12a
0642
(K12a
0642
)
A knot diagram
1
Linearized knot diagam
3 7 10 11 8 2 6 12 1 5 4 9
Solving Sequence
3,7
2
1,9
10 4 6 8 5 12 11
c
2
c
1
c
9
c
3
c
6
c
7
c
5
c
12
c
11
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.08528 × 10
21
u
65
+ 3.42453 × 10
21
u
64
+ ··· + 2.45923 × 10
21
b − 6.71501 × 10
21
,
− 7.92933 × 10
20
u
65
+ 2.33303 × 10
21
u
64
+ ··· + 7.37769 × 10
21
a − 1.03514 × 10
21
, u
66
− 2u
65
+ ··· + 5u − 3i
I
u
2
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
I
u
3
= hu
2
a − au + u
2
+ b − u + 1, 2u
2
a + a
2
− 2au + 2u
2
− u + 1, u
3
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 75 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−2.09×10
21
u
65
+3.42×10
21
u
64
+· · ·+2.46×10
21
b−6.72×10
21
, −7.93×
10
20
u
65
+2.33×10
21
u
64
+· · ·+7.38×10
21
a−1.04×10
21
, u
66
−2u
65
+· · ·+5u−3i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
9
=
0.107477u
65
− 0.316227u
64
+ ··· + 4.20310u + 0.140307
0.847942u
65
− 1.39252u
64
+ ··· + 1.15079u + 2.73053
a
10
=
−0.116848u
65
+ 0.169205u
64
+ ··· + 7.20302u − 0.938417
−0.135801u
65
− 0.253372u
64
+ ··· + 3.19813u + 0.135182
a
4
=
0.435419u
65
− 0.831909u
64
+ ··· − 2.53069u + 0.102436
−0.849654u
65
+ 0.956704u
64
+ ··· + 1.71311u − 2.32719
a
6
=
u
−u
3
+ u
a
8
=
−u
3
u
5
− u
3
+ u
a
5
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
12
=
0.808905u
65
− 0.823841u
64
+ ··· − 1.79298u + 6.02411
−0.647456u
65
+ 0.938023u
64
+ ··· + 1.14605u − 2.05947
a
11
=
−0.228635u
65
+ 0.368246u
64
+ ··· + 5.73814u − 0.217095
0.0421408u
65
− 0.505020u
64
+ ··· + 2.54989u + 1.29366
(ii) Obstruction class = −1
(iii) Cusp Shapes =
6130397275329191743669
1229615623032466676507
u
65
−
9723677310654894627572
1229615623032466676507
u
64
+ ··· +
20879266412996566941410
1229615623032466676507
u +
23149949036627936894973
1229615623032466676507
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
u
66
+ 16u
65
+ ··· + 151u + 9
c
2
, c
6
u
66
− 2u
65
+ ··· + 5u − 3
c
3
u
66
+ u
65
+ ··· + 1808u + 1480
c
4
, c
10
, c
11
u
66
− u
65
+ ··· − 32u + 8
c
8
, c
9
, c
12
u
66
− 4u
65
+ ··· − 12u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
y
66
+ 72y
65
+ ··· + 1949y + 81
c
2
, c
6
y
66
− 16y
65
+ ··· − 151y + 9
c
3
y
66
− 25y
65
+ ··· + 8713216y + 2190400
c
4
, c
10
, c
11
y
66
+ 59y
65
+ ··· − 128y + 64
c
8
, c
9
, c
12
y
66
− 66y
65
+ ··· + 210y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.940880 + 0.389587I
a = 0.801166 − 0.701373I
b = 0.261406 + 0.593445I
−5.12964 + 6.16163I −3.04372 − 8.05147I
u = −0.940880 − 0.389587I
a = 0.801166 + 0.701373I
b = 0.261406 − 0.593445I
−5.12964 − 6.16163I −3.04372 + 8.05147I
u = −1.02149
a = −1.32124
b = −0.413413
2.55529 4.40390
u = −0.796362 + 0.550324I
a = −0.924966 − 0.099092I
b = 0.43221 − 1.57388I
3.14358 + 2.21788I 4.13427 − 2.24025I
u = −0.796362 − 0.550324I
a = −0.924966 + 0.099092I
b = 0.43221 + 1.57388I
3.14358 − 2.21788I 4.13427 + 2.24025I
u = 1.046540 + 0.097833I
a = 1.322190 − 0.020742I
b = 0.397058 − 0.080498I
−1.46918 + 3.46747I 0
u = 1.046540 − 0.097833I
a = 1.322190 + 0.020742I
b = 0.397058 + 0.080498I
−1.46918 − 3.46747I 0
u = 0.921469 + 0.129566I
a = −0.661877 − 0.412132I
b = 0.514033 + 0.681154I
−6.59268 + 0.97290I −7.57653 − 0.39847I
u = 0.921469 − 0.129566I
a = −0.661877 + 0.412132I
b = 0.514033 − 0.681154I
−6.59268 − 0.97290I −7.57653 + 0.39847I
u = 0.833726 + 0.384546I
a = −0.587442 − 0.795014I
b = −0.144390 + 0.286926I
