12n
0409
(K12n
0409
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 11 3 12 6 4 8 9
Solving Sequence
8,11
12 9
1,3
7 4 6 2 5 10
c
11
c
8
c
12
c
7
c
3
c
6
c
2
c
5
c
10
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−6.16590 × 10
32
u
31
+ 1.46189 × 10
33
u
30
+ ··· + 1.24872 × 10
34
b − 5.49780 × 10
33
,
− 2.37033 × 10
33
u
31
+ 4.13166 × 10
33
u
30
+ ··· + 1.24872 × 10
34
a − 1.71464 × 10
34
, u
32
− u
31
+ ··· − 2u − 1i
I
u
2
= h−u
11
+ 7u
9
+ u
8
− 18u
7
− 5u
6
+ 21u
5
+ 7u
4
− 11u
3
− 2u
2
+ b + 2u,
u
12
− 8u
10
+ 25u
8
− 40u
6
+ u
5
+ 36u
4
− 4u
3
− 16u
2
+ a + 4u + 1,
u
13
− 8u
11
− u
10
+ 25u
9
+ 6u
8
− 39u
7
− 12u
6
+ 32u
5
+ 9u
4
− 13u
3
− 2u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 45 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−6.17×10
32
u
31
+1.46×10
33
u
30
+· · ·+1.25×10
34
b−5.50×10
33
, −2.37×
10
33
u
31
+4.13×10
33
u
30
+· · ·+1.25×10
34
a−1.71×10
34
, u
32
−u
31
+· · ·−2u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
9
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
0.189821u
31
− 0.330872u
30
+ ··· − 5.59851u + 1.37311
0.0493778u
31
− 0.117071u
30
+ ··· − 1.67061u + 0.440274
a
7
=
0.617113u
31
− 0.702924u
30
+ ··· − 2.97277u − 1.56588
0.255210u
31
− 0.230274u
30
+ ··· + 0.144769u − 0.440400
a
4
=
0.369799u
31
− 0.365325u
30
+ ··· − 1.90267u − 0.395284
0.0271602u
31
− 0.0677895u
30
+ ··· − 0.802345u − 0.0663187
a
6
=
0.361904u
31
− 0.472651u
30
+ ··· − 3.11754u − 1.12548
0.255210u
31
− 0.230274u
30
+ ··· + 0.144769u − 0.440400
a
2
=
0.661372u
31
− 0.805565u
30
+ ··· − 5.92579u − 0.0552608
0.213423u
31
− 0.264734u
30
+ ··· − 1.41893u − 0.250047
a
5
=
−0.276212u
31
+ 0.368159u
30
+ ··· + 2.60106u + 0.644536
−0.0127971u
31
+ 0.0681599u
30
+ ··· + 0.695590u + 0.0648329
a
10
=
0.209593u
31
− 0.186992u
30
+ ··· − 0.405019u + 0.305269
0.0812154u
31
− 0.0868146u
30
+ ··· + 0.236069u − 0.0317778
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2.64634u
31
− 2.11189u
30
+ ··· + 10.8149u + 2.17341
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 50u
31
+ ··· + 5873u + 169
c
2
, c
5
u
32
+ 2u
31
+ ··· + 149u + 13
c
3
, c
7
u
32
− 3u
31
+ ··· + 6u + 13
c
4
, c
10
u
32
+ u
31
+ ··· − 12u − 7
c
6
u
32
+ 2u
31
+ ··· + 2u − 11
c
8
, c
11
, c
12
u
32
− u
31
+ ··· − 2u − 1
c
9
u
32
+ 2u
31
+ ··· + 320u − 448
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 130y
31
+ ··· + 545386755y + 28561
c
2
, c
5
y
32
− 50y
31
+ ··· − 5873y + 169
c
3
, c
7
y
32
+ 25y
31
+ ··· + 484y + 169
c
4
, c
10
y
32
+ 47y
31
+ ··· + 248y + 49
c
6
y
32
+ 4y
31
+ ··· + 1272y + 121
c
8
, c
11
, c
12
y
32
− 21y
31
+ ··· − 10y + 1
c
9
y
32
+ 114y
31
+ ··· − 5478400y + 200704
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.713734 + 0.634066I
a = −1.63454 − 0.97058I
b = −1.090080 + 0.559990I
−5.38799 + 4.75007I 2.65651 − 5.98480I
u = 0.713734 − 0.634066I
a = −1.63454 + 0.97058I
b = −1.090080 − 0.559990I
−5.38799 − 4.75007I 2.65651 + 5.98480I
u = −0.780100 + 0.713092I
a = −0.578419 + 0.988520I
b = −1.43666 − 0.26099I
−5.66767 + 1.87911I 1.97848 + 0.24857I
u = −0.780100 − 0.713092I
a = −0.578419 − 0.988520I
b = −1.43666 + 0.26099I
−5.66767 − 1.87911I 1.97848 − 0.24857I
