12n
0583
(K12n
0583
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 9 2 12 6 5 4 7 8
Solving Sequence
2,6
7 3
1,9
5 10 4 8 12 11
c
6
c
2
c
1
c
5
c
9
c
4
c
8
c
12
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
14
+ u
13
+ u
12
− 2u
11
− 5u
10
+ 6u
9
− 6u
7
− 7u
6
+ 8u
5
− 2u
4
− 4u
3
− 7u
2
+ 4b + u + 1,
− u
14
+ u
13
+ u
12
− 2u
11
− 5u
10
+ 6u
9
− 6u
7
− 7u
6
+ 8u
5
− 4u
4
− 4u
3
− 5u
2
+ 2a + u − 1,
u
16
− u
15
− 2u
14
+ 3u
13
+ 6u
12
− 8u
11
− 5u
10
+ 12u
9
+ 7u
8
− 14u
7
− u
6
+ 12u
5
+ u
4
− 5u
3
+ u + 1i
I
u
2
= h−1094u
19
+ 352u
18
+ ··· + 10214b − 24990, −4277u
19
− 4179u
18
+ ··· + 40856a + 46007,
u
20
− u
19
+ ··· + 9u − 8i
I
u
3
= h2b − a − 1, a
2
+ 2a + 13, u + 1i
I
u
4
= h2b − a − 1, a
2
+ 2a + 5, u − 1i
I
u
5
= hb, a + 1, u + 1i
* 5 irreducible components of dim
C
= 0, with total 41 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−u
14
+u
13
+· · ·+4b+1, −u
14
+u
13
+· · ·+2a−1, u
16
−u
15
+· · ·+u+1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
1
2
u
14
−
1
2
u
13
+ ··· −
1
2
u +
1
2
1
4
u
14
−
1
4
u
13
+ ··· −
1
4
u −
1
4
a
5
=
−
1
2
u
15
+
7
4
u
14
+ ··· −
7
4
u −
5
4
−
1
4
u
15
+
3
4
u
14
+ ··· −
3
4
u − 1
a
10
=
−
1
4
u
15
+
1
2
u
14
+ ··· −
1
2
u −
3
4
1
2
u
13
− u
12
+ ··· +
1
2
u
2
−
1
2
a
4
=
5
4
u
15
−
3
2
u
14
+ ··· −
3
2
u +
1
4
u
15
−
5
4
u
14
+ ··· −
1
4
u +
1
4
a
8
=
1
4
u
14
−
1
4
u
13
+ ··· −
1
4
u +
3
4
1
4
u
14
−
1
4
u
13
+ ··· −
1
4
u −
1
4
a
12
=
1
4
u
15
−
1
4
u
14
+ ··· −
1
4
u
2
+
3
4
u
1
4
u
15
−
1
4
u
14
+ ··· −
1
4
u
2
+
3
4
u
a
11
=
1
4
u
15
−
1
4
u
14
+ ··· −
1
4
u
2
+
7
4
u
1
4
u
15
−
1
4
u
14
+ ··· −
1
4
u
2
+
3
4
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
15
+
9
2
u
14
−
1
2
u
13
− 10u
12
−
1
2
u
11
+ 29u
10
− 16u
9
− 31u
8
+
39
2
u
7
+ 41u
6
− 36u
5
− 21u
4
+
53
2
u
3
+
27
2
u
2
−
23
2
u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 5u
15
+ ··· + u + 1
c
2
, c
6
, c
7
c
11
, c
12
u
16
− u
15
+ ··· + u + 1
c
3
u
16
+ 3u
15
+ ··· + 146u + 58
c
4
, c
5
, c
8
c
9
, c
10
u
16
+ 3u
15
+ ··· + 6u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 19y
15
+ ··· + 23y + 1
c
2
, c
6
, c
7
c
11
, c
12
y
16
− 5y
15
+ ··· − y + 1
c
3
y
16
− 3y
15
+ ··· + 33320y + 3364
c
4
, c
5
, c
8
c
9
, c
10
y
16
+ 21y
15
+ ··· + 24y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.652641 + 0.896529I
a = 0.291031 + 1.146470I
b = 0.06994 + 1.60049I
−6.22844 − 0.56003I −3.20504 + 2.03342I
u = 0.652641 − 0.896529I
a = 0.291031 − 1.146470I
b = 0.06994 − 1.60049I
−6.22844 + 0.56003I −3.20504 − 2.03342I
u = −0.806656 + 0.293377I
a = 0.55147 − 3.56996I
b = 0.01066 − 1.75439I
−15.5290 + 1.2535I −5.93594 − 5.78561I
