12n
0109
(K12n
0109
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 3 11 12 6 1 9 8
Solving Sequence
5,10 3,6
7 2 1 11 4 9 12 8
c
5
c
6
c
2
c
1
c
10
c
4
c
9
c
11
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h6.35220 × 10
79
u
55
+ 2.03304 × 10
79
u
54
+ ··· + 6.01137 × 10
80
b + 5.14680 × 10
80
,
− 6.55161 × 10
79
u
55
− 5.38999 × 10
80
u
54
+ ··· + 6.01137 × 10
80
a − 2.37369 × 10
81
, u
56
+ 2u
55
+ ··· − u − 1i
I
u
2
= hb + 1, −2u
2
+ a + u − 4, u
3
+ 2u + 1i
I
u
3
= hb + 1, 2u
3
− u
2
+ a + 3u − 2, u
4
− u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 63 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
= h6.35×10
79
u
55
+2.03×10
79
u
54
+· · ·+6.01×10
80
b+5.15×10
80
, −6.55×
10
79
u
55
−5.39×10
80
u
54
+· · ·+6.01×10
80
a−2.37×10
81
, u
56
+2u
55
+· · ·−u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
0.108987u
55
+ 0.896633u
54
+ ··· + 0.394522u + 3.94866
−0.105670u
55
− 0.0338199u
54
+ ··· − 0.859496u − 0.856178
a
6
=
1
−u
2
a
7
=
−0.146691u
55
+ 0.118799u
54
+ ··· − 1.49357u − 0.175836
−0.0587371u
55
− 0.170653u
54
+ ··· − 0.0940050u − 0.334345
a
2
=
0.00331723u
55
+ 0.862813u
54
+ ··· − 0.464973u + 3.09249
−0.105670u
55
− 0.0338199u
54
+ ··· − 0.859496u − 0.856178
a
1
=
−0.228248u
55
− 0.101789u
54
+ ··· − 1.32208u − 0.0980010
0.0815574u
55
+ 0.220588u
54
+ ··· − 0.171485u − 0.0778350
a
11
=
−0.00788014u
55
+ 0.337184u
54
+ ··· + 0.0774386u + 0.935949
0.0508844u
55
+ 0.122185u
54
+ ··· + 0.834455u − 0.157223
a
4
=
0.0740743u
55
+ 0.802152u
54
+ ··· + 0.322672u + 3.77114
−0.134670u
55
− 0.112515u
54
+ ··· − 0.799927u − 0.831523
a
9
=
−u
u
3
+ u
a
12
=
−0.152843u
55
+ 0.00107466u
54
+ ··· + 0.363574u + 1.27464
0.288502u
55
+ 0.643234u
54
+ ··· + 0.739466u − 0.449730
a
8
=
0.103064u
55
+ 0.314464u
54
+ ··· − 0.906044u − 1.96765
−0.000335653u
55
− 0.103672u
54
+ ··· − 0.200689u − 0.573235
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2.60373u
55
+ 0.0539183u
54
+ ··· + 14.8197u + 24.5026
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
56
+ 20u
55
+ ··· + 436u + 1
c
2
, c
4
u
56
− 8u
55
+ ··· + 20u − 1
c
3
, c
6
u
56
+ 7u
55
+ ··· − 192u + 128
c
5
, c
9
u
56
− 2u
55
+ ··· + u − 1
c
7
u
56
− 2u
55
+ ··· + 24u + 36
c
8
, c
11
, c
12
u
56
+ 2u
55
+ ··· + 3u + 1
c
10
u
56
+ 14u
55
+ ··· + 663u + 99
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
56
+ 40y
55
+ ··· − 177984y + 1
c
2
, c
4
y
56
− 20y
55
+ ··· − 436y + 1
c
3
, c
6
y
56
− 45y
55
+ ··· − 749568y + 16384
c
5
, c
9
y
56
+ 14y
55
+ ··· − 5y + 1
c
7
y
56
− 6y
55
+ ··· + 8424y + 1296
c
8
, c
11
, c
12
y
56
+ 50y
55
+ ··· − 5y + 1
c
10
y
56
+ 2y
55
+ ··· − 93069y + 9801
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.434045 + 0.764937I
a = 0.226290 − 1.290380I
b = −0.870790 + 0.961543I
−5.25808 + 6.99458I −0.29764 − 8.96864I
u = 0.434045 − 0.764937I
a = 0.226290 + 1.290380I
b = −0.870790 − 0.961543I
−5.25808 − 6.99458I −0.29764 + 8.96864I
u = 0.607815 + 0.613715I
a = 0.265971 − 1.018000I
b = −0.230951 + 0.717761I
