12a
1142
(K12a
1142
)
A knot diagram
1
Linearized knot diagam
4 8 9 11 10 12 2 3 1 5 7 6
Solving Sequence
6,10
5 11
1,4
2 9 3 8 12 7
c
5
c
10
c
4
c
1
c
9
c
3
c
8
c
12
c
6
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
24
− u
23
+ ··· + 16a + 1, u
25
+ 16u
23
+ ··· − u − 1i
I
u
2
= h31265112052u
31
− 16257768219u
30
+ ··· + 84752307686b − 8778222360,
− 11111041791u
31
+ 26118385624u
30
+ ··· + 84752307686a + 287854854541,
u
32
− u
31
+ ··· − 7u + 2i
I
u
3
= hb + u, a
4
− a
3
+ a
2
+ 1, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 65 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
24
− u
23
+ · · · + 16a + 1, u
25
+ 16u
23
+ · · · − u − 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
11
=
u
u
3
+ u
a
1
=
1
16
u
24
+
1
16
u
23
+ ··· +
23
8
u −
1
16
u
a
4
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
1
16
u
24
+
1
16
u
23
+ ··· +
15
8
u −
1
16
1
16
u
24
+
1
16
u
23
+ ··· +
7
8
u −
1
16
a
9
=
1
8
u
24
+ 2u
22
+ ··· −
3
8
u −
1
8
1
16
u
24
+
1
16
u
23
+ ··· +
7
8
u −
1
16
a
3
=
3
16
u
24
+
3
16
u
23
+ ··· −
5
8
u −
5
16
−
1
4
u
24
+
3
8
u
23
+ ··· −
1
8
u −
5
8
a
8
=
3
16
u
24
−
5
16
u
23
+ ··· +
1
8
u +
23
16
−
1
8
u
23
+
1
4
u
22
+ ··· +
1
8
u +
5
8
a
12
=
1
16
u
24
+
1
16
u
23
+ ··· +
15
8
u −
1
16
u
a
7
=
−0.0625000u
24
+ 0.0625000u
23
+ ··· − 2.12500u
2
+ 0.937500
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
24
+
5
4
u
23
+ ··· +
5
4
u −
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
25
− 7u
24
+ ··· + 513u − 136
c
2
, c
3
, c
7
c
8
u
25
+ 3u
24
+ ··· − 3u + 2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
25
+ 16u
23
+ ··· − u + 1
c
9
u
25
− 15u
24
+ ··· + 2387u − 362
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
25
− 5y
24
+ ··· − 37119y −18496
c
2
, c
3
, c
7
c
8
y
25
− 29y
24
+ ··· + y −4
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
25
+ 32y
24
+ ··· − 7y −1
c
9
y
25
+ 11y
24
+ ··· − 420031y −131044
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.561043 + 0.437348I
a = −1.68535 + 0.72480I
b = −0.561043 + 0.437348I
−7.98772 − 5.54613I −2.08956 + 7.04826I
u = −0.561043 − 0.437348I
a = −1.68535 − 0.72480I
b = −0.561043 − 0.437348I
−7.98772 + 5.54613I −2.08956 − 7.04826I
u = 0.638286
a = 1.54532
b = 0.638286
−4.66650 2.55840
u = 0.521660 + 0.351950I
a = 1.52382 + 0.64449I
b = 0.521660 + 0.351950I
−0.34952 + 3.61361I 0.91371 − 9.39805I
u = 0.521660 − 0.351950I
a = 1.52382 − 0.64449I
