12n
0291
(K12n
0291
)
A knot diagram
1
Linearized knot diagam
3 6 7 8 2 5 11 5 12 1 8 10
Solving Sequence
2,6
3 1 5
7,11
8 4 10 12 9
c
2
c
1
c
5
c
6
c
7
c
4
c
10
c
12
c
9
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2.93542 × 10
17
u
55
− 6.67587 × 10
17
u
54
+ ··· + 2.93396 × 10
17
b − 4.34022 × 10
16
,
2.00229 × 10
18
u
55
− 6.17396 × 10
18
u
54
+ ··· + 2.93396 × 10
17
a − 3.02275 × 10
18
, u
56
− 4u
55
+ ··· + 2u + 1i
I
u
2
= h−u
2
a + u
2
+ b, 2u
2
a + a
2
+ au − 2u
2
− a − 2u − 1, u
3
+ u
2
− 1i
I
u
3
= hb, a − 1, u − 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
= h2.94 × 10
17
u
55
− 6.68 × 10
17
u
54
+ · · · + 2.93 × 10
17
b − 4.34 × 10
16
, 2.00 ×
10
18
u
55
−6.17×10
18
u
54
+· · ·+2.93×10
17
a−3.02×10
18
, u
56
−4u
55
+· · ·+2u+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
11
=
−6.82450u
55
+ 21.0431u
54
+ ··· + 1.50746u + 10.3026
−1.00050u
55
+ 2.27537u
54
+ ··· − 2.44477u + 0.147930
a
8
=
3.65406u
55
− 10.3520u
54
+ ··· + 5.15325u − 4.23694
4.10204u
55
− 12.0354u
54
+ ··· − 2.31411u − 2.32151
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
10
=
−6.34999u
55
+ 19.7941u
54
+ ··· − 0.691497u + 9.35095
−0.586914u
55
+ 1.14134u
54
+ ··· − 2.44719u + 0.0395175
a
12
=
−2.57455u
55
+ 8.57238u
54
+ ··· − 0.0350557u + 6.04787
3.43407u
55
− 10.9313u
54
+ ··· − 6.20880u − 2.54966
a
9
=
−3.42363u
55
+ 10.3054u
54
+ ··· − 4.48816u + 4.34550
−3.87161u
55
+ 11.9888u
54
+ ··· + 2.97921u + 2.43006
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
6957411366946544141
146698213939434107
u
55
+
43997122065882996801
293396427878868214
u
54
+ ··· +
49895915246887856523
293396427878868214
u +
19684211583463159307
293396427878868214
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
56
+ 20u
55
+ ··· + 94u + 1
c
2
, c
5
u
56
+ 4u
55
+ ··· − 2u + 1
c
3
u
56
− 2u
55
+ ··· − 37222u + 7489
c
4
, c
8
u
56
+ 4u
55
+ ··· + 416u − 64
c
7
, c
11
u
56
− 4u
55
+ ··· − 2u − 2
c
9
, c
10
, c
12
u
56
+ 5u
55
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
56
+ 36y
55
+ ··· − 6302y + 1
c
2
, c
5
y
56
− 20y
55
+ ··· − 94y + 1
c
3
y
56
− 24y
55
+ ··· − 5793307970y + 56085121
c
4
, c
8
y
56
+ 34y
55
+ ··· − 21504y + 4096
c
7
, c
11
y
56
− 18y
55
+ ··· − 80y + 4
c
9
, c
10
, c
12
y
56
− 47y
55
+ ··· − 71y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.555632 + 0.820839I
a = −1.310140 + 0.452763I
b = −0.50995 + 1.32573I
6.89840 − 1.70687I 11.04654 + 2.11039I
u = −0.555632 − 0.820839I
a = −1.310140 − 0.452763I
b = −0.50995 − 1.32573I
6.89840 + 1.70687I 11.04654 − 2.11039I
u = −0.746093 + 0.643224I
a = 1.79687 − 0.81232I
b = 0.61922 − 1.61203I
2.14654 + 0.63049I 4.81685 + 0.I
u = −0.746093 − 0.643224I
a = 1.79687 + 0.81232I
b = 0.61922 + 1.61203I
