12n
0528
(K12n
0528
)
A knot diagram
1
Linearized knot diagam
3 5 10 8 2 11 5 12 3 9 7 4
Solving Sequence
3,9
10
4,12
1 8 5 2 6 7 11
c
9
c
3
c
12
c
8
c
4
c
2
c
5
c
7
c
11
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4.69351 × 10
67
u
41
+ 3.55947 × 10
66
u
40
+ ··· + 1.96775 × 10
68
b − 6.12388 × 10
68
,
8.52173 × 10
68
u
41
+ 2.62218 × 10
67
u
40
+ ··· + 2.16453 × 10
69
a − 5.80462 × 10
68
, u
42
− u
41
+ ··· + 7u − 11i
I
u
2
= h−u
23
+ 6u
21
+ ··· + b + 1, 2u
23
− 13u
21
+ ··· + a − 8, u
24
− 7u
22
+ ··· + 2u + 1i
* 2 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−4.69×10
67
u
41
+3.56×10
66
u
40
+· · ·+1.97×10
68
b−6.12×10
68
, 8.52×
10
68
u
41
+2.62×10
67
u
40
+· · ·+2.16×10
69
a−5.80×10
68
, u
42
−u
41
+· · ·+7u−11i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
u
−u
3
+ u
a
12
=
−0.393699u
41
− 0.0121143u
40
+ ··· + 3.32268u + 0.268170
0.238522u
41
− 0.0180890u
40
+ ··· − 1.02660u + 3.11212
a
1
=
0.0278896u
41
− 0.0937965u
40
+ ··· + 3.36008u + 4.84817
0.449875u
41
+ 0.0226937u
40
+ ··· − 3.24732u + 3.95315
a
8
=
−0.270147u
41
− 0.104563u
40
+ ··· + 1.82540u − 2.97138
−0.123129u
41
+ 0.000949344u
40
+ ··· − 2.15974u − 3.88780
a
5
=
−0.173180u
41
− 0.00979843u
40
+ ··· + 0.668036u − 2.62647
0.153388u
41
− 0.142217u
40
+ ··· + 7.32309u + 5.63335
a
2
=
0.0278896u
41
− 0.0937965u
40
+ ··· + 3.36008u + 4.84817
0.296249u
41
+ 0.0542077u
40
+ ··· − 2.47919u + 3.22817
a
6
=
−0.413508u
41
+ 0.339840u
40
+ ··· − 8.97780u − 9.77944
−0.0992743u
41
− 0.0250768u
40
+ ··· + 3.47552u + 1.40486
a
7
=
0.363292u
41
− 0.398230u
40
+ ··· + 9.99588u + 9.98935
0.236749u
41
− 0.0235247u
40
+ ··· − 2.79328u + 0.767418
a
11
=
−u
2
+ 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3.35147u
41
− 0.742817u
40
+ ··· + 14.9249u + 57.1841
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 64u
41
+ ··· − 29237998u + 1857769
c
2
, c
5
u
42
+ 2u
41
+ ··· − 2362u − 1363
c
3
, c
9
u
42
+ u
41
+ ··· − 7u − 11
c
4
, c
7
u
42
+ 3u
41
+ ··· + 10u + 1
c
6
, c
11
u
42
− u
41
+ ··· + 237u − 367
c
8
u
42
+ u
40
+ ··· − 458u + 59
c
10
u
42
− 9u
41
+ ··· − 1765u + 121
c
12
u
42
+ 6u
41
+ ··· + 144391u − 29237
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
− 180y
41
+ ··· + 171645687681114y + 3451305657361
c
2
, c
5
y
42
+ 64y
41
+ ··· − 29237998y + 1857769
c
3
, c
9
y
42
− 9y
41
+ ··· − 1765y + 121
c
4
, c
7
y
42
+ 37y
41
+ ··· + 386y + 1
c
6
, c
11
y
42
+ 9y
41
+ ··· − 699887y + 134689
c
8
y
42
+ 2y
41
+ ··· − 127282y + 3481
c
10
y
42
+ 63y
41
+ ··· − 706841y + 14641
c
12
y
42
+ 66y
41
+ ··· − 9618536811y + 854802169
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.976940 + 0.189570I
a = 0.671991 − 0.140780I
b = −0.640713 + 0.564843I
0.096118 − 0.719363I 11.90225 + 0.52877I