u = 0.916172 + 0.567038I
a = 1.40079 − 1.73118I
b = 0.403288 + 0.064579I
−10.08900 + 2.24227I −4.41790 − 3.08560I
u = 0.916172 − 0.567038I
a = 1.40079 + 1.73118I
b = 0.403288 − 0.064579I
−10.08900 − 2.24227I −4.41790 + 3.08560I
u = −0.922337 + 0.681295I
a = −0.214448 + 0.331259I
b = 0.63044 − 1.91502I
−10.84610 − 2.63222I −0.52226 + 2.78850I
u = −0.922337 − 0.681295I
a = −0.214448 − 0.331259I
b = 0.63044 + 1.91502I
−10.84610 + 2.63222I −0.52226 − 2.78850I
u = −1.158300 + 0.287676I
a = −0.821536 + 1.034540I
b = −1.035360 − 0.657528I
1.13972 − 4.35971I 7.21820 + 7.90515I
u = −1.158300 − 0.287676I
a = −0.821536 − 1.034540I
b = −1.035360 + 0.657528I
1.13972 + 4.35971I 7.21820 − 7.90515I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.040580 + 0.693377I
a = 0.989786 − 0.571228I
b = 1.69133 + 0.61668I
−4.78339 − 7.28882I 3.26552 + 6.22364I
u = −1.040580 − 0.693377I
a = 0.989786 + 0.571228I
b = 1.69133 − 0.61668I
−4.78339 + 7.28882I 3.26552 − 6.22364I
u = 1.220980 + 0.289612I
a = −0.442917 − 0.682280I
b = −1.28705 + 0.65034I
1.13108 + 1.76995I 6.03589 + 2.64881I
u = 1.220980 − 0.289612I
a = −0.442917 + 0.682280I
b = −1.28705 − 0.65034I
1.13108 − 1.76995I 6.03589 − 2.64881I
u = 1.110300 + 0.663239I
a = 0.717757 + 0.805432I
b = 1.148800 + 0.137989I
−4.06902 + 0.32626I 2.59195 − 0.82777I
u = 1.110300 − 0.663239I
a = 0.717757 − 0.805432I
b = 1.148800 − 0.137989I
−4.06902 − 0.32626I 2.59195 + 0.82777I
u = 1.44139 + 0.01348I
a = −0.242696 + 0.203112I
b = −0.213818 − 0.777820I
3.51164 − 2.19651I 9.32383 + 3.81660I
u = 1.44139 − 0.01348I
a = −0.242696 − 0.203112I
b = −0.213818 + 0.777820I
3.51164 + 2.19651I 9.32383 − 3.81660I
u = 0.08148 + 1.44948I
a = −1.114080 − 0.266827I
b = −1.69713 − 0.35935I
−19.2923 − 4.4522I 1.21181 + 2.08819I
u = 0.08148 − 1.44948I
a = −1.114080 + 0.266827I
b = −1.69713 + 0.35935I
−19.2923 + 4.4522I 1.21181 − 2.08819I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.481966 + 0.257396I
a = 0.87806 + 1.29690I
b = 0.413129 + 0.311383I
−1.88542 − 1.30506I 1.44155 + 5.20789I
u = −0.481966 − 0.257396I
a = 0.87806 − 1.29690I
b = 0.413129 − 0.311383I
−1.88542 + 1.30506I 1.44155 − 5.20789I
u = 0.286902 + 0.362783I
a = 1.65914 + 1.00541I
b = 1.32318 − 0.60249I
−1.81266 + 1.48756I 1.92285 − 4.94836I
u = 0.286902 − 0.362783I
a = 1.65914 − 1.00541I
b = 1.32318 + 0.60249I
−1.81266 − 1.48756I 1.92285 + 4.94836I
u = −1.58012
a = −0.315491
b = −0.276511
7.37313 22.2100
u = 0.409982
a = 0.726284
b = −0.225298
0.661529 15.2090
u = 1.50304 + 0.73810I
a = 0.829737 + 0.686971I
b = 1.59659 − 0.82818I
−14.9102 + 12.0924I 0
u = 1.50304 − 0.73810I
a = 0.829737 − 0.686971I
b = 1.59659 + 0.82818I
−14.9102 − 12.0924I 0
u = −0.172476 + 0.197273I
a = 3.03827 − 1.51554I
b = 0.897886 − 0.510199I
−1.75865 + 1.40173I 0.26124 − 4.99544I
u = −0.172476 − 0.197273I
a = 3.03827 + 1.51554I
b = 0.897886 + 0.510199I
−1.75865 − 1.40173I 0.26124 + 4.99544I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.63318 + 0.69080I
a = 0.329691 − 0.659740I
b = 1.406370 + 0.131580I
−14.0115 − 3.2110I 0
u = −1.63318 − 0.69080I
a = 0.329691 + 0.659740I
b = 1.406370 − 0.131580I
−14.0115 + 3.2110I 0
8
II.
