12a
0836
(K12a
0836
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 3 1 6 8 7
Solving Sequence
1,7
12
3,8
9 4 10 11 6 5 2
c
12
c
7
c
8
c
3
c
9
c
11
c
6
c
5
c
2
c
1
, c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
65
+ u
64
+ ··· + b + 2u, u
65
− u
64
+ ··· + a − 2, u
68
− 2u
67
+ ··· + 2u + 1i
I
u
2
= hb − 1, −u
4
− u
3
− 2u
2
+ a − u, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 74 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
65
+u
64
+· · ·+b+2u, u
65
−u
64
+· · ·+a−2, u
68
−2u
67
+· · ·+2u+1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
12
=
1
u
2
a
3
=
−u
65
+ u
64
+ ··· − 3u + 2
u
65
− u
64
+ ··· + 7u
2
− 2u
a
8
=
u
u
3
+ u
a
9
=
u
10
+ 3u
8
+ 2u
6
− 3u
4
− 3u
2
+ 1
−u
10
− 4u
8
− 5u
6
+ 3u
2
a
4
=
u
65
− u
64
+ ··· − 4u + 1
u
65
− u
64
+ ··· + 4u
2
− 3u
a
10
=
−u
8
− 3u
6
− 3u
4
+ 1
−u
10
− 4u
8
− 5u
6
+ 3u
2
a
11
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
u
11
+ 4u
9
+ 6u
7
+ 2u
5
− 3u
3
− 2u
u
13
+ 5u
11
+ 9u
9
+ 4u
7
− 6u
5
− 5u
3
+ u
a
2
=
u
63
− u
62
+ ··· − 4u + 2
u
65
− u
64
+ ··· + 5u
2
− 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
67
+ 8u
66
+ ··· + 36u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
68
− 7u
67
+ ··· + 3u − 1
c
3
, c
8
u
68
+ u
67
+ ··· − 128u − 64
c
5
, c
6
, c
10
u
68
+ 2u
67
+ ··· + 172u + 17
c
7
, c
11
, c
12
u
68
− 2u
67
+ ··· + 2u + 1
c
9
u
68
− 6u
67
+ ··· + 19512u − 3344
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
68
− 65y
67
+ ··· + 29y + 1
c
3
, c
8
y
68
− 39y
67
+ ··· − 53248y + 4096
c
5
, c
6
, c
10
y
68
− 66y
67
+ ··· − 5682y + 289
c
7
, c
11
, c
12
y
68
+ 54y
67
+ ··· − 26y + 1
c
9
y
68
+ 18y
67
+ ··· − 465943328y + 11182336
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.274868 + 1.060340I
a = −0.526754 − 0.257503I
b = 0.172316 + 0.844015I
−7.01631 + 3.93719I 0
u = 0.274868 − 1.060340I
a = −0.526754 + 0.257503I
b = 0.172316 − 0.844015I
−7.01631 − 3.93719I 0
u = −0.895864
a = −0.191061
b = 0.633345
4.93401 −0.595830
u = 0.096240 + 1.118240I
a = 0.850399 − 0.314781I
b = −0.471781 − 0.369891I
−1.83195 + 1.81972I 0
u = 0.096240 − 1.118240I
a = 0.850399 + 0.314781I
b = −0.471781 + 0.369891I
−1.83195 − 1.81972I 0
u = 0.869205 + 0.074527I
a = 1.315300 + 0.517853I
b = 0.83012 + 1.96876I
−0.59482 + 10.16420I 1.87647 − 5.92631I
u = 0.869205 − 0.074527I
a = 1.315300 − 0.517853I
b = 0.83012 − 1.96876I
−0.59482 − 10.16420I 1.87647 + 5.92631I
u = −0.861015 + 0.021460I
a = 0.068130 − 0.155149I
b = −0.310712 + 0.559057I
7.57123 − 1.21218I 9.68961 + 0.94684I
u = −0.861015 − 0.021460I
a = 0.068130 + 0.155149I
b = −0.310712 − 0.559057I
7.57123 + 1.21218I 9.68961 − 0.94684I
u = 0.856269 + 0.057784I
a = −1.47487 − 0.56535I
b = −1.22357 − 1.80577I
5.18799 + 5.94447I 5.15329 − 5.45502I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.856269 − 0.057784I