−2.75732 + 1.48664I 3.92275 − 4.13753I
u = 0.607815 − 0.613715I
a = 0.265971 + 1.018000I
b = −0.230951 − 0.717761I
−2.75732 − 1.48664I 3.92275 + 4.13753I
u = −0.451362 + 0.714754I
a = 0.234053 + 1.284310I
b = −0.750580 − 0.842936I
−0.27969 − 3.79973I 5.42430 + 9.11034I
u = −0.451362 − 0.714754I
a = 0.234053 − 1.284310I
b = −0.750580 + 0.842936I
−0.27969 + 3.79973I 5.42430 − 9.11034I
u = −0.862715 + 0.810503I
a = −0.398734 + 0.878095I
b = 0.522743 − 1.197170I
1.24825 − 8.12321I 0
u = −0.862715 − 0.810503I
a = −0.398734 − 0.878095I
b = 0.522743 + 1.197170I
1.24825 + 8.12321I 0
u = 0.892853 + 0.801583I
a = −0.414906 − 0.801769I
b = 0.592527 + 1.126870I
6.21297 + 4.29286I 0
u = 0.892853 − 0.801583I
a = −0.414906 + 0.801769I
b = 0.592527 − 1.126870I
6.21297 − 4.29286I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.278333 + 0.741597I
a = 0.099886 + 1.151350I
b = −1.217460 − 0.625912I
−6.70409 + 0.23714I −3.85920 + 2.27316I
u = −0.278333 − 0.741597I
a = 0.099886 − 1.151350I
b = −1.217460 + 0.625912I
−6.70409 − 0.23714I −3.85920 − 2.27316I
u = 1.056250 + 0.600176I
a = −0.185355 − 0.398040I
b = 0.633974 + 0.569529I
−3.26116 + 0.99617I 0
u = 1.056250 − 0.600176I
a = −0.185355 + 0.398040I
b = 0.633974 − 0.569529I
−3.26116 − 0.99617I 0
u = −0.073699 + 0.770744I
a = −0.052889 + 0.368979I
b = −1.55528 − 0.19232I
−7.63905 − 3.70925I −4.91263 + 4.06920I
u = −0.073699 − 0.770744I
a = −0.052889 − 0.368979I
b = −1.55528 + 0.19232I
−7.63905 + 3.70925I −4.91263 − 4.06920I
u = −0.942964 + 0.786108I
a = −0.426418 + 0.678802I
b = 0.687117 − 1.007030I
3.85768 − 0.35095I 0
u = −0.942964 − 0.786108I
a = −0.426418 − 0.678802I
b = 0.687117 + 1.007030I
3.85768 + 0.35095I 0
u = −0.749144 + 1.024530I
a = 0.848834 − 1.108890I
b = 0.771803 + 0.803578I
0.55186 + 2.03907I 0
u = −0.749144 − 1.024530I
a = 0.848834 + 1.108890I
b = 0.771803 − 0.803578I
0.55186 − 2.03907I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.014910 + 0.770101I
a = −0.429292 + 0.522295I
b = 0.792637 − 0.851387I
3.49549 − 0.01257I 0
u = −1.014910 − 0.770101I
a = −0.429292 − 0.522295I
b = 0.792637 + 0.851387I
3.49549 + 0.01257I 0
u = 0.354062 + 0.595793I
a = 0.21013 − 1.55671I
b = −0.850160 + 0.434176I
−1.72486 + 1.19747I −0.82877 − 2.04827I
u = 0.354062 − 0.595793I
a = 0.21013 + 1.55671I
b = −0.850160 − 0.434176I
−1.72486 − 1.19747I −0.82877 + 2.04827I
u = 0.784983 + 1.045670I
a = 0.735614 + 1.168520I
b = 0.864080 − 0.816507I
5.42839 + 1.98837I 0
u = 0.784983 − 1.045670I
a = 0.735614 − 1.168520I
b = 0.864080 + 0.816507I
5.42839 − 1.98837I 0
u = 0.085970 + 0.684763I
a = −0.488414 − 0.527247I
b = −1.353840 + 0.169567I
−2.62828 + 1.16505I 0.23653 − 4.88495I
u = 0.085970 − 0.684763I
a = −0.488414 + 0.527247I
b = −1.353840 − 0.169567I
−2.62828 − 1.16505I 0.23653 + 4.88495I
u = 1.064270 + 0.803296I
a = −0.499399 − 0.427068I
b = 0.922982 + 0.802570I
5.24801 − 4.07030I 0
u = 1.064270 − 0.803296I
a = −0.499399 + 0.427068I
b = 0.922982 − 0.802570I
5.24801 + 4.07030I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.413382 + 0.518807I
