12a
0839
(K12a
0839
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 1 3 6 8 7
Solving Sequence
8,11
12 7
1,4
2 9 3 6 10 5
c
11
c
7
c
12
c
1
c
8
c
3
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
42
+ u
41
+ ··· + b + u, −u
28
− 11u
26
+ ··· + a − 1, u
47
+ 2u
46
+ ··· − 2u − 1i
I
u
2
= hu
3
+ b + u, u
2
+ a + 1, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 53 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
= hu
42
+u
41
+· · ·+b+u, −u
28
−11u
26
+· · ·+a−1, u
47
+2u
46
+· · ·−2u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
u
28
+ 11u
26
+ ··· + 8u + 1
−u
42
− u
41
+ ··· − 9u
2
− u
a
2
=
−u
46
− u
45
+ ··· − 11u
2
− 6u
−u
46
− 2u
45
+ ··· + 3u + 1
a
9
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
2u
46
+ 2u
45
+ ··· + 14u
2
+ 7u
2u
46
+ 4u
45
+ ··· − 4u − 2
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
5
=
−u
9
− 4u
7
− 5u
5
+ 3u
−u
9
− 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
46
+ 8u
45
+ ··· + 8u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
47
− 7u
46
+ ··· − 7u + 1
c
3
, c
9
u
47
− u
46
+ ··· + 128u + 64
c
5
, c
6
, c
8
c
10
u
47
− 2u
46
+ ··· − 18u − 9
c
7
, c
11
, c
12
u
47
+ 2u
46
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
47
− 51y
46
+ ··· + 47y −1
c
3
, c
9
y
47
+ 39y
46
+ ··· + 36864y −4096
c
5
, c
6
, c
8
c
10
y
47
− 60y
46
+ ··· + 1494y −81
c
7
, c
11
, c
12
y
47
+ 36y
46
+ ··· + 22y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.931573 + 0.033964I
a = −3.62555 − 1.06085I
b = −4.04479 − 1.25290I
18.8163 + 7.6066I −17.4102 − 3.4010I
u = −0.931573 − 0.033964I
a = −3.62555 + 1.06085I
b = −4.04479 + 1.25290I
18.8163 − 7.6066I −17.4102 + 3.4010I
u = 0.925379
a = 4.48125
b = 5.06824
−15.5609 −16.5480
u = −0.921953 + 0.011274I
a = 1.352420 − 0.172739I
b = 1.42411 + 0.22950I
−13.30020 + 3.04123I −15.7591 − 2.6303I
u = −0.921953 − 0.011274I
a = 1.352420 + 0.172739I
b = 1.42411 − 0.22950I
−13.30020 − 3.04123I −15.7591 + 2.6303I
u = 0.350823 + 1.049320I
a = 1.24839 − 0.95940I
b = 1.86804 + 0.49348I
−8.18690 + 1.30928I −14.5225 + 0.I
u = 0.350823 − 1.049320I
a = 1.24839 + 0.95940I
b = 1.86804 − 0.49348I
−8.18690 − 1.30928I −14.5225 + 0.I
u = 0.885281
a = −1.61021
b = −1.68165
−8.45625 −8.69640
u = 0.044109 + 1.177400I
a = −0.897270 + 0.427390I
b = 0.26502 + 1.43818I
1.19605 − 0.94580I −9.30810 + 0.I
u = 0.044109 − 1.177400I
a = −0.897270 − 0.427390I
b = 0.26502 − 1.43818I
1.19605 + 0.94580I −9.30810 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.262817 + 1.164060I
a = −0.677836 + 0.380418I
b = 0.236782 − 0.642439I
−0.606425 − 1.119520I −12.48597 + 0.I
u = 0.262817 − 1.164060I
a = −0.677836 − 0.380418I
b = 0.236782 + 0.642439I
−0.606425 + 1.119520I −12.48597 + 0.I
u = −0.173486 + 1.230470I
a = 0.394455 + 0.372411I
b = 0.781097 + 0.132649I
2.76219 + 2.33868I 0
