12a
0636
(K12a
0636
)
A knot diagram
1
Linearized knot diagam
3 7 10 9 11 2 1 12 4 6 5 8
Solving Sequence
2,6
7 3 1
8,11
5 12 10 4 9
c
6
c
2
c
1
c
7
c
5
c
11
c
10
c
3
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
25
− 2u
24
+ ··· + b − 1, 7u
25
+ 15u
24
+ ··· + 2a + 11, u
26
+ 3u
25
+ ··· + 11u + 2i
I
u
2
= h−u
13
a − 8u
14
+ ··· − a − 10, −2u
13
a + 3u
14
+ ··· + a − 2,
u
15
− u
14
− 4u
13
+ 5u
12
+ 6u
11
− 10u
10
+ 7u
8
− 8u
7
+ 4u
6
+ 6u
5
− 8u
4
+ 2u
3
+ 2u
2
− 2u + 1i
I
u
3
= h−u
9
+ 2u
7
− u
5
− 2u
3
+ b + u, −u
8
− u
7
+ 3u
6
+ 2u
5
− 3u
4
− u
3
− u
2
+ a − u + 2,
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 66 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
25
−2u
24
+· · ·+b−1, 7u
25
+15u
24
+· · ·+2a+11, u
26
+3u
25
+· · ·+11u+2i
(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
8
=
u
6
− u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
11
=
−
7
2
u
25
−
15
2
u
24
+ ··· − 31u −
11
2
u
25
+ 2u
24
+ ··· + 8u + 1
a
5
=
−
1
2
u
25
−
3
2
u
24
+ ··· − 6u −
1
2
−u
24
− u
23
+ ··· − 5u − 1
a
12
=
u
9
− 2u
7
+ u
5
+ 2u
3
− u
u
11
− 3u
9
+ 4u
7
− u
5
− u
3
+ u
a
10
=
−
9
2
u
25
−
19
2
u
24
+ ··· − 39u −
13
2
u
25
+ 2u
24
+ ··· + 8u + 1
a
4
=
−
1
2
u
25
−
1
2
u
24
+ ··· − 2u −
1
2
−u
24
− u
23
+ ··· − 4u − 1
a
9
=
u
12
− 3u
10
+ 3u
8
+ 2u
6
− 4u
4
+ u
2
+ 1
u
14
− 4u
12
+ 7u
10
− 4u
8
− 2u
6
+ 4u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
25
− 4u
24
+ 12u
23
+ 34u
22
− 22u
21
− 122u
20
− 30u
19
+
220u
18
+ 202u
17
− 140u
16
− 356u
15
− 190u
14
+ 202u
13
+ 434u
12
+ 198u
11
− 234u
10
−
362u
9
− 134u
8
+ 126u
7
+ 202u
6
+ 86u
5
− 38u
4
− 78u
3
− 32u
2
+ 4u + 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 15u
25
+ ··· + 17u + 4
c
2
, c
6
u
26
− 3u
25
+ ··· − 11u + 2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
26
+ 18u
24
+ ··· − u + 1
c
7
, c
8
, c
12
u
26
− 9u
25
+ ··· − 215u + 26
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 7y
25
+ ··· + 207y + 16
c
2
, c
6
y
26
− 15y
25
+ ··· − 17y + 4
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
26
+ 36y
25
+ ··· + 5y + 1
c
7
, c
8
, c
12
y
26
+ 29y
25
+ ··· − 257y + 676
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.956817 + 0.396091I
a = −0.651697 + 0.991591I
b = 0.487266 + 0.268935I
−0.58961 + 3.47100I 5.92481 − 9.23773I
u = −0.956817 − 0.396091I
a = −0.651697 − 0.991591I
b = 0.487266 − 0.268935I
−0.58961 − 3.47100I 5.92481 + 9.23773I
u = −0.751125 + 0.602479I
a = −1.187320 − 0.581977I
b = 0.03078 + 1.53000I
−8.48939 + 2.35165I 0.61282 − 3.28103I
u = −0.751125 − 0.602479I
a = −1.187320 + 0.581977I
b = 0.03078 − 1.53000I
−8.48939 − 2.35165I 0.61282 + 3.28103I