b = 0.521660 − 0.351950I
−0.34952 − 3.61361I 0.91371 + 9.39805I
u = −0.200130 + 0.529662I
a = −1.05021 + 1.68712I
b = −0.200130 + 0.529662I
−8.41951 + 2.51161I −3.21376 + 1.03660I
u = −0.200130 − 0.529662I
a = −1.05021 − 1.68712I
b = −0.200130 − 0.529662I
−8.41951 − 2.51161I −3.21376 − 1.03660I
u = −0.09775 + 1.44695I
a = −0.581028 − 1.096330I
b = −0.09775 + 1.44695I
−13.25460 − 4.79321I −7.77691 + 3.39473I
u = −0.09775 − 1.44695I
a = −0.581028 + 1.096330I
b = −0.09775 − 1.44695I
−13.25460 + 4.79321I −7.77691 − 3.39473I
u = 0.02846 + 1.45258I
a = 0.171580 − 1.155960I
b = 0.02846 + 1.45258I
−6.93212 + 2.05900I −4.36476 − 3.38495I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.02846 − 1.45258I
a = 0.171580 + 1.155960I
b = 0.02846 − 1.45258I
−6.93212 − 2.05900I −4.36476 + 3.38495I
u = −0.464935 + 0.201716I
a = −1.289760 + 0.413307I
b = −0.464935 + 0.201716I
0.912372 − 0.622752I 7.19624 + 2.63067I
u = −0.464935 − 0.201716I
a = −1.289760 − 0.413307I
b = −0.464935 − 0.201716I
0.912372 + 0.622752I 7.19624 − 2.63067I
u = 0.166735 + 0.405495I
a = 0.68732 + 1.26329I
b = 0.166735 + 0.405495I
−1.020640 − 0.969548I −2.52284 + 1.35231I
u = 0.166735 − 0.405495I
a = 0.68732 − 1.26329I
b = 0.166735 − 0.405495I
−1.020640 + 0.969548I −2.52284 − 1.35231I
u = 0.27079 + 1.54625I
a = 1.040560 − 0.311629I
b = 0.27079 + 1.54625I
−11.13940 + 6.44106I −3.32238 − 2.71115I
u = 0.27079 − 1.54625I
a = 1.040560 + 0.311629I
b = 0.27079 − 1.54625I
−11.13940 − 6.44106I −3.32238 + 2.71115I
u = −0.31312 + 1.55281I
a = −1.136180 − 0.202662I
b = −0.31312 + 1.55281I
−13.0092 − 10.4518I −6.37304 + 7.21331I
u = −0.31312 − 1.55281I
a = −1.136180 + 0.202662I
b = −0.31312 − 1.55281I
−13.0092 + 10.4518I −6.37304 − 7.21331I
u = −0.21804 + 1.58247I
a = −0.814184 − 0.307816I
b = −0.21804 + 1.58247I
−14.4736 − 3.0731I −8.36964 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.21804 − 1.58247I
a = −0.814184 + 0.307816I
b = −0.21804 − 1.58247I
−14.4736 + 3.0731I −8.36964 + 0.I
u = 0.34476 + 1.56544I
a = 1.182290 − 0.106671I
b = 0.34476 + 1.56544I
18.3511 + 13.0389I −8.43014 − 5.93322I
u = 0.34476 − 1.56544I
a = 1.182290 + 0.106671I
b = 0.34476 − 1.56544I
18.3511 − 13.0389I −8.43014 + 5.93322I
u = 0.20347 + 1.64400I
a = 0.678475 − 0.143846I
b = 0.20347 + 1.64400I
16.0652 + 1.5629I −9.92611 − 0.62064I
u = 0.20347 − 1.64400I
a = 0.678475 + 0.143846I
b = 0.20347 − 1.64400I
16.0652 − 1.5629I −9.92611 + 0.62064I
7
II.