2.14654 − 0.63049I 4.81685 + 0.I
u = 0.611315 + 0.811190I
a = 0.82986 + 1.25644I
b = −0.84573 + 1.32108I
−1.11438 + 4.39097I 2.00000 − 3.05982I
u = 0.611315 − 0.811190I
a = 0.82986 − 1.25644I
b = −0.84573 − 1.32108I
−1.11438 − 4.39097I 2.00000 + 3.05982I
u = 0.639496 + 0.745500I
a = 0.21432 + 1.59797I
b = −0.18371 + 1.40658I
2.28095 + 2.17813I 5.60335 − 1.04644I
u = 0.639496 − 0.745500I
a = 0.21432 − 1.59797I
b = −0.18371 − 1.40658I
2.28095 − 2.17813I 5.60335 + 1.04644I
u = −1.057330 + 0.031394I
a = 1.059980 + 0.396752I
b = 1.037210 − 0.754122I
−3.27157 + 1.64123I 0
u = −1.057330 − 0.031394I
a = 1.059980 − 0.396752I
b = 1.037210 + 0.754122I
−3.27157 − 1.64123I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.815754 + 0.715389I
a = 2.13178 − 1.95768I
b = 0.80101 − 3.45506I
4.64235 + 1.91730I 0
u = −0.815754 − 0.715389I
a = 2.13178 + 1.95768I
b = 0.80101 + 3.45506I
4.64235 − 1.91730I 0
u = 0.646892 + 0.638776I
a = −0.850743 − 0.918967I
b = 0.826318 − 0.495030I
1.49600 − 0.77880I 5.52001 + 0.97967I
u = 0.646892 − 0.638776I
a = −0.850743 + 0.918967I
b = 0.826318 + 0.495030I
1.49600 + 0.77880I 5.52001 − 0.97967I
u = 0.655547 + 0.884678I
a = −1.01004 − 1.55997I
b = 0.57473 − 1.83740I
3.85590 + 9.25734I 0
u = 0.655547 − 0.884678I
a = −1.01004 + 1.55997I
b = 0.57473 + 1.83740I
3.85590 − 9.25734I 0
u = 0.859653 + 0.688129I
a = 0.664066 − 0.017009I
b = −0.054388 − 0.658388I
11.29430 − 2.64795I 0
u = 0.859653 − 0.688129I
a = 0.664066 + 0.017009I
b = −0.054388 + 0.658388I
11.29430 + 2.64795I 0
u = 0.876983 + 0.151494I
a = 0.1160180 + 0.0212256I
b = −0.393762 − 0.405166I
−1.49543 − 0.33054I −5.58521 + 0.41922I
u = 0.876983 − 0.151494I
a = 0.1160180 − 0.0212256I
b = −0.393762 + 0.405166I
−1.49543 + 0.33054I −5.58521 − 0.41922I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.115310 + 0.073858I
a = −0.302761 + 0.507491I
b = −0.541213 − 0.678094I
−7.34462 + 3.52834I 0
u = −1.115310 − 0.073858I
a = −0.302761 − 0.507491I
b = −0.541213 + 0.678094I
−7.34462 − 3.52834I 0
u = −0.947166 + 0.640358I
a = −1.40216 + 1.01867I
b = −0.74124 + 2.06542I
1.51907 + 4.39807I 0
u = −0.947166 − 0.640358I
a = −1.40216 − 1.01867I
b = −0.74124 − 2.06542I
1.51907 − 4.39807I 0
u = −0.904546 + 0.704031I
a = −2.66914 + 2.18937I
b = −0.75000 + 3.34742I
4.37132 + 3.51272I 0
u = −0.904546 − 0.704031I
a = −2.66914 − 2.18937I
b = −0.75000 − 3.34742I
4.37132 − 3.51272I 0
u = 0.199557 + 0.823827I
a = −0.188450 + 0.800184I
b = −0.896805 − 0.162123I
1.27100 − 5.53243I 6.88478 + 5.95128I
u = 0.199557 − 0.823827I
a = −0.188450 − 0.800184I
b = −0.896805 + 0.162123I
1.27100 + 5.53243I 6.88478 − 5.95128I
u = −1.149520 + 0.171385I
a = −0.352409 − 0.425484I
b = 0.086994 + 0.749495I
−3.37562 + 8.61142I 0
