12a
1283
(K12a
1283
)
A knot diagram
1
Linearized knot diagam
5 10 8 9 11 12 1 4 2 3 6 7
Solving Sequence
2,9
10 3
5,11
6 1 4 8 7 12
c
9
c
2
c
10
c
5
c
1
c
4
c
8
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
18
+ u
17
+ ··· + 4a − 13u, u
19
− u
18
+ ··· − u − 1i
I
u
2
= h538u
23
+ 1753u
22
+ ··· + 3334b + 10456, 432u
23
+ 12271u
22
+ ··· + 23338a + 126536,
u
24
− 10u
22
+ ··· − 16u − 7i
I
u
3
= hb − 1, a
2
− 3, u + 1i
I
u
4
= hb − 1, a, u + 1i
I
u
5
= hb, a − 1, u − 1i
I
u
6
= hb + 1, a − 1, u − 1i
I
u
7
= hb + 1, a + 1, u − 1i
I
v
1
= ha, b − 1, v + 1i
* 8 irreducible components of dim
C
= 0, with total 50 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
= hb − u, −u
18
+ u
17
+ · · · + 4a − 13u, u
19
− u
18
+ · · · − u − 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
u
−u
3
+ u
a
5
=
1
4
u
18
−
1
4
u
17
+ ··· +
1
4
u
2
+
13
4
u
u
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
6
=
1
4
u
18
−
1
4
u
17
+ ··· +
1
4
u
2
+
9
4
u
1
4
u
18
−
1
4
u
17
+ ··· +
1
4
u
2
+
5
4
u
a
1
=
−
1
4
u
17
−
1
4
u
16
+ ··· +
1
4
u −
1
4
1
4
u
18
−
1
4
u
17
+ ··· +
1
4
u
2
+
5
4
u
a
4
=
1
4
u
18
−
1
4
u
17
+ ··· +
1
4
u
2
+
9
4
u
u
a
8
=
1
4
u
17
−
1
4
u
16
+ ··· +
1
4
u +
5
4
u
2
a
7
=
1
2
u
18
−
3
2
u
17
+ ··· −
1
2
u −
1
2
1
4
u
18
−
11
4
u
16
+ ··· −
1
2
u
2
−
1
4
a
12
=
−
1
4
u
18
−
1
4
u
17
+ ··· −
1
4
u + 1
−
1
4
u
18
+
1
2
u
17
+ ··· +
1
2
u +
3
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
18
+
5
2
u
17
+ 17u
16
−
51
2
u
15
− 81u
14
+ 104u
13
+ 206u
12
−
411
2
u
11
−
583
2
u
10
+
351
2
u
9
+ 215u
8
−
9
2
u
7
−
149
2
u
6
−
103
2
u
5
+ 36u
4
−
29
2
u
3
−
59
2
u
2
+
15
2
u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 15u
18
+ ··· + 1586u + 218
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
19
+ u
18
+ ··· − u + 1
c
5
, c
6
, c
7
c
11
, c
12
u
19
+ 3u
18
+ ··· − 6u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− y
18
+ ··· + 65512y −47524
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
19
− 23y
18
+ ··· + y −1
c
5
, c
6
, c
7
c
11
, c
12
y
19
− 25y
18
+ ··· − 24y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.310639 + 0.700279I
a = −0.43101 + 1.74263I
b = −0.310639 + 0.700279I
−12.66640 − 3.70859I −6.97096 + 4.51080I
u = −0.310639 − 0.700279I
a = −0.43101 − 1.74263I
b = −0.310639 − 0.700279I
−12.66640 + 3.70859I −6.97096 − 4.51080I
u = −1.34431
a = 1.98662
b = −1.34431
−7.55334 2.37220
u = 0.264676 + 0.585091I
a = 0.44609 + 1.54901I
b = 0.264676 + 0.585091I
−3.30479 + 2.71809I −7.05035 − 6.50802I
u = 0.264676 − 0.585091I
a = 0.44609 − 1.54901I
b = 0.264676 − 0.585091I
−3.30479 − 2.71809I −7.05035 + 6.50802I
u = −0.583241
a = −2.35110
b = −0.583241
−11.3004 −5.80070
u = 1.47018
a = −1.06148
b = 1.47018
3.59702 2.15400
u = 1.48891 + 0.35458I
a = −0.245253 + 1.382770I
b = 1.48891 + 0.35458I
−1.12008 + 11.80290I 0.70242 − 5.53182I
u = 1.48891 − 0.35458I
a = −0.245253 − 1.382770I
b = 1.48891 − 0.35458I
−1.12008 − 11.80290I 0.70242 + 5.53182I
u = −1.52435 + 0.16917I
a = 0.597354 + 0.752907I
b = −1.52435 + 0.16917I