u = −0.806656 − 0.293377I
a = 0.55147 + 3.56996I
b = 0.01066 + 1.75439I
−15.5290 − 1.2535I −5.93594 + 5.78561I
u = −0.863049 + 0.777771I
a = −0.003821 − 0.394263I
b = 0.459429 − 0.680536I
1.40791 + 2.31178I −2.22810 − 1.02238I
u = −0.863049 − 0.777771I
a = −0.003821 + 0.394263I
b = 0.459429 + 0.680536I
1.40791 − 2.31178I −2.22810 + 1.02238I
u = 0.699263 + 0.372224I
a = 0.54747 + 2.11955I
b = 0.023041 + 1.137640I
−5.08177 − 1.44001I −5.73865 + 4.77821I
u = 0.699263 − 0.372224I
a = 0.54747 − 2.11955I
b = 0.023041 − 1.137640I
−5.08177 + 1.44001I −5.73865 − 4.77821I
u = 1.000520 + 0.770635I
a = −0.326142 − 0.537879I
b = 0.646604 − 0.125765I
3.08984 − 6.02737I 0.18925 + 5.84414I
u = 1.000520 − 0.770635I
a = −0.326142 + 0.537879I
b = 0.646604 + 0.125765I
3.08984 + 6.02737I 0.18925 − 5.84414I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.097620 + 0.735783I
a = −0.99995 + 1.38493I
b = 0.416151 + 0.956386I
−0.22230 + 9.61368I −4.84554 − 8.00895I
u = −1.097620 − 0.735783I
a = −0.99995 − 1.38493I
b = 0.416151 − 0.956386I
−0.22230 − 9.61368I −4.84554 + 8.00895I
u = 1.172200 + 0.710728I
a = −1.65989 − 2.17594I
b = 0.11516 − 1.70265I
−9.5315 − 11.7377I −6.63835 + 6.61183I
u = 1.172200 − 0.710728I
a = −1.65989 + 2.17594I
b = 0.11516 + 1.70265I
−9.5315 + 11.7377I −6.63835 − 6.61183I
u = −0.257289 + 0.427551I
a = 0.599831 − 0.510753I
b = −0.240981 − 0.391034I
0.019028 + 0.937186I 0.40237 − 7.45995I
u = −0.257289 − 0.427551I
a = 0.599831 + 0.510753I
b = −0.240981 + 0.391034I
0.019028 − 0.937186I 0.40237 + 7.45995I
6
II. I
u
2
= h−1094u
19
+ 352u
18
+ · · · + 10214b − 24990, −4277u
19
− 4179u
18
+
· · · + 40856a + 46007, u
20
− u
19
+ · · · + 9u − 8i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
0.104685u
19
+ 0.102286u
18
+ ··· − 0.766717u − 1.12608
0.107108u
19
− 0.0344625u
18
+ ··· − 0.680537u + 2.44664
a
5
=
−0.0908312u
19
+ 0.201757u
18
+ ··· − 0.154714u − 1.48343
0.265126u
19
+ 0.108479u
18
+ ··· − 1.49990u + 1.58273
a
10
=
−0.466370u
19
− 0.00396515u
18
+ ··· + 1.09002u − 2.43014
−0.122968u
19
+ 0.202271u
18
+ ··· − 0.110828u − 1.64989
a
4
=
0.305830u
19
− 0.198722u
18
+ ··· − 0.0623409u + 2.07194
−0.154690u
19
+ 0.0378892u
18
+ ··· + 1.80644u − 1.05639
a
8
=
−0.00242314u
19
+ 0.136749u
18
+ ··· − 0.0861807u − 3.57272
0.107108u
19
− 0.0344625u
18
+ ··· − 0.680537u + 2.44664
a
12
=
0.171505u
19
− 0.289872u
18
+ ··· + 4.48857u + 2.09132
−0.227335u
19
+ 0.238594u
18
+ ··· + 0.323771u − 1.91326
a
11
=
1
8
u
19
−
1
8
u
18
+ ··· +
19
8
u +
9
8
−0.0465048u
19
+ 0.164872u
18
+ ··· − 1.11357u − 0.966321
(ii) Obstruction class = −1
(iii) Cusp Shapes =