u = −0.173486 − 1.230470I
a = 0.394455 − 0.372411I
b = 0.781097 − 0.132649I
2.76219 − 2.33868I 0
u = −0.283890 + 1.214980I
a = −0.93375 − 1.68089I
b = −2.46189 − 0.25027I
−2.16231 + 3.49023I 0
u = −0.283890 − 1.214980I
a = −0.93375 + 1.68089I
b = −2.46189 + 0.25027I
−2.16231 − 3.49023I 0
u = 0.729642 + 0.166193I
a = −2.35770 + 1.36778I
b = −1.47388 + 0.31386I
−10.80020 − 5.29604I −17.2800 + 4.5994I
u = 0.729642 − 0.166193I
a = −2.35770 − 1.36778I
b = −1.47388 − 0.31386I
−10.80020 + 5.29604I −17.2800 − 4.5994I
u = −0.057596 + 1.253360I
a = 0.339761 − 0.116633I
b = 0.339456 − 1.125610I
3.89474 + 1.72506I 0
u = −0.057596 − 1.253360I
a = 0.339761 + 0.116633I
b = 0.339456 + 1.125610I
3.89474 − 1.72506I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.271270 + 1.256580I
a = 0.044945 + 0.350451I
b = −0.929623 + 1.047440I
0.20223 − 5.63538I 0
u = 0.271270 − 1.256580I
a = 0.044945 − 0.350451I
b = −0.929623 − 1.047440I
0.20223 + 5.63538I 0
u = −0.681129
a = 3.55601
b = 1.84966
−5.84226 −16.3910
u = 0.663598 + 0.066636I
a = 0.818243 − 0.611101I
b = 0.281216 − 0.780752I
−3.85427 − 2.27055I −16.0301 + 4.5719I
u = 0.663598 − 0.066636I
a = 0.818243 + 0.611101I
b = 0.281216 + 0.780752I
−3.85427 + 2.27055I −16.0301 − 4.5719I
u = 0.418616 + 1.278440I
a = 0.422426 − 0.961243I
b = 1.59423 + 0.48650I
−4.48545 − 4.66586I 0
u = 0.418616 − 1.278440I
a = 0.422426 + 0.961243I
b = 1.59423 − 0.48650I
−4.48545 + 4.66586I 0
u = −0.098317 + 1.342180I
a = 0.247608 + 0.502928I
b = −1.198420 + 0.642791I
−1.25656 + 3.25255I 0
u = −0.098317 − 1.342180I
a = 0.247608 − 0.502928I
b = −1.198420 − 0.642791I
−1.25656 − 3.25255I 0
u = −0.467449 + 1.264530I
a = 1.55167 + 2.04635I
b = 3.15396 − 2.12723I
−16.8558 − 2.6238I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.467449 − 1.264530I
a = 1.55167 − 2.04635I
b = 3.15396 + 2.12723I
−16.8558 + 2.6238I 0
u = 0.290224 + 1.319350I
a = 0.22664 − 1.55655I
b = 1.55061 − 0.91105I
−6.15563 − 8.92874I 0
u = 0.290224 − 1.319350I
a = 0.22664 + 1.55655I
b = 1.55061 + 0.91105I
−6.15563 + 8.92874I 0
u = −0.450761 + 1.279910I
a = −0.262410 − 0.919173I
b = −0.848073 + 0.408862I
−9.36462 + 1.85590I 0
u = −0.450761 − 1.279910I
a = −0.262410 + 0.919173I
b = −0.848073 − 0.408862I
−9.36462 − 1.85590I 0
u = 0.449749 + 1.289880I
a = −1.11666 + 2.79890I
b = −4.69994 − 1.43155I
−11.55410 − 4.90637I 0
u = 0.449749 − 1.289880I
a = −1.11666 − 2.79890I
b = −4.69994 + 1.43155I
−11.55410 + 4.90637I 0
u = −0.430024 + 0.463380I
a = 0.073435 − 0.954770I
b = 0.627166 + 0.371706I
−6.80327 + 1.66591I −14.4251 − 3.8260I
u = −0.430024 − 0.463380I
a = 0.073435 + 0.954770I
b = 0.627166 − 0.371706I
−6.80327 − 1.66591I −14.4251 + 3.8260I
u = −0.443644 + 1.297270I
a = −0.442064 − 0.748189I
b = −1.82872 + 0.23677I
−9.23009 + 7.91648I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.443644 − 1.297270I