10.36490 − 2.09930I 4.61522 − 0.98931I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.52435 − 0.16917I
a = 0.597354 − 0.752907I
b = −1.52435 − 0.16917I
10.36490 + 2.09930I 4.61522 + 0.98931I
u = −1.50399 + 0.30349I
a = 0.345723 + 1.228060I
b = −1.50399 + 0.30349I
8.30594 − 9.64872I 2.30074 + 6.54307I
u = −1.50399 − 0.30349I
a = 0.345723 − 1.228060I
b = −1.50399 − 0.30349I
8.30594 + 9.64872I 2.30074 − 6.54307I
u = 1.54004
a = −0.705382
b = 1.54004
3.55474 2.45220
u = 1.52397 + 0.24269I
a = −0.439996 + 1.008280I
b = 1.52397 + 0.24269I
11.80370 + 6.00675I 6.85446 − 4.30060I
u = 1.52397 − 0.24269I
a = −0.439996 − 1.008280I
b = 1.52397 − 0.24269I
11.80370 − 6.00675I 6.85446 + 4.30060I
u = 0.440690
a = 1.53450
b = 0.440690
−1.88440 −3.40050
u = −0.200256 + 0.361761I
a = −0.474490 + 1.061500I
b = −0.200256 + 0.361761I
−0.010314 − 0.815704I −0.34015 + 8.34541I
u = −0.200256 − 0.361761I
a = −0.474490 − 1.061500I
b = −0.200256 − 0.361761I
−0.010314 + 0.815704I −0.34015 − 8.34541I
6
II. I
u
2
= h538u
23
+ 1753u
22
+ · · · + 3334b + 10456, 432u
23
+ 12271u
22
+ · · · +
23338a + 126536, u
24
− 10u
22
+ · · · − 16u − 7i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
u
−u
3
+ u
a
5
=
−0.0185106u
23
− 0.525795u
22
+ ··· − 3.14153u − 5.42189
−0.161368u
23
− 0.525795u
22
+ ··· − 5.71296u − 3.13617
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
6
=
−0.0101123u
23
− 0.227055u
22
+ ··· − 0.0356500u − 2.14714
−0.155369u
23
− 0.0266947u
22
+ ··· − 0.708758u − 1.08278
a
1
=
0.166252u
23
+ 0.0464907u
22
+ ··· + 1.68781u − 4.87750
0.184763u
23
+ 0.572286u
22
+ ··· + 5.82933u + 0.544391
a
4
=
1
7
u
23
−
10
7
u
21
+ ··· +
18
7
u −
16
7
−0.161368u
23
− 0.525795u
22
+ ··· − 5.71296u − 3.13617
a
8
=
−0.448025u
23
− 0.161368u
22
+ ··· − 3.84219u + 2.45544
0.525795u
23
+ 0.346131u
22
+ ··· + 5.71806u + 2.12957
a
7
=
−0.0533893u
23
− 0.0419916u
22
+ ··· + 0.862627u − 5.17516
−0.0929814u
23
+ 0.263947u
22
+ ··· + 4.43491u − 0.327534
a
12
=
−0.268960u
23
− 0.363227u
22
+ ··· − 3.56684u + 3.09911
0.134673u
23
+ 0.00479904u
22
+ ··· − 2.85573u + 0.548590
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
654
1667
u
23
−
3736
1667
u
22
+ ··· −
38132
1667
u −
26158
1667
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
12
− 4u
11
+ ··· − 6u + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
24
− 10u
22
+ ··· + 16u − 7
c
5
, c
6
, c
7
c
11
, c
12
(u
12
− 2u
11
+ ··· − 4u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
12
+ 8y
11
+ ··· − 14y + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
24
− 20y
23
+ ··· − 508y + 49
c
5
, c
6
, c
7
c
11
, c
12
(y
12
− 16y
11
+ ··· − 6y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.358425 + 0.917120I
a = 1.06378 − 1.20789I
b = 1.43345 − 0.26200I
−7.05914 − 7.20360I −2.08749 + 4.71657I
u = −0.358425 − 0.917120I
a = 1.06378 + 1.20789I
b = 1.43345 + 0.26200I
−7.05914 + 7.20360I −2.08749 − 4.71657I
u = 0.726659 + 0.612159I
a = −0.556760 − 0.649993I
b = −1.361680 + 0.028095I
3.08210 − 0.49850I 1.36863 + 1.38008I
u = 0.726659 − 0.612159I
a = −0.556760 + 0.649993I