u = −1.149520 − 0.171385I
a = −0.352409 + 0.425484I
b = 0.086994 − 0.749495I
−3.37562 − 8.61142I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877750 + 0.767459I
a = 0.61912 + 1.30040I
b = 1.71436 + 0.44120I
3.73144 + 2.89531I 0
u = −0.877750 − 0.767459I
a = 0.61912 − 1.30040I
b = 1.71436 − 0.44120I
3.73144 − 2.89531I 0
u = 0.996417 + 0.644457I
a = 1.098620 − 0.312059I
b = 1.46572 + 1.16153I
0.45076 − 4.30103I 0
u = 0.996417 − 0.644457I
a = 1.098620 + 0.312059I
b = 1.46572 − 1.16153I
0.45076 + 4.30103I 0
u = 0.807123
a = 3.25049
b = −1.20792
0.339779 61.8040
u = 1.041510 + 0.583028I
a = 1.391730 − 0.045593I
b = 1.52856 + 0.47260I
−4.20570 − 3.27627I 0
u = 1.041510 − 0.583028I
a = 1.391730 + 0.045593I
b = 1.52856 − 0.47260I
−4.20570 + 3.27627I 0
u = 0.386849 + 0.703387I
a = 0.080082 − 1.229030I
b = 0.502755 − 0.488230I
−2.39037 − 1.55268I 1.61575 + 2.61232I
u = 0.386849 − 0.703387I
a = 0.080082 + 1.229030I
b = 0.502755 + 0.488230I
−2.39037 + 1.55268I 1.61575 − 2.61232I
u = 1.115730 + 0.439932I
a = −0.739493 + 0.517959I
b = −1.131660 + 0.027778I
−1.67584 + 0.98983I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.115730 − 0.439932I
a = −0.739493 − 0.517959I
b = −1.131660 − 0.027778I
−1.67584 − 0.98983I 0
u = 1.21660
a = −0.482061
b = 0.154807
0.661022 0
u = 1.013820 + 0.677594I
a = −1.87407 − 0.28208I
b = −1.80895 − 0.83661I
1.16708 − 7.61471I 0
u = 1.013820 − 0.677594I
a = −1.87407 + 0.28208I
b = −1.80895 + 0.83661I
1.16708 + 7.61471I 0
u = 1.043410 + 0.693114I
a = −1.80141 − 0.10454I
b = −1.81609 − 1.65261I
−2.41199 − 10.04440I 0
u = 1.043410 − 0.693114I
a = −1.80141 + 0.10454I
b = −1.81609 + 1.65261I
−2.41199 + 10.04440I 0
u = −1.063310 + 0.686319I
a = 1.20482 − 1.11113I
b = 0.57272 − 1.88306I
5.39990 + 7.34868I 0
u = −1.063310 − 0.686319I
a = 1.20482 + 1.11113I
b = 0.57272 + 1.88306I
5.39990 − 7.34868I 0
u = −0.906500 + 0.883744I
a = 0.426122 + 0.260836I
b = 0.259723 + 0.288378I
8.38533 + 3.24583I 0
u = −0.906500 − 0.883744I
a = 0.426122 − 0.260836I
b = 0.259723 − 0.288378I
8.38533 − 3.24583I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.055930 + 0.737690I
a = 2.11012 + 0.58896I
b = 1.89812 + 2.08623I
2.6209 − 15.2730I 0
u = 1.055930 − 0.737690I
a = 2.11012 − 0.58896I
b = 1.89812 − 2.08623I
2.6209 + 15.2730I 0
u = −0.665378
a = −2.70313
b = −1.90693
7.93809 29.4940
u = 0.372658 + 0.279589I
a = −1.53134 − 0.64420I
b = 1.017650 − 0.002132I
1.155810 − 0.800051I 6.92184 − 0.19721I
u = 0.372658 − 0.279589I
a = −1.53134 + 0.64420I
b = 1.017650 + 0.002132I
1.155810 + 0.800051I 6.92184 + 0.19721I
u = −0.112048
a = 5.51199
b = 0.496888
0.859867 11.9670
10
II. I
u
2
= h−u
2
a + u
2
+ b, 2u
2
a + a
2
+ au − 2u
2
− a − 2u − 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