a = −0.442064 + 0.748189I
b = −1.82872 − 0.23677I
−9.23009 − 7.91648I 0
u = −0.443422 + 1.315870I
a = 0.30539 + 2.51704I
b = 4.35921 + 0.13601I
−16.4532 + 12.5153I 0
u = −0.443422 − 1.315870I
a = 0.30539 − 2.51704I
b = 4.35921 − 0.13601I
−16.4532 − 12.5153I 0
u = −0.475692
a = −1.09705
b = −0.304127
−0.943106 −10.1730
u = −0.220025 + 0.246765I
a = −0.83102 + 1.27964I
b = −0.099600 + 0.370710I
−0.433930 + 0.816857I −9.44363 − 8.26201I
u = −0.220025 − 0.246765I
a = −0.83102 − 1.27964I
b = −0.099600 − 0.370710I
−0.433930 − 0.816857I −9.44363 + 8.26201I
u = 0.228742
a = 2.90776
b = −0.724057
−2.00058 0.109480
9
II. I
u
2
= hu
3
+ b + u, u
2
+ a + 1, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
−u
2
− 1
−u
3
− u
a
2
=
0
u
4
− u
3
+ 2u
2
− u
a
9
=
−u
5
− 2u
3
− u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
3
=
−u
2
− 1
−u
3
− u
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
5
− 2u
3
− u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
5
=
−u
2
− 1
−u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
4
+ 6u
3
− 11u
2
+ 6u − 17
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
9
u
6
c
4
(u + 1)
6
c
5
, c
6
, c
8
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
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
9
y
6
c
5
, c
6
, c
8
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
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
a = −1.76250
b = −1.53904
−9.30502 −19.0600
u = −0.138835 + 1.234450I
a = 0.504580 + 0.342767I
b = −0.493180 + 0.575288I
1.31531 + 1.97241I −8.22189 − 4.83849I
u = −0.138835 − 1.234450I
a = 0.504580 − 0.342767I
b = −0.493180 − 0.575288I
1.31531 − 1.97241I −8.22189 + 4.83849I
u = 0.408802 + 1.276380I
a = 0.462019 − 1.043570I
b = 1.52087 + 0.16310I
−5.34051 − 4.59213I −15.2853 + 2.7994I
u = 0.408802 − 1.276380I
a = 0.462019 + 1.043570I
b = 1.52087 − 0.16310I
−5.34051 + 4.59213I −15.2853 − 2.7994I
u = −0.413150
a = −1.17069
b = 0.483672
−2.38379 −21.9250
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
47
− 7u
46
+ ··· − 7u + 1)
c
3
, c
9
u
6
(u
47
− u
46
+ ··· + 128u + 64)
c
4
((u + 1)
6
)(u
47
− 7u
46
+ ··· − 7u + 1)
c
5
, c
6
, c
8
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
47
− 2u
46
+ ··· − 18u − 9)
c
7
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
47
+ 2u
46
+ ··· − 2u − 1)
c
10
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
47
− 2u
46
+ ··· − 18u − 9)
c
11
, c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
47
+ 2u
46
+ ··· − 2u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
47
− 51y
46
+ ··· + 47y −1)
c
3
, c
9
y
6
(y
47
+ 39y
46
+ ··· + 36864y −4096)
c
5
, c
6
, c
8
c
10
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
47
− 60y
46
+ ··· + 1494y −81)
c
7
, c
11
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
47
+ 36y
46
+ ··· + 22y −1)
15