I
u
2
= h3.13 × 10
10
u
31
− 1.63 × 10
10
u
30
+ · · · + 8.48× 10
10
b − 8.78 ×10
9
, −1.11×
10
10
u
31
+2.61×10
10
u
30
+· · ·+8.48×10
10
a+2.88×10
11
, u
32
−u
31
+· · ·−7u+2i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
11
=
u
u
3
+ u
a
1
=
0.131100u
31
− 0.308173u
30
+ ··· + 2.64660u − 3.39642
−0.368900u
31
+ 0.191827u
30
+ ··· − 2.85340u + 0.103575
a
4
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
0.257094u
31
− 0.416002u
30
+ ··· + 3.69092u − 3.68254
−0.652251u
31
+ 0.451316u
30
+ ··· − 4.12999u + 0.211450
a
9
=
0.0315636u
31
− 0.0452649u
30
+ ··· + 0.291482u − 2.93870
−0.0995366u
31
+ 0.262908u
30
+ ··· − 1.35511u + 0.457721
a
3
=
−0.458023u
31
+ 0.342355u
30
+ ··· − 6.00927u + 2.47313
0.0217054u
31
− 0.180440u
30
+ ··· + 4.15577u − 0.559030
a
8
=
0.220688u
31
− 0.0368277u
30
+ ··· − 0.676663u − 2.74045
0.269259u
31
+ 0.0522139u
30
+ ··· − 2.98024u + 1.30099
a
12
=
1
2
u
31
−
1
2
u
30
+ ··· +
11
2
u −
7
2
−0.368900u
31
+ 0.191827u
30
+ ··· − 2.85340u + 0.103575
a
7
=
−0.0517875u
31
− 0.317112u
30
+ ··· − 10.4579u − 1.49089
−0.177073u
31
+ 0.446436u
30
+ ··· − 2.47872u + 1.73780
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
27044210812
42376153843
u
31
−
8739385902
42376153843
u
30
+ ··· −
295063087682
42376153843
u +
92212569266
42376153843
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
16
− 5u
15
+ ··· + 8u − 7)
2
c
2
, c
3
, c
7
c
8
(u
16
− u
15
+ ··· + 2u
2
− 1)
2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
32
+ u
31
+ ··· + 7u + 2
c
9
(u
16
+ 5u
15
+ ··· − 4u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
16
− 7y
15
+ ··· − 344y + 49)
2
c
2
, c
3
, c
7
c
8
(y
16
− 19y
15
+ ··· − 4y + 1)
2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
32
+ 27y
31
+ ··· − 5y + 4
c
9
(y
16
+ 13y
15
+ ··· − 48y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.880391 + 0.506625I
a = 1.017130 − 0.972732I
b = 0.16383 − 1.46376I
−6.29225 − 6.07197I −4.61575 + 7.02814I
u = −0.880391 − 0.506625I
a = 1.017130 + 0.972732I
b = 0.16383 + 1.46376I
−6.29225 + 6.07197I −4.61575 − 7.02814I
u = −0.774157 + 0.692338I
a = 0.741176 − 0.809846I
b = 0.02347 − 1.45170I
−6.89084 + 0.48968I −6.35607 − 1.43137I
u = −0.774157 − 0.692338I
a = 0.741176 + 0.809846I
b = 0.02347 + 1.45170I
−6.89084 − 0.48968I −6.35607 + 1.43137I
u = 0.777840 + 0.542265I
a = −0.977635 − 0.812403I
b = −0.11249 − 1.41553I
−4.30716 + 2.57669I −0.69244 − 2.71681I
u = 0.777840 − 0.542265I
a = −0.977635 + 0.812403I
b = −0.11249 + 1.41553I
−4.30716 − 2.57669I −0.69244 + 2.71681I
u = 0.192406 + 1.054070I
a = 0.0248167 − 0.1359550I
b = 0.192406 − 1.054070I
−4.05396 −9.09362 + 0.I
u = 0.192406 − 1.054070I
a = 0.0248167 + 0.1359550I