a = 1.54616 − 0.28428I
b = −0.174530 − 0.207759I
−2.93194 + 2.06182I 3.92769 − 4.29642I
u = 0.413382 − 0.518807I
a = 1.54616 + 0.28428I
b = −0.174530 + 0.207759I
−2.93194 − 2.06182I 3.92769 + 4.29642I
u = −0.824055 + 1.071620I
a = 0.590241 − 1.211660I
b = 0.975048 + 0.809784I
2.94925 − 6.19699I 0
u = −0.824055 − 1.071620I
a = 0.590241 + 1.211660I
b = 0.975048 − 0.809784I
2.94925 + 6.19699I 0
u = −0.468063 + 0.427077I
a = 2.06154 + 1.32565I
b = −0.599437 + 0.131295I
0.420464 + 0.590909I 7.13336 + 0.51074I
u = −0.468063 − 0.427077I
a = 2.06154 − 1.32565I
b = −0.599437 − 0.131295I
0.420464 − 0.590909I 7.13336 − 0.51074I
u = 0.509445 + 0.375315I
a = 2.90388 − 1.33826I
b = −0.746189 − 0.253774I
−4.18225 − 3.70043I 1.67113 − 1.36683I
u = 0.509445 − 0.375315I
a = 2.90388 + 1.33826I
b = −0.746189 + 0.253774I
−4.18225 + 3.70043I 1.67113 + 1.36683I
u = −1.096180 + 0.817244I
a = −0.518579 + 0.361029I
b = 0.983579 − 0.750622I
−0.10153 + 7.89751I 0
u = −1.096180 − 0.817244I
a = −0.518579 − 0.361029I
b = 0.983579 + 0.750622I
−0.10153 − 7.89751I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873027 + 1.098490I
a = 0.401362 − 1.247710I
b = 1.115410 + 0.785925I
2.46724 − 6.89451I 0
u = −0.873027 − 1.098490I
a = 0.401362 + 1.247710I
b = 1.115410 − 0.785925I
2.46724 + 6.89451I 0
u = 0.90047 + 1.09827I
a = 0.313656 + 1.301590I
b = 1.19115 − 0.80027I
4.29861 + 11.19740I 0
u = 0.90047 − 1.09827I
a = 0.313656 − 1.301590I
b = 1.19115 + 0.80027I
4.29861 − 11.19740I 0
u = −0.10217 + 1.42460I
a = 0.637507 − 0.076993I
b = 0.711327 + 0.052901I
−4.30900 − 2.14834I 0
u = −0.10217 − 1.42460I
a = 0.637507 + 0.076993I
b = 0.711327 − 0.052901I
−4.30900 + 2.14834I 0
u = −0.91547 + 1.10371I
a = 0.250236 − 1.309310I
b = 1.23792 + 0.78854I
−1.0410 − 15.1593I 0
u = −0.91547 − 1.10371I
a = 0.250236 + 1.309310I
b = 1.23792 − 0.78854I
−1.0410 + 15.1593I 0
u = 0.85543 + 1.15949I
a = 0.352837 + 1.051270I
b = 1.096290 − 0.637339I
−4.89061 + 5.92736I 0
u = 0.85543 − 1.15949I
a = 0.352837 − 1.051270I
b = 1.096290 + 0.637339I
−4.89061 − 5.92736I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.481256
a = 0.741225
b = 0.0486633
0.741508 13.5170
u = −0.453091 + 0.130268I
a = 6.23400 + 2.08890I
b = −1.045430 + 0.085047I
−4.97667 − 2.52844I 15.0645 + 21.3531I
u = −0.453091 − 0.130268I
a = 6.23400 − 2.08890I
b = −1.045430 − 0.085047I
−4.97667 + 2.52844I 15.0645 − 21.3531I
u = 0.20928 + 1.56362I
a = 0.526789 + 0.134917I
b = 0.787380 − 0.084461I
−10.62760 + 5.32282I 0
u = 0.20928 − 1.56362I
a = 0.526789 − 0.134917I
b = 0.787380 + 0.084461I
−10.62760 − 5.32282I 0
u = 0.355132
a = 10.2088
b = −1.03133
−0.754394 76.5300
10
II. I
u
2
= hb + 1, −2u
2
+ a + u − 4, u
3
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
2u
2
− u + 4
−1
a
6
=
1
−u
2
a
7
=
1
−u
2
a
2
=
2u
2
− u + 3
−1
a
1
=
−1
0
a
11
=
u
u
a
4
=
2u
2
− u + 4
−1
a
9
=
−u
−u − 1
a
12
=
−u
2
+ u
−u
2
a
8
=
u
2
+ u + 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
2
+ u − 14