u = −0.976940 − 0.189570I
a = 0.671991 + 0.140780I
b = −0.640713 − 0.564843I
0.096118 + 0.719363I 11.90225 − 0.52877I
u = 0.996611 + 0.295480I
a = −1.090500 − 0.450071I
b = 0.717959 + 0.455088I
0.95171 + 5.32146I 14.6843 − 4.3710I
u = 0.996611 − 0.295480I
a = −1.090500 + 0.450071I
b = 0.717959 − 0.455088I
0.95171 − 5.32146I 14.6843 + 4.3710I
u = −0.271254 + 0.920102I
a = −0.032414 + 0.826443I
b = −0.107773 + 1.208280I
−3.16398 − 3.11599I 6.81507 + 2.35566I
u = −0.271254 − 0.920102I
a = −0.032414 − 0.826443I
b = −0.107773 − 1.208280I
−3.16398 + 3.11599I 6.81507 − 2.35566I
u = 0.742652 + 0.598949I
a = 0.581796 − 1.131730I
b = −0.114376 − 0.798928I
−1.67032 + 2.18141I 6.55140 − 5.10092I
u = 0.742652 − 0.598949I
a = 0.581796 + 1.131730I
b = −0.114376 + 0.798928I
−1.67032 − 2.18141I 6.55140 + 5.10092I
u = 0.901422 + 0.608862I
a = 0.41986 − 1.91727I
b = −0.595052 − 1.146190I
−4.01965 + 5.81935I 2.88718 − 6.41779I
u = 0.901422 − 0.608862I
a = 0.41986 + 1.91727I
b = −0.595052 + 1.146190I
−4.01965 − 5.81935I 2.88718 + 6.41779I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.026110 + 0.457658I
a = −0.89076 − 1.58817I
b = 1.201650 − 0.209885I
3.18164 − 4.85238I 16.3570 + 7.8732I
u = −1.026110 − 0.457658I
a = −0.89076 + 1.58817I
b = 1.201650 + 0.209885I
3.18164 + 4.85238I 16.3570 − 7.8732I
u = 0.762100 + 0.321278I
a = 3.04433 − 0.63560I
b = 0.139498 − 0.544927I
−4.53677 − 1.72545I 2.47718 − 1.65314I
u = 0.762100 − 0.321278I
a = 3.04433 + 0.63560I
b = 0.139498 + 0.544927I
−4.53677 + 1.72545I 2.47718 + 1.65314I
u = 0.709703 + 0.385831I
a = −1.34330 + 1.34754I
b = 1.088080 + 0.084929I
3.75865 + 0.52675I 15.2045 − 4.6516I
u = 0.709703 − 0.385831I
a = −1.34330 − 1.34754I
b = 1.088080 − 0.084929I
3.75865 − 0.52675I 15.2045 + 4.6516I
u = 0.055992 + 0.739207I
a = −0.59055 − 1.78054I
b = 0.966097 − 0.562046I
−5.24825 − 1.98551I 5.33983 + 2.52760I
u = 0.055992 − 0.739207I
a = −0.59055 + 1.78054I
b = 0.966097 + 0.562046I
−5.24825 + 1.98551I 5.33983 − 2.52760I
u = −0.928890 + 0.877770I
a = 0.52280 + 1.42327I
b = −0.086373 + 0.854209I
−9.64609 − 3.25422I −3.44885 + 0.I
u = −0.928890 − 0.877770I
a = 0.52280 − 1.42327I
b = −0.086373 − 0.854209I
−9.64609 + 3.25422I −3.44885 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.110845 + 0.666061I
a = −0.619219 − 0.873228I
b = −1.045990 − 0.541356I
0.96220 + 1.18700I 9.79126 − 0.55178I
u = −0.110845 − 0.666061I
a = −0.619219 + 0.873228I
b = −1.045990 + 0.541356I
0.96220 − 1.18700I 9.79126 + 0.55178I
u = 0.626245 + 0.157018I
a = 1.13195 − 0.98921I
b = −0.984773 − 0.837523I
−0.58346 + 4.36237I 9.67579 − 7.91348I
u = 0.626245 − 0.157018I
a = 1.13195 + 0.98921I
b = −0.984773 + 0.837523I
−0.58346 − 4.36237I 9.67579 + 7.91348I