b = 0.192406 + 1.054070I
−4.05396 −9.09362 + 0.I
u = 0.949812 + 0.504302I
a = −1.00934 − 1.06834I
b = −0.18803 − 1.50441I
−14.4043 + 8.2886I −6.57708 − 5.27135I
u = 0.949812 − 0.504302I
a = −1.00934 + 1.06834I
b = −0.18803 + 1.50441I
−14.4043 − 8.2886I −6.57708 + 5.27135I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.060795 + 1.080160I
a = 0.211903 + 0.762923I
b = 0.159960 − 0.159944I
−1.40282 − 1.52971I 2.72737 + 5.08772I
u = −0.060795 − 1.080160I
a = 0.211903 − 0.762923I
b = 0.159960 + 0.159944I
−1.40282 + 1.52971I 2.72737 − 5.08772I
u = 0.195301 + 1.117820I
a = −0.601834 + 0.773671I
b = −0.450162 − 0.094431I
−7.98944 + 3.12434I −1.94060 − 3.66013I
u = 0.195301 − 1.117820I
a = −0.601834 − 0.773671I
b = −0.450162 + 0.094431I
−7.98944 − 3.12434I −1.94060 + 3.66013I
u = 0.840396 + 0.765707I
a = −0.642609 − 0.918714I
b = 0.00756 − 1.51110I
−15.1904 − 2.2836I −7.92472 + 0.30826I
u = 0.840396 − 0.765707I
a = −0.642609 + 0.918714I
b = 0.00756 + 1.51110I
−15.1904 + 2.2836I −7.92472 − 0.30826I
u = −0.344556 + 1.164540I
a = −0.110939 − 0.374956I
b = −0.344556 − 1.164540I
−11.2964 −8.14780 + 0.I
u = −0.344556 − 1.164540I
a = −0.110939 + 0.374956I
b = −0.344556 + 1.164540I
−11.2964 −8.14780 + 0.I
u = −0.11249 + 1.41553I
a = 0.833628 + 0.159750I
b = 0.777840 − 0.542265I
−4.30716 − 2.57669I −0.69244 + 2.71681I
u = −0.11249 − 1.41553I
a = 0.833628 − 0.159750I
b = 0.777840 + 0.542265I
−4.30716 + 2.57669I −0.69244 − 2.71681I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.02347 + 1.45170I
a = −0.785289 − 0.003672I
b = −0.774157 − 0.692338I
−6.89084 − 0.48968I −6.35607 + 1.43137I
u = 0.02347 − 1.45170I
a = −0.785289 + 0.003672I
b = −0.774157 + 0.692338I
−6.89084 + 0.48968I −6.35607 − 1.43137I
u = 0.16383 + 1.46376I
a = −0.955911 + 0.168093I
b = −0.880391 − 0.506625I
−6.29225 + 6.07197I −4.61575 − 7.02814I
u = 0.16383 − 1.46376I
a = −0.955911 − 0.168093I
b = −0.880391 + 0.506625I
−6.29225 − 6.07197I −4.61575 + 7.02814I
u = 0.00756 + 1.51110I
a = 0.837083 − 0.103955I
b = 0.840396 − 0.765707I
−15.1904 + 2.2836I −7.92472 − 0.30826I
u = 0.00756 − 1.51110I
a = 0.837083 + 0.103955I
b = 0.840396 + 0.765707I
−15.1904 − 2.2836I −7.92472 + 0.30826I
u = −0.18803 + 1.50441I
a = 1.031610 + 0.150184I
b = 0.949812 − 0.504302I
−14.4043 − 8.2886I −6.57708 + 5.27135I
u = −0.18803 − 1.50441I
a = 1.031610 − 0.150184I
b = 0.949812 + 0.504302I
−14.4043 + 8.2886I −6.57708 − 5.27135I
u = −0.450162 + 0.094431I
a = 2.32309 − 0.67146I
b = 0.195301 − 1.117820I
−7.98944 − 3.12434I −1.94060 + 3.66013I
u = −0.450162 − 0.094431I
a = 2.32309 + 0.67146I
b = 0.195301 + 1.117820I
−7.98944 + 3.12434I −1.94060 − 3.66013I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.159960 + 0.159944I