a = −1.47487 + 0.56535I
b = −1.22357 + 1.80577I
5.18799 − 5.94447I 5.15329 + 5.45502I
u = −0.842334 + 0.053972I
a = −0.054941 + 0.246831I
b = 0.457743 − 0.884595I
2.36110 − 3.67346I 3.13855 + 2.67186I
u = −0.842334 − 0.053972I
a = −0.054941 − 0.246831I
b = 0.457743 + 0.884595I
2.36110 + 3.67346I 3.13855 − 2.67186I
u = 0.832992 + 0.037249I
a = 1.70552 + 0.38792I
b = 1.61773 + 1.11342I
3.56303 + 1.00362I 2.39038 − 0.55408I
u = 0.832992 − 0.037249I
a = 1.70552 − 0.38792I
b = 1.61773 − 1.11342I
3.56303 − 1.00362I 2.39038 + 0.55408I
u = −0.039755 + 1.193740I
a = −1.47104 + 1.52287I
b = 1.210760 − 0.091966I
−4.53839 − 0.93955I 0
u = −0.039755 − 1.193740I
a = −1.47104 − 1.52287I
b = 1.210760 + 0.091966I
−4.53839 + 0.93955I 0
u = 0.726273 + 0.104581I
a = −1.340580 − 0.221368I
b = −0.223564 − 0.553926I
−4.17485 − 0.25355I 0.106562 − 0.720449I
u = 0.726273 − 0.104581I
a = −1.340580 + 0.221368I
b = −0.223564 + 0.553926I
−4.17485 + 0.25355I 0.106562 + 0.720449I
u = 0.139307 + 1.264580I
a = 0.387719 + 0.377472I
b = 0.052113 − 0.152925I
−3.21418 + 2.27728I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.139307 − 1.264580I
a = 0.387719 − 0.377472I
b = 0.052113 + 0.152925I
−3.21418 − 2.27728I 0
u = 0.420834 + 1.201420I
a = −1.004800 + 0.249172I
b = 0.82821 − 1.85021I
−4.06146 − 5.54434I 0
u = 0.420834 − 1.201420I
a = −1.004800 − 0.249172I
b = 0.82821 + 1.85021I
−4.06146 + 5.54434I 0
u = −0.385541 + 1.220900I
a = −1.264540 − 0.576743I
b = 0.601184 + 0.839021I
−1.23320 − 0.73928I 0
u = −0.385541 − 1.220900I
a = −1.264540 + 0.576743I
b = 0.601184 − 0.839021I
−1.23320 + 0.73928I 0
u = 0.402134 + 1.217770I
a = 1.050320 + 0.234054I
b = −1.21789 + 1.63529I
1.61428 − 1.42992I 0
u = 0.402134 − 1.217770I
a = 1.050320 − 0.234054I
b = −1.21789 − 1.63529I
1.61428 + 1.42992I 0
u = 0.376637 + 1.240150I
a = −0.656599 − 1.012060I
b = 1.64633 − 0.88359I
−0.15109 + 3.34387I 0
u = 0.376637 − 1.240150I
a = −0.656599 + 1.012060I
b = 1.64633 + 0.88359I
−0.15109 − 3.34387I 0
u = −0.402053 + 1.254660I
a = 0.816153 + 0.355455I
b = −0.409003 − 0.522193I
3.75268 − 3.31527I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.402053 − 1.254660I
a = 0.816153 − 0.355455I
b = −0.409003 + 0.522193I
3.75268 + 3.31527I 0
u = −0.104897 + 1.326030I
a = 0.59543 + 3.16636I
b = 0.56443 − 1.71708I
−6.63847 − 0.67362I 0
u = −0.104897 − 1.326030I
a = 0.59543 − 3.16636I
b = 0.56443 + 1.71708I
−6.63847 + 0.67362I 0
u = 0.126734 + 1.333640I
a = −0.924830 − 0.807120I
b = 0.106274 + 0.311282I
−8.45190 + 3.15667I 0
u = 0.126734 − 1.333640I
a = −0.924830 + 0.807120I
b = 0.106274 − 0.311282I
−8.45190 − 3.15667I 0
u = −0.146516 + 1.332920I
a = −0.99913 − 3.23738I
b = −0.36801 + 1.96714I
−6.11073 − 5.54221I 0
u = −0.146516 − 1.332920I
a = −0.99913 + 3.23738I
b = −0.36801 − 1.96714I
−6.11073 + 5.54221I 0
u = −0.427740 + 1.279080I