I
u
2
= h−u
11
+7u
9
+· · ·+b+2u, u
12
−8u
10
+· · ·+a+1, u
13
−8u
11
+· · ·+2u+1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
9
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−u
12
+ 8u
10
− 25u
8
+ 40u
6
− u
5
− 36u
4
+ 4u
3
+ 16u
2
− 4u − 1
u
11
− 7u
9
− u
8
+ 18u
7
+ 5u
6
− 21u
5
− 7u
4
+ 11u
3
+ 2u
2
− 2u
a
7
=
−u
12
+ 7u
10
+ ··· + 5u − 3
−u
12
+ 7u
10
+ ··· + u − 1
a
4
=
2u
12
− u
11
+ ··· + u + 4
u
12
− 7u
10
− u
9
+ 19u
8
+ 5u
7
− 26u
6
− 7u
5
+ 19u
4
+ 2u
3
− 7u
2
+ u + 1
a
6
=
u
9
− 6u
7
+ 12u
5
− 10u
3
+ u
2
+ 4u − 2
−u
12
+ 7u
10
+ ··· + u − 1
a
2
=
−u
8
+ 6u
6
− 12u
4
+ 8u
2
− 1
u
11
− 7u
9
− u
8
+ 18u
7
+ 5u
6
− 21u
5
− 7u
4
+ 11u
3
+ 3u
2
− 2u − 1
a
5
=
u
11
− 7u
9
+ 18u
7
− 20u
5
+ 7u
3
+ u
2
+ 2u − 2
−u
8
+ 5u
6
− 8u
4
+ 5u
2
− 1
a
10
=
2u
12
− u
11
+ ··· + 7u + 2
−u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −2u
12
−2u
11
+ 16u
10
+ 15u
9
−47u
8
−42u
7
+ 64u
6
+ 54u
5
−46u
4
−29u
3
+ 17u
2
+ 2u + 5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
− 13u
12
+ ··· + 7u − 1
c
2
u
13
+ 3u
12
+ ··· − 3u − 1
c
3
u
13
+ 3u
11
+ ··· + 6u − 1
c
4
u
13
+ 8u
11
+ 24u
9
+ u
8
+ 33u
7
+ 4u
6
+ 20u
5
+ 5u
4
+ 5u
3
+ u
2
+ 2u − 1
c
5
u
13
− 3u
12
+ ··· − 3u + 1
c
6
u
13
+ u
12
− u
11
− u
10
− 2u
7
− 9u
6
− 2u
5
− 7u
4
− 3u
3
− 3u
2
− 1
c
7
u
13
+ 3u
11
+ ··· + 6u + 1
c
8
u
13
− 8u
11
+ ··· + 2u − 1
c
9
u
13
+ u
12
+ ··· − u + 1
c
10
u
13
+ 8u
11
+ 24u
9
− u
8
+ 33u
7
− 4u
6
+ 20u
5
− 5u
4
+ 5u
3
− u
2
+ 2u + 1
c
11
, c
12
u
13
− 8u
11
+ ··· + 2u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
− 21y
12
+ ··· − 21y −1
c
2
, c
5
y
13
− 13y
12
+ ··· + 7y −1
c
3
, c
7
y
13
+ 6y
12
+ ··· + 22y −1
c
4
, c
10
y
13
+ 16y
12
+ ··· + 6y −1
c
6
y
13
− 3y
12
+ ··· − 6y −1
c
8
, c
11
, c
12
y
13
− 16y
12
+ ··· + 8y −1
c
9
y
13
+ 23y
12
+ ··· + 15y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.001170 + 0.552293I
a = 0.875628 + 0.971745I
b = 0.230247 − 0.841237I
−9.35275 − 2.09354I 7.20188 + 0.46928I
u = −1.001170 − 0.552293I
a = 0.875628 − 0.971745I
b = 0.230247 + 0.841237I
−9.35275 + 2.09354I 7.20188 − 0.46928I
u = 1.229660 + 0.182594I
a = −0.338886 − 0.829343I
b = −1.11117 + 1.04233I
0.99221 + 2.74184I 3.93696 − 4.65266I
u = 1.229660 − 0.182594I
a = −0.338886 + 0.829343I
b = −1.11117 − 1.04233I
0.99221 − 2.74184I 3.93696 + 4.65266I
u = −1.298440 + 0.119369I
a = −0.766018 + 1.016630I
b = −1.227350 − 0.566412I
−1.21383 − 4.55409I 3.28508 + 3.98345I