6596
5107
u
19
+
828
5107
u
18
+ ··· +
15248
5107
u +
25134
5107
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 9u
19
+ ··· + 385u + 64
c
2
, c
6
, c
7
c
11
, c
12
u
20
− u
19
+ ··· + 9u − 8
c
3
(u
10
− u
9
− 5u
8
+ 4u
7
+ 8u
6
− 3u
5
− 5u
4
− 2u
3
+ 3u
2
− u − 1)
2
c
4
, c
5
, c
8
c
9
, c
10
(u
10
− u
9
+ 7u
8
− 6u
7
+ 16u
6
− 11u
5
+ 13u
4
− 6u
3
+ 3u
2
− u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 3y
19
+ ··· + 2431y + 4096
c
2
, c
6
, c
7
c
11
, c
12
y
20
− 9y
19
+ ··· − 385y + 64
c
3
(y
10
− 11y
9
+ ··· − 7y + 1)
2
c
4
, c
5
, c
8
c
9
, c
10
(y
10
+ 13y
9
+ ··· − 7y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.844382 + 0.577422I
a = −1.82553 − 0.56071I
b = 0.03425 − 1.67211I
−14.0102 − 2.2863I −7.60221 + 2.91176I
u = 0.844382 − 0.577422I
a = −1.82553 + 0.56071I
b = 0.03425 + 1.67211I
−14.0102 + 2.2863I −7.60221 − 2.91176I
u = −0.598296 + 0.894963I
a = 0.237969 + 0.532497I
b = −0.420834 + 0.842935I
1.29172 − 3.55946I −2.35774 + 4.06361I
u = −0.598296 − 0.894963I
a = 0.237969 − 0.532497I
b = −0.420834 − 0.842935I
1.29172 + 3.55946I −2.35774 − 4.06361I
u = 0.471623 + 0.985477I
a = 0.169899 − 1.229670I
b = −0.10787 − 1.66265I
−7.38803 + 5.55652I −4.20810 − 2.88175I
u = 0.471623 − 0.985477I
a = 0.169899 + 1.229670I
b = −0.10787 + 1.66265I
−7.38803 − 5.55652I −4.20810 + 2.88175I
u = −0.789879 + 0.382389I
a = −1.61085 − 0.05163I
b = 0.153406 + 0.833677I
−5.14913 + 1.60532I −7.05654 − 5.03395I
u = −0.789879 − 0.382389I
a = −1.61085 + 0.05163I
b = 0.153406 − 0.833677I
−5.14913 − 1.60532I −7.05654 + 5.03395I
u = 0.754802 + 0.842126I
a = 0.394821 + 0.277239I
b = −0.635590
3.84350 2.04859 + 0.I
u = 0.754802 − 0.842126I
a = 0.394821 − 0.277239I
b = −0.635590
3.84350 2.04859 + 0.I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.14103
a = 0.159161
b = 0.317683
−2.68035 4.40060
u = −0.902493 + 0.781683I
a = 0.910195 − 1.019170I
b = −0.420834 − 0.842935I
1.29172 + 3.55946I −2.35774 − 4.06361I
u = −0.902493 − 0.781683I
a = 0.910195 + 1.019170I
b = −0.420834 + 0.842935I
1.29172 − 3.55946I −2.35774 + 4.06361I
u = 1.239710 + 0.076961I
a = 0.06009 + 1.78443I
b = 0.153406 + 0.833677I
−5.14913 + 1.60532I −7.05654 − 5.03395I
u = 1.239710 − 0.076961I
a = 0.06009 − 1.78443I
b = 0.153406 − 0.833677I
−5.14913 − 1.60532I −7.05654 + 5.03395I
u = 0.723928
a = −1.56044
b = 0.317683
−2.68035 4.40060
u = 1.043190 + 0.765280I
a = 1.48424 + 1.76241I
b = −0.10787 + 1.66265I
−7.38803 − 5.55652I −4.20810 + 2.88175I
u = 1.043190 − 0.765280I
a = 1.48424 − 1.76241I
b = −0.10787 − 1.66265I
−7.38803 + 5.55652I −4.20810 − 2.88175I
u = −1.354480 + 0.103519I
a = 0.06730 − 3.11843I