a = −0.709037 + 0.446061I
b = 0.629160 − 0.054561I
0.96243 − 4.73147I 0
u = −0.427740 − 1.279080I
a = −0.709037 − 0.446061I
b = 0.629160 + 0.054561I
0.96243 + 4.73147I 0
u = −0.396517 + 1.289790I
a = −0.091140 − 0.825123I
b = −0.217786 + 0.572163I
3.48887 − 5.72199I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.396517 − 1.289790I
a = −0.091140 + 0.825123I
b = −0.217786 − 0.572163I
3.48887 + 5.72199I 0
u = 0.375212 + 1.299320I
a = 0.99882 − 2.11483I
b = 1.59088 + 1.31222I
−0.60813 + 5.34727I 0
u = 0.375212 − 1.299320I
a = 0.99882 + 2.11483I
b = 1.59088 − 1.31222I
−0.60813 − 5.34727I 0
u = 0.325026 + 1.312790I
a = −1.12164 + 1.07523I
b = −0.598332 − 0.646888I
−8.57365 + 3.57514I 0
u = 0.325026 − 1.312790I
a = −1.12164 − 1.07523I
b = −0.598332 + 0.646888I
−8.57365 − 3.57514I 0
u = −0.355477 + 0.540754I
a = −1.56452 − 0.66386I
b = 0.109402 + 1.371900I
−7.83604 + 3.56277I −3.48429 − 0.26987I
u = −0.355477 − 0.540754I
a = −1.56452 + 0.66386I
b = 0.109402 − 1.371900I
−7.83604 − 3.56277I −3.48429 + 0.26987I
u = −0.536593 + 0.355337I
a = 1.184830 + 0.517192I
b = 0.32245 − 1.66008I
−7.14400 − 6.91551I −1.34390 + 7.27933I
u = −0.536593 − 0.355337I
a = 1.184830 − 0.517192I
b = 0.32245 + 1.66008I
−7.14400 + 6.91551I −1.34390 − 7.27933I
u = −0.380136 + 1.310130I
a = 0.137470 + 1.319400I
b = 0.338862 − 0.902883I
−1.90173 − 8.06695I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.380136 − 1.310130I
a = 0.137470 − 1.319400I
b = 0.338862 + 0.902883I
−1.90173 + 8.06695I 0
u = −0.076307 + 1.363730I
a = −0.76099 − 2.76851I
b = −0.28084 + 1.50736I
−13.64730 + 2.32372I 0
u = −0.076307 − 1.363730I
a = −0.76099 + 2.76851I
b = −0.28084 − 1.50736I
−13.64730 − 2.32372I 0
u = −0.165622 + 1.358730I
a = 1.04285 + 3.06303I
b = 0.23955 − 1.93117I
−12.5196 − 9.3121I 0
u = −0.165622 − 1.358730I
a = 1.04285 − 3.06303I
b = 0.23955 + 1.93117I
−12.5196 + 9.3121I 0
u = 0.388285 + 1.314120I
a = −1.57085 + 2.17185I
b = −1.20114 − 1.94252I
0.89978 + 10.41270I 0
u = 0.388285 − 1.314120I
a = −1.57085 − 2.17185I
b = −1.20114 + 1.94252I
0.89978 − 10.41270I 0
u = 0.393601 + 1.326560I
a = 1.80874 − 2.01322I
b = 0.80659 + 2.06065I
−4.9810 + 14.6950I 0
u = 0.393601 − 1.326560I
a = 1.80874 + 2.01322I
b = 0.80659 − 2.06065I
−4.9810 − 14.6950I 0
u = −0.461201 + 0.310199I
a = −1.342820 − 0.347618I
b = −0.39492 + 1.52052I
−1.02729 − 3.44858I 1.47745 + 8.39807I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.461201 − 0.310199I
a = −1.342820 + 0.347618I
b = −0.39492 − 1.52052I
−1.02729 + 3.44858I 1.47745 − 8.39807I
u = 0.397825 + 0.339088I
a = −1.159450 − 0.792131I
b = 0.421810 + 0.207688I
−3.31331 + 1.34942I −0.69445 − 4.81797I
u = 0.397825 − 0.339088I
a = −1.159450 + 0.792131I
b = 0.421810 − 0.207688I
−3.31331 − 1.34942I −0.69445 + 4.81797I
u = −0.295735 + 0.374812I
a = 1.64151 + 0.53692I
b = 0.199141 − 1.171910I