u = 0.904845 + 0.273734I
a = 0.298839 + 0.531776I
b = 0.141020 + 0.376831I
−1.42636 − 1.11233I 0.498012 + 0.385891I
u = 0.904845 − 0.273734I
a = 0.298839 − 0.531776I
b = 0.141020 − 0.376831I
−1.42636 + 1.11233I 0.498012 − 0.385891I
u = −0.069667 + 0.918847I
a = 0.772660 + 0.797604I
b = −0.33361 + 1.64747I
−18.8709 − 8.2243I −1.19263 + 3.34279I
u = −0.069667 − 0.918847I
a = 0.772660 − 0.797604I
b = −0.33361 − 1.64747I
−18.8709 + 8.2243I −1.19263 − 3.34279I
u = −0.006971 + 0.822667I
a = −0.236714 − 0.504478I
b = 0.401873 − 0.547293I
−4.02599 − 1.44616I 4.11158 + 4.76185I
u = −0.006971 − 0.822667I
a = −0.236714 + 0.504478I
b = 0.401873 + 0.547293I
−4.02599 + 1.44616I 4.11158 − 4.76185I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.341936 + 0.725717I
a = −0.612752 + 0.286096I
b = 0.14424 − 1.56542I
−10.39290 − 4.06497I 0.32404 + 2.28928I
u = −0.341936 − 0.725717I
a = −0.612752 − 0.286096I
b = 0.14424 + 1.56542I
−10.39290 + 4.06497I 0.32404 − 2.28928I
u = −1.068130 + 0.547532I
a = 2.11393 − 1.03569I
b = −0.20016 − 1.55835I
−12.4889 + 8.8626I −2.32670 − 6.75099I
u = −1.068130 − 0.547532I
a = 2.11393 + 1.03569I
b = −0.20016 + 1.55835I
−12.4889 − 8.8626I −2.32670 + 6.75099I
u = 1.201580 + 0.188741I
a = −0.30204 − 2.23637I
b = −0.11408 − 1.65028I
−15.2170 + 1.3980I −5.67569 − 0.13534I
u = 1.201580 − 0.188741I
a = −0.30204 + 2.23637I
b = −0.11408 + 1.65028I
−15.2170 − 1.3980I −5.67569 + 0.13534I
u = 1.223020 + 0.456500I
a = −0.578173 − 0.689340I
b = −0.367731 − 0.587274I
−7.67362 − 3.10658I 0.75309 − 1.41288I
u = 1.223020 − 0.456500I
a = −0.578173 + 0.689340I
b = −0.367731 + 0.587274I
−7.67362 + 3.10658I 0.75309 + 1.41288I
u = −1.225690 + 0.460987I
a = 0.339499 − 1.317720I
b = −0.447158 − 0.574007I
−7.64611 + 6.04329I 0.84178 − 7.93478I
u = −1.225690 − 0.460987I
a = 0.339499 + 1.317720I
b = −0.447158 + 0.574007I
−7.64611 − 6.04329I 0.84178 + 7.93478I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.288060 + 0.430073I
a = 0.67479 + 2.08726I
b = 0.32678 + 1.67472I
16.3921 + 3.4886I −4.76966 − 0.45734I
u = 1.288060 − 0.430073I
a = 0.67479 − 2.08726I
b = 0.32678 − 1.67472I
16.3921 − 3.4886I −4.76966 + 0.45734I
u = −1.258780 + 0.509888I
a = −1.71248 + 2.27956I
b = 0.35659 + 1.64288I
16.9890 + 13.3492I −4.07199 − 6.35165I
u = −1.258780 − 0.509888I
a = −1.71248 − 2.27956I
b = 0.35659 − 1.64288I
16.9890 − 13.3492I −4.07199 + 6.35165I
u = −0.438383 + 0.308301I
a = 0.831459 − 0.008336I
b = −0.425812 + 0.076831I
0.801816 − 0.109645I 12.97054 + 1.47787I
u = −0.438383 − 0.308301I
a = 0.831459 + 0.008336I
b = −0.425812 − 0.076831I
0.801816 + 0.109645I 12.97054 − 1.47787I
7
II. I
u
2
=
h−u
13
a−8u
14
+· · ·−a−10, −2u