−0.18506 − 3.20503I 2.36874 + 8.90166I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.833726 − 0.384546I
a = −0.587442 + 0.795014I
b = −0.144390 − 0.286926I
−0.18506 + 3.20503I 2.36874 − 8.90166I
u = 0.990925 + 0.475204I
a = 0.428686 + 0.120652I
b = −0.10012 − 1.70913I
5.36492 − 5.88722I 0
u = 0.990925 − 0.475204I
a = 0.428686 − 0.120652I
b = −0.10012 + 1.70913I
5.36492 + 5.88722I 0
u = −0.498795 + 0.739168I
a = −1.153080 − 0.711498I
b = 0.622959 − 1.122100I
4.16450 + 2.72632I 6.92245 − 3.37892I
u = −0.498795 − 0.739168I
a = −1.153080 + 0.711498I
b = 0.622959 + 1.122100I
4.16450 − 2.72632I 6.92245 + 3.37892I
u = −1.029240 + 0.415785I
a = −0.304326 + 0.230817I
b = 0.01201 − 1.76238I
0.44075 + 9.86566I 0
u = −1.029240 − 0.415785I
a = −0.304326 − 0.230817I
b = 0.01201 + 1.76238I
0.44075 − 9.86566I 0
u = −0.793049 + 0.384644I
a = −1.47843 + 0.13514I
b = −0.776296 − 0.191562I
−3.25737 + 1.93313I 0.42721 − 3.82999I
u = −0.793049 − 0.384644I
a = −1.47843 − 0.13514I
b = −0.776296 + 0.191562I
−3.25737 − 1.93313I 0.42721 + 3.82999I
u = −0.959975 + 0.624161I
a = −0.499021 − 0.214013I
b = 0.15361 − 1.51024I
2.82571 + 2.19205I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.959975 − 0.624161I
a = −0.499021 + 0.214013I
b = 0.15361 + 1.51024I
2.82571 − 2.19205I 0
u = −0.886053 + 0.741340I
a = −0.89572 − 1.29867I
b = 0.177451 − 1.335260I
−1.92250 + 2.81816I 0
u = −0.886053 − 0.741340I
a = −0.89572 + 1.29867I
b = 0.177451 + 1.335260I
−1.92250 − 2.81816I 0
u = 0.882438 + 0.780869I
a = 0.699653 − 0.793108I
b = −0.180230 − 1.231910I
4.00172 − 2.93689I 0
u = 0.882438 − 0.780869I
a = 0.699653 + 0.793108I
b = −0.180230 + 1.231910I
4.00172 + 2.93689I 0
u = 0.356962 + 0.735559I
a = 1.16102 − 0.91617I
b = −0.668874 − 1.010960I
7.41551 + 1.48844I 10.63091 − 0.75571I
u = 0.356962 − 0.735559I
a = 1.16102 + 0.91617I
b = −0.668874 + 1.010960I
7.41551 − 1.48844I 10.63091 + 0.75571I
u = 0.727878 + 0.303493I
a = 1.48101 + 0.77779I
b = −0.88271 − 2.24176I
−3.78561 − 1.23006I 1.98020 + 5.99169I
u = 0.727878 − 0.303493I
a = 1.48101 − 0.77779I
b = −0.88271 + 2.24176I
−3.78561 + 1.23006I 1.98020 − 5.99169I
u = −0.248411 + 0.746724I
a = −1.07892 − 1.06631I
b = 0.658592 − 0.929096I
2.98364 − 5.69446I 6.38560 + 3.41468I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.248411 − 0.746724I
a = −1.07892 + 1.06631I
b = 0.658592 + 0.929096I
2.98364 + 5.69446I 6.38560 − 3.41468I
u = −0.762533 + 0.185034I
a = 0.400057 − 0.427963I
b = −0.335136 + 0.241912I
−1.264720 + 0.575310I −4.44845 − 0.83683I
u = −0.762533 − 0.185034I
a = 0.400057 + 0.427963I
b = −0.335136 − 0.241912I
−1.264720 − 0.575310I −4.44845 + 0.83683I
u = 0.849055 + 0.877486I
a = 0.144726 − 0.873025I
b = −0.025755 − 1.136960I
2.96927 + 3.52515I 0
u = 0.849055 − 0.877486I
a = 0.144726 + 0.873025I
b = −0.025755 + 1.136960I
2.96927 − 3.52515I 0
u = 0.887168 + 0.853480I
a = 0.936403 − 0.683518I
b = −0.229264 − 1.107390I
4.14950 − 1.92316I 0
u = 0.887168 − 0.853480I
a = 0.936403 + 0.683518I
b = −0.229264 + 1.107390I
4.14950 + 1.92316I 0
u = 0.825576 + 0.913463I
a = −0.59888 + 3.00432I
b = 1.19791 + 3.03712I
9.28839 + 8.00052I 0
u = 0.825576 − 0.913463I
a = −0.59888 − 3.00432I
b = 1.19791 − 3.03712I
9.28839 − 8.00052I 0
u = −0.884770 + 0.860589I
a = −0.212411 − 0.810921I
b = 0.040113 − 1.187130I
7.38355 + 0.47295I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.884770 − 0.860589I
a = −0.212411 + 0.810921I
b = 0.040113 + 1.187130I