−1.52842 + 0.74751I −1.82763 + 0.40847I
u = −0.295735 − 0.374812I
a = 1.64151 − 0.53692I
b = 0.199141 + 1.171910I
−1.52842 − 0.74751I −1.82763 − 0.40847I
u = 0.437462 + 0.102734I
a = 0.720475 + 0.489361I
b = −0.1275350 − 0.0350142I
0.917664 + 0.265375I 10.32983 − 1.81252I
u = 0.437462 − 0.102734I
a = 0.720475 − 0.489361I
b = −0.1275350 + 0.0350142I
0.917664 − 0.265375I 10.32983 + 1.81252I
u = −0.227060
a = 2.62077
b = 0.966677
−1.29008 −12.1470
11
II. I
u
2
= hb − 1, −u
4
− u
3
− 2u
2
+ a − u, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
12
=
1
u
2
a
3
=
u
4
+ u
3
+ 2u
2
+ u
1
a
8
=
u
u
3
+ u
a
9
=
u
u
3
+ u
a
4
=
u
4
+ u
3
+ 2u
2
+ u
1
a
10
=
u
3
+ 2u
u
3
+ u
a
11
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
−u
5
− 2u
3
− u
−u
5
− u
4
− 2u
3
− u
2
− u + 1
a
5
=
−1
0
a
2
=
u
4
+ u
3
+ 2u
2
+ u + 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 6u
3
+ 11u
2
+ 6u + 5
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
, c
6
, c
9
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
7
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
, c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
6
, c
9
c
10
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
7
, c
11
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = 0.567375
b = 1.00000
6.01515 7.06030
u = 0.138835 + 1.234450I
a = −1.35607 − 0.92119I
b = 1.00000
−4.60518 + 1.97241I −3.77811 − 4.83849I
u = 0.138835 − 1.234450I
a = −1.35607 + 0.92119I
b = 1.00000
−4.60518 − 1.97241I −3.77811 + 4.83849I
u = −0.408802 + 1.276380I
a = −0.354716 + 0.801205I
b = 1.00000
2.05064 − 4.59213I 3.28527 + 2.79936I
u = −0.408802 − 1.276380I
a = −0.354716 − 0.801205I
b = 1.00000
2.05064 + 4.59213I 3.28527 − 2.79936I
u = 0.413150
a = 0.854195
b = 1.00000
−0.906083 9.92530
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
68
− 7u
67
+ ··· + 3u − 1)
c
3
, c
8
u
6
(u
68
+ u
67
+ ··· − 128u − 64)
c
4
((u + 1)
6
)(u
68
− 7u
67
+ ··· + 3u − 1)
c
5
, c
6
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
68
+ 2u
67
+ ··· + 172u + 17)
c
7
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
68
− 2u
67
+ ··· + 2u + 1)
c
9
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
68
− 6u
67
+ ··· + 19512u − 3344)
c
10
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
68
+ 2u
67
+ ··· + 172u + 17)
c
11
, c
12
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
68
− 2u
67
+ ··· + 2u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
68
− 65y
67
+ ··· + 29y + 1)
c
3
, c
8
y
6
(y
68
− 39y
67
+ ··· − 53248y + 4096)
c
5
, c
6
, c
10
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
68
− 66y
67
+ ··· − 5682y + 289)
c
7
, c
11
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
68
+ 54y
67
+ ··· − 26y + 1)
c
9
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
68
+ 18y
67
+ ··· − 465943328y + 11182336)
17