13
a+3u
14
+· · ·+a−2, u
15
−u
14
+· · ·−2u+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
8
=
u
6
− u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
11
=
a
0.727273u
14
+ 0.0909091au
13
+ ··· + 0.0909091a + 0.909091
a
5
=
0.272727au
14
− 0.363636u
14
+ ··· − 0.909091a + 0.727273
−0.363636au
14
− 0.181818au
13
+ ··· − 0.181818a − 0.909091
a
12
=
u
9
− 2u
7
+ u
5
+ 2u
3
− u
u
11
− 3u
9
+ 4u
7
− u
5
− u
3
+ u
a
10
=
−0.727273u
14
− 0.0909091au
13
+ ··· + 0.909091a − 0.909091
0.727273u
14
+ 0.0909091au
13
+ ··· + 0.0909091a + 0.909091
a
4
=
0.727273au
14
+ 0.363636u
14
+ ··· + 0.909091a − 0.727273
1
a
9
=
u
12
− 3u
10
+ 3u
8
+ 2u
6
− 4u
4
+ u
2
+ 1
u
14
− 4u
12
+ 7u
10
− 4u
8
− 2u
6
+ 4u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
13
− 16u
11
+ 4u
10
+ 28u
9
− 12u
8
− 12u
7
+ 16u
6
− 16u
5
+ 24u
3
− 8u
2
+ 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
15
+ 9u
14
+ ··· − 4u
2
+ 1)
2
c
2
, c
6
(u
15
+ u
14
+ ··· − 2u − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
30
+ u
29
+ ··· + 54u + 17
c
7
, c
8
, c
12
(u
15
+ 3u
14
+ ··· + 8u
2
− 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
15
− 5y
14
+ ··· + 8y −1)
2
c
2
, c
6
(y
15
− 9y
14
+ ··· + 4y
2
− 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
30
+ 27y
29
+ ··· + 4428y + 289
c
7
, c
8
, c
12
(y
15
+ 19y
14
+ ··· + 16y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.023100 + 0.900040I
a = −0.360717 + 1.290430I
b = 0.10773 + 1.57610I
−11.31470 + 3.25615I 0.32867 − 2.40088I
u = 0.023100 + 0.900040I
a = 0.518581 − 0.298234I
b = −0.988185 − 0.651753I
−11.31470 + 3.25615I 0.32867 − 2.40088I
u = 0.023100 − 0.900040I
a = −0.360717 − 1.290430I
b = 0.10773 − 1.57610I
−11.31470 − 3.25615I 0.32867 + 2.40088I
u = 0.023100 − 0.900040I
a = 0.518581 + 0.298234I
b = −0.988185 + 0.651753I
−11.31470 − 3.25615I 0.32867 + 2.40088I
u = −0.863978
a = −1.75727 + 1.74904I
b = 0.234017 + 1.079020I
−4.54552 −4.48380
u = −0.863978
a = −1.75727 − 1.74904I
b = 0.234017 − 1.079020I
−4.54552 −4.48380
u = −1.093890 + 0.311098I
a = −0.115665 + 0.228205I
b = −0.603738 + 0.781085I
−6.68965 + 1.10849I −3.51398 − 0.68443I
u = −1.093890 + 0.311098I
a = 0.99897 − 2.63611I
b = −0.061421 − 1.364080I
−6.68965 + 1.10849I −3.51398 − 0.68443I
u = −1.093890 − 0.311098I
a = −0.115665 − 0.228205I
b = −0.603738 − 0.781085I
−6.68965 − 1.10849I −3.51398 + 0.68443I
u = −1.093890 − 0.311098I
a = 0.99897 + 2.63611I
b = −0.061421 + 1.364080I
−6.68965 − 1.10849I −3.51398 + 0.68443I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.747479 + 0.391613I
a = 1.293270 − 0.129521I
b = −0.046233 + 1.126590I
−2.04760 − 1.75942I 6.85085 + 5.01461I