7.38355 − 0.47295I 0
u = −0.907494 + 0.841916I
a = 2.68354 + 3.61420I
b = −0.25692 + 4.58473I
2.86142 + 3.13275I 0
u = −0.907494 − 0.841916I
a = 2.68354 − 3.61420I
b = −0.25692 − 4.58473I
2.86142 − 3.13275I 0
u = −0.854780 + 0.913777I
a = 0.95509 + 2.99116I
b = −0.97675 + 3.23830I
14.5393 − 3.2940I 0
u = −0.854780 − 0.913777I
a = 0.95509 − 2.99116I
b = −0.97675 − 3.23830I
14.5393 + 3.2940I 0
u = 0.928525 + 0.839216I
a = 0.267321 − 0.699993I
b = −0.044132 − 1.250160I
4.02096 − 4.36836I 0
u = 0.928525 − 0.839216I
a = 0.267321 + 0.699993I
b = −0.044132 + 1.250160I
4.02096 + 4.36836I 0
u = −0.935049 + 0.841580I
a = −0.997222 − 0.708350I
b = 0.164677 − 1.077480I
7.22562 + 5.84884I 0
u = −0.935049 − 0.841580I
a = −0.997222 + 0.708350I
b = 0.164677 + 1.077480I
7.22562 − 5.84884I 0
u = 0.886666 + 0.900213I
a = −1.43168 + 3.00902I
b = 0.70096 + 3.51971I
12.21740 − 1.91088I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.886666 − 0.900213I
a = −1.43168 − 3.00902I
b = 0.70096 − 3.51971I
12.21740 + 1.91088I 0
u = 0.965870 + 0.830951I
a = 1.048900 − 0.703382I
b = −0.122028 − 1.049300I
2.60149 − 9.86130I 0
u = 0.965870 − 0.830951I
a = 1.048900 + 0.703382I
b = −0.122028 + 1.049300I
2.60149 + 9.86130I 0
u = 0.957311 + 0.866384I
a = −2.46807 + 2.30758I
b = −0.30755 + 3.64760I
11.99000 − 4.61174I 0
u = 0.957311 − 0.866384I
a = −2.46807 − 2.30758I
b = −0.30755 − 3.64760I
11.99000 + 4.61174I 0
u = 0.998275 + 0.835366I
a = −2.75742 + 1.52819I
b = −0.88652 + 3.29956I
8.7382 − 14.4571I 0
u = 0.998275 − 0.835366I
a = −2.75742 − 1.52819I
b = −0.88652 − 3.29956I
8.7382 + 14.4571I 0
u = −0.983999 + 0.853365I
a = 2.61145 + 1.85077I
b = 0.63135 + 3.43344I
14.1256 + 9.8142I 0
u = −0.983999 − 0.853365I
a = 2.61145 − 1.85077I
b = 0.63135 − 3.43344I
14.1256 − 9.8142I 0
u = −0.482668 + 0.473305I
a = 0.026115 − 1.148390I
b = −0.243945 − 0.319913I
−2.37826 + 1.37049I 3.17735 − 4.52175I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.482668 − 0.473305I
a = 0.026115 + 1.148390I
b = −0.243945 + 0.319913I
−2.37826 − 1.37049I 3.17735 + 4.52175I
u = −0.268428 + 0.572165I
a = −0.433288 + 0.777956I
b = −0.434835 + 0.673053I
−3.09183 − 2.58712I 2.42181 + 3.21581I
u = −0.268428 − 0.572165I
a = −0.433288 − 0.777956I
b = −0.434835 − 0.673053I
−3.09183 + 2.58712I 2.42181 − 3.21581I
u = 0.438013 + 0.335978I
a = 0.778089 + 0.240873I
b = 0.555725 + 0.212651I
0.967621 + 0.114990I 10.14067 − 0.04829I
u = 0.438013 − 0.335978I
a = 0.778089 − 0.240873I
b = 0.555725 − 0.212651I
0.967621 − 0.114990I 10.14067 + 0.04829I
u = 0.493664
a = 1.46260
b = 0.604228
0.957505 14.1910
11
II. I
u
2
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
9
=
u
2
+ u
u
2
a
10
=
u + 1
0
a
4
=
1
0
a
6
=
u
u
2
+ u − 1
a
8
=
u
2
− 1
u
2
a
5
=
1
0
a
12
=
u + 1
0
a
11
=
u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 10u + 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
3
, c
4
, c
10
c
11
u
3
c
6
u
3
− u
2
+ 1
c
7
u
3
+ u
2
+ 2u + 1
c
8
, c
9
(u + 1)
3
c
12
(u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
y
3
+ 3y
2
+ 2y −1
c
2
, c
6
y
3
− y
2
+ 2y −1
c
3
, c
4
, c
10
c
11
y
3
c
8
, c
9
, c
12
(y −1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.662359 − 0.562280I
b = 0.215080 − 1.307140I
4.66906 + 2.82812I 11.91407 − 2.22005I
u = −0.877439 − 0.744862I
a = −0.662359 + 0.562280I
b = 0.215080 + 1.307140I
4.66906 − 2.82812I 11.91407 + 2.22005I
u = 0.754878
a = 1.32472
b = 0.569840
0.531480 −5.82810
15
III.