b = −1.361680 − 0.028095I
3.08210 + 0.49850I 1.36863 − 1.38008I
u = 0.421897 + 0.830088I
a = −0.92080 − 1.15289I
b = −1.40739 − 0.19551I
2.05779 + 5.52285I −0.56374 − 6.48307I
u = 0.421897 − 0.830088I
a = −0.92080 + 1.15289I
b = −1.40739 + 0.19551I
2.05779 − 5.52285I −0.56374 + 6.48307I
u = −0.539453 + 0.732545I
a = 0.737853 − 0.986740I
b = 1.389660 − 0.101631I
5.05906 − 2.46907I 5.52253 + 3.95252I
u = −0.539453 − 0.732545I
a = 0.737853 + 0.986740I
b = 1.389660 + 0.101631I
5.05906 + 2.46907I 5.52253 − 3.95252I
u = −0.914759 + 0.672614I
a = 0.716633 − 0.381724I
b = 1.42619 + 0.14001I
−5.38423 + 1.70959I 0.128193 − 0.167200I
u = −0.914759 − 0.672614I
a = 0.716633 + 0.381724I
b = 1.42619 − 0.14001I
−5.38423 − 1.70959I 0.128193 + 0.167200I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.14845
a = −0.192638
b = 0.678097
2.62918 −3.06920
u = 0.678097
a = 0.326261
b = −1.14845
2.62918 −3.06920
u = −1.361680 + 0.028095I
a = −0.007416 + 0.597014I
b = 0.726659 + 0.612159I
3.08210 − 0.49850I 1.36863 + 1.38008I
u = −1.361680 − 0.028095I
a = −0.007416 − 0.597014I
b = 0.726659 − 0.612159I
3.08210 + 0.49850I 1.36863 − 1.38008I
u = 1.389660 + 0.101631I
a = 0.176320 − 0.784892I
b = −0.539453 − 0.732545I
5.05906 + 2.46907I 5.52253 − 3.95252I
u = 1.389660 − 0.101631I
a = 0.176320 + 0.784892I
b = −0.539453 + 0.732545I
5.05906 − 2.46907I 5.52253 + 3.95252I
u = −0.580967 + 0.112101I
a = −2.24045 − 0.43231I
b = −0.580967 − 0.112101I
−11.2998 −5.66710 + 0.I
u = −0.580967 − 0.112101I
a = −2.24045 + 0.43231I
b = −0.580967 + 0.112101I
−11.2998 −5.66710 + 0.I
u = −1.40739 + 0.19551I
a = −0.275184 − 0.926925I
b = 0.421897 − 0.830088I
2.05779 − 5.52285I −0.56374 + 6.48307I
u = −1.40739 − 0.19551I
a = −0.275184 + 0.926925I
b = 0.421897 + 0.830088I
2.05779 + 5.52285I −0.56374 − 6.48307I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.42619 + 0.14001I
a = −0.220284 + 0.604438I
b = −0.914759 + 0.672614I
−5.38423 + 1.70959I 0.128193 − 0.167200I
u = 1.42619 − 0.14001I
a = −0.220284 − 0.604438I
b = −0.914759 − 0.672614I
−5.38423 − 1.70959I 0.128193 + 0.167200I
u = 1.43345 + 0.26200I
a = 0.316640 − 1.040500I
b = −0.358425 − 0.917120I
−7.05914 + 7.20360I −2.08749 − 4.71657I
u = 1.43345 − 0.26200I
a = 0.316640 + 1.040500I
b = −0.358425 + 0.917120I
−7.05914 − 7.20360I −2.08749 + 4.71657I
12
III. I
u
3
= hb − 1, a
2
− 3, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
9
=
1
0
a
10
=
1
−1
a
3
=
−1
0
a
5
=
a
1
a
11
=
0
−1
a
6
=
a
a + 1
a
1
=
−3
−a − 1
a
4
=
a − 1
1
a
8
=
a
1
a
7
=
−2a
−a − 2
a
12
=
3
a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u
2
− 3
c
2
, c
8
(u − 1)
2
c
3
, c
4
, c
9
c
10
(u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
(y − 3)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y − 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.73205
b = 1.00000
−9.86960 0
u = −1.00000
a = −1.73205
b = 1.00000
−9.86960 0
16
IV. I
u
4
= hb − 1, a, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
9
=
1
0
a
10
=
1
−1
a
3
=
−1
0
a
5
=
0
1
a
11
=
0
−1
a
6
=
0
1
a
1
=
0
−1
a
4
=
−1