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
8
, c
10
u
3
+ 2u + 1
c
7
u
3
+ 3u
2
+ 5u + 2
c
9
, c
11
, c
12
u
3
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
8
, c
9
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
7
y
3
+ y
2
+ 13y −4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.22670 + 1.46771I
a = −0.432268 − 0.136798I
b = −1.00000
−11.08570 + 5.13794I −7.46495 − 0.52866I
u = 0.22670 − 1.46771I
a = −0.432268 + 0.136798I
b = −1.00000
−11.08570 − 5.13794I −7.46495 + 0.52866I
u = −0.453398
a = 4.86454
b = −1.00000
−0.857735 −15.0700
14
III. I
u
3
= hb + 1, 2u
3
− u
2
+ a + 3u − 2, u
4
− u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−2u
3
+ u
2
− 3u + 2
−1
a
6
=
1
−u
2
a
7
=
1
−u
2
a
2
=
−2u
3
+ u
2
− 3u + 1
−1
a
1
=
−1
0
a
11
=
u
u
a
4
=
−2u
3
+ u
2
− 3u + 2
−1
a
9
=
−u
u
3
+ u
a
12
=
u
3
+ 2u − 1
−u
3
+ u
2
− u + 2
a
8
=
−u
3
+ u
2
− 2u + 2
−u
3
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
3
+ 3u
2
− 4u + 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
6
u
4
c
4
(u + 1)
4
c
5
, c
8
, c
10
u
4
− u
3
+ 2u
2
− 2u + 1
c
7
(u
2
− u + 1)
2
c
9
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
8
, c
9
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
7
(y
2
+ y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 0.57070 − 1.62477I
b = −1.00000
−4.93480 + 2.02988I 2.57732 − 1.82047I
u = 0.621744 − 0.440597I
a = 0.57070 + 1.62477I
b = −1.00000
−4.93480 − 2.02988I 2.57732 + 1.82047I
u = −0.121744 + 1.306620I
a = −0.570696 + 0.107280I
b = −1.00000
−4.93480 − 2.02988I −3.07732 + 2.50966I
u = −0.121744 − 1.306620I
a = −0.570696 − 0.107280I
b = −1.00000
−4.93480 + 2.02988I −3.07732 − 2.50966I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
56
+ 20u
55
+ ··· + 436u + 1)
c
2
((u − 1)
7
)(u
56
− 8u
55
+ ··· + 20u − 1)
c
3
, c
6
u
7
(u
56
+ 7u
55
+ ··· − 192u + 128)
c
4
((u + 1)
7
)(u
56
− 8u
55
+ ··· + 20u − 1)
c
5
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
56
− 2u
55
+ ··· + u − 1)
c
7
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
56
− 2u
55
+ ··· + 24u + 36)
c
8
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
56
+ 2u
55
+ ··· + 3u + 1)
c
9
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
56
− 2u
55
+ ··· + u − 1)
c
10
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
56
+ 14u
55
+ ··· + 663u + 99)
c
11
, c
12
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
56
+ 2u
55
+ ··· + 3u + 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
56
+ 40y
55
+ ··· − 177984y + 1)
c
2
, c
4
((y −1)
7
)(y
56
− 20y
55
+ ··· − 436y + 1)
c
3
, c
6
y
7
(y
56
− 45y
55
+ ··· − 749568y + 16384)
c
5
, c
9
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
56
+ 14y
55
+ ··· − 5y + 1)
c
7
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
56
− 6y
55
+ ··· + 8424y + 1296)
c
8
, c
11
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
56
+ 50y
55
+ ··· − 5y + 1)
c
10
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
56
+ 2y
55
+ ··· − 93069y + 9801)
20