a = −3.18688 + 2.04561I
b = −0.060795 − 1.080160I
−1.40282 + 1.52971I 2.72737 − 5.08772I
u = 0.159960 − 0.159944I
a = −3.18688 − 2.04561I
b = −0.060795 + 1.080160I
−1.40282 − 1.52971I 2.72737 + 5.08772I
14
III. I
u
3
= hb + u, a
4
− a
3
+ a
2
+ 1, u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
5
=
1
−1
a
11
=
u
0
a
1
=
a
−u
a
4
=
0
−1
a
2
=
a
a − u
a
9
=
−a
2
u
−a + u
a
3
=
a
3
− a
2
− 1
−a
3
u − a
2
− 1
a
8
=
a
3
u + au
a
3
u + a
2
+ 1
a
12
=
a + u
−u
a
7
=
au
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
2
− 4a − 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
+ u
3
+ u
2
+ 1)
2
c
2
, c
3
, c
7
c
8
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
4
, c
5
, c
6
c
10
, c
11
, c
12
(u
2
+ 1)
4
c
9
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
2
, c
3
, c
7
c
8
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
(y + 1)
8
c
9
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.351808 + 0.720342I
b = − 1.000000I
−3.07886 + 1.41510I −4.17326 − 4.90874I
u = 1.000000I
a = −0.351808 − 0.720342I
b = − 1.000000I
−3.07886 − 1.41510I −4.17326 + 4.90874I
u = 1.000000I
a = 0.851808 + 0.911292I
b = − 1.000000I
−10.08060 − 3.16396I −7.82674 + 2.56480I
u = 1.000000I
a = 0.851808 − 0.911292I
b = − 1.000000I
−10.08060 + 3.16396I −7.82674 − 2.56480I
u = − 1.000000I
a = −0.351808 + 0.720342I
b = 1.000000I
−3.07886 + 1.41510I −4.17326 − 4.90874I
u = − 1.000000I
a = −0.351808 − 0.720342I
b = 1.000000I
−3.07886 − 1.41510I −4.17326 + 4.90874I
u = − 1.000000I
a = 0.851808 + 0.911292I
b = 1.000000I
−10.08060 − 3.16396I −7.82674 + 2.56480I
u = − 1.000000I
a = 0.851808 − 0.911292I
b = 1.000000I
−10.08060 + 3.16396I −7.82674 − 2.56480I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
16
− 5u
15
+ ··· + 8u − 7)
2
· (u
25
− 7u
24
+ ··· + 513u − 136)
c
2
, c
3
, c
7
c
8
(u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)(u
16
− u
15
+ ··· + 2u
2
− 1)
2
· (u
25
+ 3u
24
+ ··· − 3u + 2)
c
4
, c
5
, c
6
c
10
, c
11
, c
12
((u
2
+ 1)
4
)(u
25
+ 16u
23
+ ··· − u + 1)(u
32
+ u
31
+ ··· + 7u + 2)
c
9
(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)(u
16
+ 5u
15
+ ··· − 4u + 1)
2
· (u
25
− 15u
24
+ ··· + 2387u − 362)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
16
− 7y
15
+ ··· − 344y + 49)
2
· (y
25
− 5y
24
+ ··· − 37119y −18496)
c
2
, c
3
, c
7
c
8
((y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
)(y
16
− 19y
15
+ ··· − 4y + 1)
2
· (y
25
− 29y
24
+ ··· + y −4)
c
4
, c
5
, c
6
c
10
, c
11
, c
12
((y + 1)
8
)(y
25
+ 32y
24
+ ··· − 7y −1)(y
32
+ 27y
31
+ ··· − 5y + 4)
c
9
((y
4
− y
3
+ 3y
2
− 2y + 1)
2
)(y
16
+ 13y
15
+ ··· − 48y + 1)
2
· (y
25
+ 11y
24
+ ··· − 420031y −131044)
20