u = −0.666149 + 1.210860I
a = −0.050312 − 1.150930I
b = 0.87990 − 1.25839I
−6.84658 − 4.87263I 10.00000 + 0.I
u = −0.666149 − 1.210860I
a = −0.050312 + 1.150930I
b = 0.87990 + 1.25839I
−6.84658 + 4.87263I 10.00000 + 0.I
u = −0.530306 + 0.145723I
a = −0.925303 + 0.897460I
b = −1.388070 − 0.265720I
1.33067 + 1.52042I 12.91104 − 5.67422I
u = −0.530306 − 0.145723I
a = −0.925303 − 0.897460I
b = −1.388070 + 0.265720I
1.33067 − 1.52042I 12.91104 + 5.67422I
u = −0.99651 + 1.16113I
a = 0.510029 + 0.364618I
b = 1.30183 + 1.39531I
−16.7577 − 4.3210I 0
u = −0.99651 − 1.16113I
a = 0.510029 − 0.364618I
b = 1.30183 − 1.39531I
−16.7577 + 4.3210I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.461304
a = 0.359897
b = −0.288008
0.654622 15.3870
u = 1.56280
a = −0.170963
b = −0.172890
7.95511 0
u = −1.17709 + 1.02908I
a = 0.669839 + 1.234860I
b = −1.49268 + 1.06354I
−16.1321 − 3.7400I 0
u = −1.17709 − 1.02908I
a = 0.669839 − 1.234860I
b = −1.49268 − 1.06354I
−16.1321 + 3.7400I 0
u = 1.23722 + 0.98615I
a = −0.674732 + 1.221550I
b = 1.28035 + 1.24557I
−16.5397 + 14.0085I 0
u = 1.23722 − 0.98615I
a = −0.674732 − 1.221550I
b = 1.28035 − 1.24557I
−16.5397 − 14.0085I 0
u = −1.44741 + 0.67124I
a = −0.550696 − 0.105785I
b = −0.154517 − 1.136070I
−4.06097 − 2.48143I 0
u = −1.44741 − 0.67124I
a = −0.550696 + 0.105785I
b = −0.154517 + 1.136070I
−4.06097 + 2.48143I 0
u = 0.91039 + 1.31239I
a = −0.494881 + 0.556209I
b = −1.04089 + 1.47723I
−17.7875 − 5.7506I 0
u = 0.91039 − 1.31239I
a = −0.494881 − 0.556209I
b = −1.04089 − 1.47723I
−17.7875 + 5.7506I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.13842 + 1.22014I
a = −0.338929 + 0.751534I
b = 0.306293 + 1.180910I
−11.22360 + 4.40183I 0
u = 1.13842 − 1.22014I
a = −0.338929 − 0.751534I
b = 0.306293 − 1.180910I
−11.22360 − 4.40183I 0
9
II.
I
u
2
= h−u
23
+6u
21
+· · ·+b+1, 2u
23
−13u
21
+· · ·+a−8, u
24
−7u
22
+· · ·+2u+1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
u
−u
3
+ u
a
12
=
−2u
23
+ 13u
21
+ ··· + 7u + 8
u
23
− 6u
21
+ ··· + u
3
− 1
a
1
=
u
23
− u
22
+ ··· + 5u + 8
3u
23
− u
22
+ ··· − u − 2
a
8
=
2u
23
− 11u
21
+ ··· − 6u − 8
−2u
21
+ 2u
20
+ ··· + u + 3
a
5
=
−5u
23
+ u
22
+ ··· + 8u − 2
6u
23
− u
22
+ ··· + 3u + 10
a
2
=
u
23
− u
22
+ ··· + 5u + 8
3u
23
− u
22
+ ··· − u
2
− 1
a
6
=
−2u
23
+ 3u
22
+ ··· + 57u
2
− 11
10u
23
− 5u
22
+ ··· + 2u + 11
a
7
=
u
23
+ u
22
+ ··· − u − 11
6u
23
− 4u
22
+ ··· + 3u + 8
a
11
=
−u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 7u
23
−2u
22
−35u
21
+15u
20
+93u
19
−35u
18
−173u
17
+60u
16
+216u
15
−55u
14
−151u
13
−
21u
12
−17u
11
+202u
10
+137u
9
−350u
8
−118u
7
+346u
6
+38u
5
−193u
4
+7u
3
+69u
2
−3u−1
10
(iv) u-Polynomials at the component
11