u = 0.747479 + 0.391613I
a = −0.75255 + 1.32580I
b = 0.169565 − 0.296554I
−2.04760 − 1.75942I 6.85085 + 5.01461I
u = 0.747479 − 0.391613I
a = 1.293270 + 0.129521I
b = −0.046233 − 1.126590I
−2.04760 + 1.75942I 6.85085 − 5.01461I
u = 0.747479 − 0.391613I
a = −0.75255 − 1.32580I
b = 0.169565 + 0.296554I
−2.04760 + 1.75942I 6.85085 − 5.01461I
u = 1.070290 + 0.443484I
a = 0.982216 + 0.740855I
b = −0.692609 + 0.458051I
−5.70338 − 5.68434I −0.20490 + 7.47679I
u = 1.070290 + 0.443484I
a = −1.93478 − 1.79741I
b = 0.133299 − 1.346070I
−5.70338 − 5.68434I −0.20490 + 7.47679I
u = 1.070290 − 0.443484I
a = 0.982216 − 0.740855I
b = −0.692609 − 0.458051I
−5.70338 + 5.68434I −0.20490 − 7.47679I
u = 1.070290 − 0.443484I
a = −1.93478 + 1.79741I
b = 0.133299 + 1.346070I
−5.70338 + 5.68434I −0.20490 − 7.47679I
u = −1.268720 + 0.457284I
a = 0.836002 − 0.171074I
b = 1.000150 − 0.693082I
−15.2659 + 1.5494I −3.09602 − 0.66420I
u = −1.268720 + 0.457284I
a = −0.92571 + 2.56606I
b = −0.08422 + 1.59670I
−15.2659 + 1.5494I −3.09602 − 0.66420I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.268720 − 0.457284I
a = 0.836002 + 0.171074I
b = 1.000150 + 0.693082I
−15.2659 − 1.5494I −3.09602 + 0.66420I
u = −1.268720 − 0.457284I
a = −0.92571 − 2.56606I
b = −0.08422 − 1.59670I
−15.2659 − 1.5494I −3.09602 + 0.66420I
u = 1.260410 + 0.482704I
a = −0.16128 − 1.54115I
b = 1.020220 − 0.627490I
−15.0770 − 8.1923I −2.69502 + 5.35870I
u = 1.260410 + 0.482704I
a = 1.45407 + 2.60668I
b = −0.13370 + 1.58929I
−15.0770 − 8.1923I −2.69502 + 5.35870I
u = 1.260410 − 0.482704I
a = −0.16128 + 1.54115I
b = 1.020220 + 0.627490I
−15.0770 + 8.1923I −2.69502 − 5.35870I
u = 1.260410 − 0.482704I
a = 1.45407 − 2.60668I
b = −0.13370 − 1.58929I
−15.0770 + 8.1923I −2.69502 − 5.35870I
u = 0.193328 + 0.557909I
a = 0.736955 − 0.543574I
b = −0.057344 − 1.272060I
−3.31411 + 1.73642I 3.57231 − 4.08118I
u = 0.193328 + 0.557909I
a = −1.31209 − 0.67705I
b = 0.502458 + 0.520559I
−3.31411 + 1.73642I 3.57231 − 4.08118I
u = 0.193328 − 0.557909I
a = 0.736955 + 0.543574I
b = −0.057344 + 1.272060I
−3.31411 − 1.73642I 3.57231 + 4.08118I
u = 0.193328 − 0.557909I
a = −1.31209 + 0.67705I
b = 0.502458 − 0.520559I
−3.31411 − 1.73642I 3.57231 + 4.08118I
13
III. I
u
3
=
h−u
9
+2u
7
−u
5
−2u
3
+b+u, −u
8
−u
7
+· · ·+a+2, u
10
−3u
8
+4u
6
−u
4
−u
2
+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
8
=
u
6
− u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
11
=
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ u
3
+ u
2
+ u − 2
u
9
− 2u
7
+ u
5
+ 2u
3
− u
a
5
=
u
8
− u
7
− 2u
6
+ 2u
5
+ 2u
4
− 2u
3
+ u
2
− u
−1
a
12
=
u
9
− 2u
7
+ u
5
+ 2u
3
− u
0
a
10
=
−u
9
+ u