I
u
3
= hu
2
a − au + u
2
+ b − u + 1, 2u
2
a + a
2
− 2au + 2u
2
− u + 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
9
=
a
−u
2
a + au − u
2
+ u − 1
a
10
=
u
2
+ a − 1
−u
2
a + au + u − 1
a
4
=
−u
2
a + au − u
2
− a + 4u
2
a
6
=
u
−u
2
+ u + 1
a
8
=
−u
2
+ 1
−u
2
a
5
=
−1
0
a
12
=
−u
2
− a + 1
u
2
a − au − u + 1
a
11
=
−u
2
a + au + u
2
+ a + u − 2
−u
2
a + au + u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
− u
2
+ 1)
2
c
3
, c
4
, c
10
c
11
(u
2
+ 2)
3
c
6
(u
3
+ u
2
− 1)
2
c
7
(u
3
+ u
2
+ 2u + 1)
2
c
8
, c
9
(u − 1)
6
c
12
(u + 1)
6
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
6
(y
3
− y
2
+ 2y −1)
2
c
3
, c
4
, c
10
c
11
(y + 2)
6
c
8
, c
9
, c
12
(y −1)
6
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.391035 + 0.678606I
b = −0.215080 + 0.107072I
−0.26574 − 2.82812I 3.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 1.71575 − 1.80317I
b = −0.21508 − 2.72135I
−0.26574 − 2.82812I 3.50976 + 2.97945I
u = 0.877439 − 0.744862I
a = −0.391035 − 0.678606I
b = −0.215080 − 0.107072I
−0.26574 + 2.82812I 3.50976 − 2.97945I
u = 0.877439 − 0.744862I
a = 1.71575 + 1.80317I
b = −0.21508 + 2.72135I
−0.26574 + 2.82812I 3.50976 − 2.97945I
u = −0.754878
a = −1.32472 + 1.06756I
b = −0.56984 − 1.41421I
−4.40332 −3.01950
u = −0.754878
a = −1.32472 − 1.06756I
b = −0.56984 + 1.41421I
−4.40332 −3.01950
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
((u
3
− u
2
+ 2u − 1)
3
)(u
66
+ 16u
65
+ ··· + 151u + 9)
c
2
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
66
− 2u
65
+ ··· + 5u − 3)
c
3
u
3
(u
2
+ 2)
3
(u
66
+ u
65
+ ··· + 1808u + 1480)
c
4
, c
10
, c
11
u
3
(u
2
+ 2)
3
(u
66
− u
65
+ ··· − 32u + 8)
c
6
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
66
− 2u
65
+ ··· + 5u − 3)
c
7
((u
3
+ u
2
+ 2u + 1)
3
)(u
66
+ 16u
65
+ ··· + 151u + 9)
c
8
, c
9
((u − 1)
6
)(u + 1)
3
(u
66
− 4u
65
+ ··· − 12u − 1)
c
12
((u − 1)
3
)(u + 1)
6
(u
66
− 4u
65
+ ··· − 12u − 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
((y
3
+ 3y
2
+ 2y −1)
3
)(y
66
+ 72y
65
+ ··· + 1949y + 81)
c
2
, c
6
((y
3
− y
2
+ 2y −1)
3
)(y
66
− 16y
65
+ ··· − 151y + 9)
c
3
y
3
(y + 2)
6
(y
66
− 25y
65
+ ··· + 8713216y + 2190400)
c
4
, c
10
, c
11
y
3
(y + 2)
6
(y
66
+ 59y
65
+ ··· − 128y + 64)
c
8
, c
9
, c
12
((y −1)
9
)(y
66
− 66y
65
+ ··· + 210y + 1)
21