u = −1.298440 − 0.119369I
a = −0.766018 − 1.016630I
b = −1.227350 + 0.566412I
−1.21383 + 4.55409I 3.28508 − 3.98345I
u = 0.572766 + 0.333551I
a = 0.953439 + 0.550346I
b = 1.031350 + 0.503157I
−1.38489 − 0.74935I 5.96891 − 3.49027I
u = 0.572766 − 0.333551I
a = 0.953439 − 0.550346I
b = 1.031350 − 0.503157I
−1.38489 + 0.74935I 5.96891 + 3.49027I
u = −0.294410 + 0.263773I
a = 1.38207 − 3.00576I
b = 0.992614 − 0.156850I
−4.69980 + 3.16875I 5.12846 − 3.30315I
u = −0.294410 − 0.263773I
a = 1.38207 + 3.00576I
b = 0.992614 + 0.156850I
−4.69980 − 3.16875I 5.12846 + 3.30315I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.60888 + 0.07753I
a = −0.048946 − 0.541592I
b = −0.616598 + 0.097202I
2.26514 − 1.76839I 2.54626 + 1.07577I
u = 1.60888 − 0.07753I
a = −0.048946 + 0.541592I
b = −0.616598 − 0.097202I
2.26514 + 1.76839I 2.54626 − 1.07577I
u = −1.63455
a = −0.114567
b = −0.598176
7.04862 −3.13510
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
13
− 13u
12
+ ··· + 7u − 1)(u
32
+ 50u
31
+ ··· + 5873u + 169)
c
2
(u
13
+ 3u
12
+ ··· − 3u − 1)(u
32
+ 2u
31
+ ··· + 149u + 13)
c
3
(u
13
+ 3u
11
+ ··· + 6u − 1)(u
32
− 3u
31
+ ··· + 6u + 13)
c
4
(u
13
+ 8u
11
+ 24u
9
+ u
8
+ 33u
7
+ 4u
6
+ 20u
5
+ 5u
4
+ 5u
3
+ u
2
+ 2u − 1)
· (u
32
+ u
31
+ ··· − 12u − 7)
c
5
(u
13
− 3u
12
+ ··· − 3u + 1)(u
32
+ 2u
31
+ ··· + 149u + 13)
c
6
(u
13
+ u
12
− u
11
− u
10
− 2u
7
− 9u
6
− 2u
5
− 7u
4
− 3u
3
− 3u
2
− 1)
· (u
32
+ 2u
31
+ ··· + 2u − 11)
c
7
(u
13
+ 3u
11
+ ··· + 6u + 1)(u
32
− 3u
31
+ ··· + 6u + 13)
c
8
(u
13
− 8u
11
+ ··· + 2u − 1)(u
32
− u
31
+ ··· − 2u − 1)
c
9
(u
13
+ u
12
+ ··· − u + 1)(u
32
+ 2u
31
+ ··· + 320u − 448)
c
10
(u
13
+ 8u
11
+ 24u
9
− u
8
+ 33u
7
− 4u
6
+ 20u
5
− 5u
4
+ 5u
3
− u
2
+ 2u + 1)
· (u
32
+ u
31
+ ··· − 12u − 7)
c
11
, c
12
(u
13
− 8u
11
+ ··· + 2u + 1)(u
32
− u
31
+ ··· − 2u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
13
− 21y
12
+ ··· − 21y −1)
· (y
32
− 130y
31
+ ··· + 545386755y + 28561)
c
2
, c
5
(y
13
− 13y
12
+ ··· + 7y −1)(y
32
− 50y
31
+ ··· − 5873y + 169)
c
3
, c
7
(y
13
+ 6y
12
+ ··· + 22y −1)(y
32
+ 25y
31
+ ··· + 484y + 169)
c
4
, c
10
(y
13
+ 16y
12
+ ··· + 6y −1)(y
32
+ 47y
31
+ ··· + 248y + 49)
c
6
(y
13
− 3y
12
+ ··· − 6y −1)(y
32
+ 4y
31
+ ··· + 1272y + 121)
c
8
, c
11
, c
12
(y
13
− 16y
12
+ ··· + 8y −1)(y
32
− 21y
31
+ ··· − 10y + 1)
c
9
(y
13
+ 23y
12
+ ··· + 15y −1)
· (y
32
+ 114y
31
+ ··· − 5478400y + 200704)
15