11
=
a
u
2
a − u
2
a
8
=
−au + 2u
2
+ 2u − 1
−au + 2u
2
+ 2u − 1
a
4
=
u
u
a
10
=
au − 2u
2
− u + 2
au − 2u
2
− 2u + 2
a
12
=
−2au + 3u
2
+ a + 3u − 2
u
2
a − 2au + 2u
2
+ 3u − 2
a
9
=
−au + 2u
2
+ 2u − 1
−au + 2u
2
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
a − 4u
2
+ 3a − 10u + 3
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
8
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
9
, c
10
(u
2
− u − 1)
3
c
11
, c
12
(u
2
+ u − 1)
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
8
y
6
c
7
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.586612 + 0.101930I
b = 0.044325 + 0.562280I
11.90680 + 2.82812I 13.45212 − 4.14885I
u = −0.877439 + 0.744862I
a = 0.86067 + 1.76749I
b = 2.28039 + 0.56228I
4.01109 + 2.82812I 20.9825 + 0.8478I
u = −0.877439 − 0.744862I
a = 0.586612 − 0.101930I
b = 0.044325 − 0.562280I
11.90680 − 2.82812I 13.45212 + 4.14885I
u = −0.877439 − 0.744862I
a = 0.86067 − 1.76749I
b = 2.28039 − 0.56228I
4.01109 − 2.82812I 20.9825 − 0.8478I
u = 0.754878
a = 1.51473
b = 0.293316
−0.126494 0.305530
u = 0.754878
a = −2.40929
b = −1.94275
7.76919 −18.1750
14
III. I
u
3
= hb, a − 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
1
=
0
−1
a
5
=
1
1
a
7
=
−1
0
a
11
=
1
0
a
8
=
−1
0
a
4
=
0
1
a
10
=
1
1
a
12
=
1
0
a
9
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
12
u − 1
c
5
, c
6
, c
8
c
9
, c
10
u + 1
c
7
, c
11
u
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
c
12
y −1
c
7
, c
11
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
0 0
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
3
− u
2
+ 2u − 1)
2
(u
56
+ 20u
55
+ ··· + 94u + 1)
c
2
(u − 1)(u
3
+ u
2
− 1)
2
(u
56
+ 4u
55
+ ··· − 2u + 1)
c
3
(u − 1)(u
3
− u
2
+ 2u − 1)
2
(u
56
− 2u
55
+ ··· − 37222u + 7489)
c
4
u
6
(u − 1)(u
56
+ 4u
55
+ ··· + 416u − 64)
c
5
(u + 1)(u
3
− u
2
+ 1)
2
(u
56
+ 4u
55
+ ··· − 2u + 1)
c
6
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
56
+ 20u
55
+ ··· + 94u + 1)
c
7
u(u
2
− u − 1)
3
(u
56
− 4u
55
+ ··· − 2u − 2)
c
8
u
6
(u + 1)(u
56
+ 4u
55
+ ··· + 416u − 64)
c
9
, c
10
(u + 1)(u
2
− u − 1)
3
(u
56
+ 5u
55
+ ··· + 3u + 1)
c
11
u(u
2
+ u − 1)
3
(u
56
− 4u
55
+ ··· − 2u − 2)
c
12
(u − 1)(u
2
+ u − 1)
3
(u
56
+ 5u
55
+ ··· + 3u + 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
(y
56
+ 36y
55
+ ··· − 6302y + 1)
c
2
, c
5
(y −1)(y
3
− y
2
+ 2y −1)
2
(y
56
− 20y
55
+ ··· − 94y + 1)
c
3
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
· (y
56
− 24y
55
+ ··· − 5793307970y + 56085121)
c
4
, c
8
y
6
(y −1)(y
56
+ 34y
55
+ ··· − 21504y + 4096)
c
7
, c
11
y(y
2
− 3y + 1)
3
(y
56
− 18y
55
+ ··· − 80y + 4)
c
9
, c
10
, c
12
(y −1)(y
2
− 3y + 1)
3
(y
56
− 47y
55
+ ··· − 71y + 1)
20