1
a
8
=
0
1
a
7
=
0
1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u
c
2
, c
8
u − 1
c
3
, c
4
, c
9
c
10
u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
y
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y − 1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
3.28987 12.0000
20
V. I
u
5
= hb, a − 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
3
=
1
0
a
5
=
1
0
a
11
=
0
−1
a
6
=
1
1
a
1
=
1
1
a
4
=
1
0
a
8
=
1
0
a
7
=
0
−1
a
12
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u − 1
c
2
, c
5
, c
6
c
7
, c
9
, c
10
c
11
, c
12
u + 1
c
3
, c
4
, c
8
u
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
c
10
, c
11
, c
12
y − 1
c
3
, c
4
, c
8
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
−1.64493 −6.00000
24
VI. I
u
6
= hb + 1, a − 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
3
=
1
0
a
5
=
1
−1
a
11
=
0
−1
a
6
=
1
0
a
1
=
1
0
a
4
=
2
−1
a
8
=
−1
1
a
7
=
0
1
a
12
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
25
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
8
u + 1
c
3
, c
4
, c
9
c
10
, c
11
, c
12
u − 1
26
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
27
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = −1.00000
0 0
28
VII. I
u
7
= hb + 1, a + 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
3
=
1
0
a
5
=
−1
−1
a
11
=
0
−1
a
6
=
−1
−2
a
1
=
1
2
a
4
=
0
−1
a
8
=
1
1
a
7
=
0
−1
a
12
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
29
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
u − 1
c
2
, c
8
, c
11
c
12
u + 1
30
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
31
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
0 0
32
VIII. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
2
=
−1
0
a
9
=
1
0
a
10
=
1
0
a
3
=
−1
0
a
5
=
0
1
a
11
=
1
0
a
6
=
−1
1
a
1
=
−1
1
a
4
=
−1
1
a
8
=
0
1
a
7
=
1
0
a
12
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
33
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u − 1
c
2
, c
9
, c
10
u
c
3
, c
4
, c
5
c
6
, c
7
, c
8
c
11
, c
12
u + 1
34
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
11
, c
12
y − 1
c
2
, c
9
, c
10
y
35
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−1.64493 −6.00000
36
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
3
(u + 1)(u
2
− 3)(u
12
− 4u
11
+ ··· − 6u + 1)
2
· (u
19
+ 15u
18
+ ··· + 1586u + 218)
c
2
, c
8
u(u − 1)
3
(u + 1)
3
(u
19
+ u
18
+ ··· − u + 1)
· (u
24
− 10u
22
+ ··· + 16u − 7)
c
3
, c
4
, c
9
c
10
u(u − 1)
2
(u + 1)
4
(u
19
+ u
18
+ ··· − u + 1)
· (u
24
− 10u
22
+ ··· + 16u − 7)
c
5
, c
6
, c
7
c
11
, c
12
u(u − 1)(u + 1)
3
(u
2
− 3)(u
12
− 2u
11
+ ··· − 4u + 1)
2
· (u
19
+ 3u
18
+ ··· − 6u − 2)
37
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y − 3)
2
(y − 1)
4
(y
12
+ 8y
11
+ ··· − 14y + 1)
2
· (y
19
− y
18
+ ··· + 65512y − 47524)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y(y − 1)
6
(y
19
− 23y
18
+ ··· + y − 1)(y
24
− 20y
23
+ ··· − 508y + 49)
c
5
, c
6
, c
7
c
11
, c
12
y(y − 3)
2
(y − 1)
4
(y
12
− 16y
11
+ ··· − 6y + 1)
2
· (y
19
− 25y
18
+ ··· − 24y − 4)
38