b = 0.03425 − 1.67211I
−14.0102 − 2.2863I −7.60221 + 2.91176I
u = −1.354480 − 0.103519I
a = 0.06730 + 3.11843I
b = 0.03425 + 1.67211I
−14.0102 + 2.2863I −7.60221 − 2.91176I
11
III. I
u
3
= h2b − a − 1, a
2
+ 2a + 13, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
6
=
1
0
a
7
=
1
1
a
3
=
1
0
a
1
=
−1
−1
a
9
=
a
1
2
a +
1
2
a
5
=
−
1
2
a −
11
2
−3
a
10
=
−
3
2
a +
1
2
−a − 1
a
4
=
1
2
a +
9
2
3
a
8
=
1
2
a −
1
2
1
2
a +
1
2
a
12
=
1
2
a −
3
2
1
2
a −
1
2
a
11
=
1
2
a −
1
2
1
2
a +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 3
c
6
, c
11
, c
12
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 3)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000 + 3.46410I
b = 1.73205I
−16.4493 −12.0000
u = −1.00000
a = −1.00000 − 3.46410I
b = − 1.73205I
−16.4493 −12.0000
15
IV. I
u
4
= h2b − a − 1, a
2
+ 2a + 5, u − 1i
(i) Arc colorings
a
2
=
0
1
a
6
=
1
0
a
7
=
1
1
a
3
=
−1
0
a
1
=
1
1
a
9
=
a
1
2
a +
1
2
a
5
=
−
1
2
a −
3
2
−1
a
10
=
1
2
a +
1
2
0
a
4
=
−
1
2
a −
5
2
−1
a
8
=
1
2
a −
1
2
1
2
a +
1
2
a
12
=
−
1
2
a +
3
2
−
1
2
a +
1
2
a
11
=
−
1
2
a +
1
2
−
1
2
a −
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
2
c
2
, c
7
(u + 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000 + 2.00000I
b = 1.000000I
−6.57974 −12.0000
u = 1.00000
a = −1.00000 − 2.00000I
b = − 1.000000I
−6.57974 −12.0000
19
V. I
u
5
= hb, a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
6
=
1
0
a
7
=
1
1
a
3
=
1
0
a
1
=
−1
−1
a
9
=
−1
0
a
5
=
1
0
a
10
=
−1
0
a
4
=
1
0
a
8
=
−1
0
a
12
=
−2
−1
a
11
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
11
, c
12
u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
16
+ 5u
15
+ ··· + u + 1)(u
20
+ 9u
19
+ ··· + 385u + 64)
c
2
, c
7
((u − 1)
3
)(u + 1)
2
(u
16
− u
15
+ ··· + u + 1)(u
20
− u
19
+ ··· + 9u − 8)
c
3
u(u
2
+ 1)(u
2
+ 3)
· (u
10
− u
9
− 5u
8
+ 4u
7
+ 8u
6
− 3u
5
− 5u
4
− 2u
3
+ 3u
2
− u − 1)
2
· (u
16
+ 3u
15
+ ··· + 146u + 58)
c
4
, c
5
, c
8
c
9
, c
10
u(u
2
+ 1)(u
2
+ 3)
· (u
10
− u
9
+ 7u
8
− 6u
7
+ 16u
6
− 11u
5
+ 13u
4
− 6u
3
+ 3u
2
− u − 1)
2
· (u
16
+ 3u
15
+ ··· + 6u + 2)
c
6
, c
11
, c
12
((u − 1)
2
)(u + 1)
3
(u
16
− u
15
+ ··· + u + 1)(u
20
− u
19
+ ··· + 9u − 8)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
16
+ 19y
15
+ ··· + 23y + 1)
· (y
20
+ 3y
19
+ ··· + 2431y + 4096)
c
2
, c
6
, c
7
c
11
, c
12
((y −1)
5
)(y
16
− 5y
15
+ ··· − y + 1)(y
20
− 9y
19
+ ··· − 385y + 64)
c
3
y(y + 1)
2
(y + 3)
2
(y
10
− 11y
9
+ ··· − 7y + 1)
2
· (y
16
− 3y
15
+ ··· + 33320y + 3364)
c
4
, c
5
, c
8
c
9
, c
10
y(y + 1)
2
(y + 3)
2
(y
10
+ 13y
9
+ ··· − 7y + 1)
2
· (y
16
+ 21y
15
+ ··· + 24y + 4)
25