8
+ 3u
7
− 3u
6
− 3u
5
+ 3u
4
− u
3
+ u
2
+ 2u − 2
u
9
− 2u
7
+ u
5
+ 2u
3
− u
a
4
=
u
8
− u
7
− 2u
6
+ 2u
5
+ 2u
4
− 2u
3
+ u
2
− 2u
−u
3
+ u − 1
a
9
=
−u
8
+ 3u
6
− 3u
4
+ 1
u
8
− 2u
6
+ 2u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
8
− 8u
6
+ 8u
4
+ 4u
2
− 4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
, c
6
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u
2
+ 1)
5
c
7
, c
8
, c
12
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
2
, c
6
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y + 1)
10
c
7
, c
8
, c
12
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.822375 + 0.339110I
a = −1.88547 − 1.25135I
b = 1.000000I
−3.61897 + 1.53058I −0.51511 − 4.43065I
u = −0.822375 − 0.339110I
a = −1.88547 + 1.25135I
b = − 1.000000I
−3.61897 − 1.53058I −0.51511 + 4.43065I
u = 0.822375 + 0.339110I
a = 0.32986 + 1.50891I
b = 1.000000I
−3.61897 − 1.53058I −0.51511 + 4.43065I
u = 0.822375 − 0.339110I
a = 0.32986 − 1.50891I
b = − 1.000000I
−3.61897 + 1.53058I −0.51511 − 4.43065I
u = 0.766826I
a = −0.821196 − 0.370286I
b = − 1.000000I
−5.69095 −1.48110
u = − 0.766826I
a = −0.821196 + 0.370286I
b = 1.000000I
−5.69095 −1.48110
u = −1.200150 + 0.455697I
a = 1.56305 − 1.07974I
b = − 1.000000I
−9.16243 + 4.40083I −4.74431 − 3.49859I
u = −1.200150 − 0.455697I
a = 1.56305 + 1.07974I
b = 1.000000I
−9.16243 − 4.40083I −4.74431 + 3.49859I
u = 1.200150 + 0.455697I
a = −0.186244 − 1.292420I
b = − 1.000000I
−9.16243 − 4.40083I −4.74431 + 3.49859I
u = 1.200150 − 0.455697I
a = −0.186244 + 1.292420I
b = 1.000000I
−9.16243 + 4.40083I −4.74431 − 3.49859I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
15
+ 9u
14
+ ··· − 4u
2
+ 1)
2
· (u
26
+ 15u
25
+ ··· + 17u + 4)
c
2
, c
6
(u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1)(u
15
+ u
14
+ ··· − 2u − 1)
2
· (u
26
− 3u
25
+ ··· − 11u + 2)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
((u
2
+ 1)
5
)(u
26
+ 18u
24
+ ··· − u + 1)(u
30
+ u
29
+ ··· + 54u + 17)
c
7
, c
8
, c
12
(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)(u
15
+ 3u
14
+ ··· + 8u
2
− 1)
2
· (u
26
− 9u
25
+ ··· − 215u + 26)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
15
− 5y
14
+ ··· + 8y −1)
2
· (y
26
− 7y
25
+ ··· + 207y + 16)
c
2
, c
6
((y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
)(y
15
− 9y
14
+ ··· + 4y
2
− 1)
2
· (y
26
− 15y
25
+ ··· − 17y + 4)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
((y + 1)
10
)(y
26
+ 36y
25
+ ··· + 5y + 1)
· (y
30
+ 27y
29
+ ··· + 4428y + 289)
c
7
, c
8
, c
12
((y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
)(y
15
+ 19y
14
+ ··· + 16y −1)
2
· (y
26
+ 29y
25
+ ··· − 257y + 676)
19
