
8
− 3u
7
− 2u
6
+ 9u
5
+ 5u
4
− 13u
3
− 3u
2
+ 3u − 1)
2
· (u
10
− 2u
9
− 5u
8
+ 7u
7
+ 14u
6
− 36u
4
− 8u
3
+ 42u
2
− 17u − 4)
c
5
, c
6
, c
7
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
10
+ 4u
8
− 2u
7
+ 6u
6
− 7u
5
+ 3u
4
− 7u
3
+ u
2
− 2u − 1)
· (u
16
+ 2u
15
+ ··· + 2u + 1)
c
9
(u
2
− u + 1)
2
(u
3
+ 3u
2
+ 5u + 2)
· (u
8
+ 2u
7
− 7u
6
− 12u
5
+ 5u
4
− 3u
3
− 2u
2
− 2u + 1)
2
· (u
10
− 6u
9
+ 9u
8
+ 2u
7
+ 7u
6
− 51u
5
+ 50u
4
− 20u
3
+ 7u
2
− 4u − 4)
c
10
, c
11
, c
12
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
10
+ 4u
8
− 2u
7
+ 6u
6
− 7u
5
+ 3u
4
− 7u
3
+ u
2
− 2u − 1)
· (u
16
+ 2u
15
+ ··· + 2u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
8
− 33y
7
+ ··· + 145y + 1)
2
· (y
10
− 34y
9
+ ··· − 333665y + 256)
c
2
, c
4
(y −1)
7
· (y
8
− 13y
7
+ 68y
6
− 185y
5
+ 287y
4
− 249y
3
+ 77y
2
− 3y + 1)
2
· (y
10
− 14y
9
+ ··· − 625y + 16)
c
3
, c
8
y
7
(y
8
− 15y
7
+ 89y
6
− 252y
5
+ 366y
4
− 305y
3
− 95y
2
+ 8y + 16)
2
· (y
10
− 15y
9
+ ··· − 3392y + 1024)
c
5
, c
6
, c
7
c
10
, c
11
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
10
+ 8y
9
+ ··· − 6y + 1)
· (y
16
+ 10y
15
+ ··· + 12y
2
+ 1)
c
9
(y
2
+ y + 1)
2
(y
3
+ y
2
+ 13y −4)
· (y
8
− 18y
7
+ 107y
6
− 206y
5
− 9y
4
− 91y
3
+ 2y
2
− 8y + 1)
2
· (y
10
− 18y
9
+ ··· − 72y + 16)
20