Crossings u-Polynomials at each crossing
c
1
u
24
− 19u
23
+ ··· − 13u + 1
c
2
u
24
+ u
23
+ ··· − u − 1
c
3
u
24
− 7u
22
+ ··· − 2u + 1
c
4
u
24
+ 4u
23
+ ··· + u + 1
c
5
u
24
− u
23
+ ··· + u − 1
c
6
u
24
− 4u
22
+ ··· − 2u − 1
c
7
u
24
− 4u
23
+ ··· − u + 1
c
8
u
24
+ 3u
23
+ ··· + 3u − 1
c
9
u
24
− 7u
22
+ ··· + 2u + 1
c
10
u
24
− 14u
23
+ ··· − 18u + 1
c
11
u
24
− 4u
22
+ ··· + 2u − 1
c
12
u
24
+ u
23
+ ··· + 46u − 103
12
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
− 37y
23
+ ··· + 17y + 1
c
2
, c
5
y
24
+ 19y
23
+ ··· + 13y + 1
c
3
, c
9
y
24
− 14y
23
+ ··· − 18y + 1
c
4
, c
7
y
24
+ 8y
23
+ ··· + 17y + 1
c
6
, c
11
y
24
− 8y
23
+ ··· − 16y + 1
c
8
y
24
− 11y
23
+ ··· − 19y + 1
c
10
y
24
+ 6y
23
+ ··· − 26y + 1
c
12
y
24
+ 5y
23
+ ··· − 49084y + 10609
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.801340 + 0.736116I
a = 0.21096 − 1.56855I
b = −0.605197 − 1.263280I
−3.00301 + 5.73538I 10.39480 − 5.83398I
u = 0.801340 − 0.736116I
a = 0.21096 + 1.56855I
b = −0.605197 + 1.263280I
−3.00301 − 5.73538I 10.39480 + 5.83398I
u = −1.030300 + 0.360014I
a = −1.118250 − 0.241019I
b = 0.802224 − 0.589756I
0.37671 − 6.12039I 9.63076 + 10.31907I
u = −1.030300 − 0.360014I
a = −1.118250 + 0.241019I
b = 0.802224 + 0.589756I
0.37671 + 6.12039I 9.63076 − 10.31907I
u = −1.033250 + 0.446927I
a = −1.21979 − 1.90987I
b = 1.346880 − 0.258537I
2.46070 − 4.50580I 6.63131 + 3.28501I
u = −1.033250 − 0.446927I
a = −1.21979 + 1.90987I
b = 1.346880 + 0.258537I
2.46070 + 4.50580I 6.63131 − 3.28501I
u = 0.601908 + 0.599038I
a = 0.182147 − 0.835112I
b = −1.33970 − 0.50009I
0.31347 + 2.55641I 8.68969 − 4.76226I
u = 0.601908 − 0.599038I
a = 0.182147 + 0.835112I
b = −1.33970 + 0.50009I
0.31347 − 2.55641I 8.68969 + 4.76226I
u = 1.063590 + 0.527317I
a = −0.632358 + 0.883549I
b = 1.219720 − 0.405042I
1.85831 + 1.96827I 9.06368 − 2.03092I
u = 1.063590 − 0.527317I
a = −0.632358 − 0.883549I
b = 1.219720 + 0.405042I
1.85831 − 1.96827I 9.06368 + 2.03092I
15
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.745990 + 0.297929I
a = 0.433446 + 0.100374I
b = −1.000640 − 0.751291I
−0.72691 + 3.28719I 7.46239 − 1.13118I
u = −0.745990 − 0.297929I
a = 0.433446 − 0.100374I
b = −1.000640 + 0.751291I
−0.72691 − 3.28719I 7.46239 + 1.13118I
u = 1.171030 + 0.265644I
a = 0.443531 − 0.797924I
b = 0.460708 + 0.162456I
−2.23108 + 3.70583I 9.26047 − 4.04767I
u = 1.171030 − 0.265644I
a = 0.443531 + 0.797924I
b = 0.460708 − 0.162456I
−2.23108 − 3.70583I 9.26047 + 4.04767I
u = 0.758754 + 0.127493I
a = 3.92397 − 0.99022I
b = −0.605895 + 0.289636I
−3.98670 − 2.05045I 14.4656 + 4.5172I
u = 0.758754 − 0.127493I
a = 3.92397 + 0.99022I
b = −0.605895 − 0.289636I
−3.98670 + 2.05045I 14.4656 − 4.5172I
u = 0.975256 + 0.756613I
a = 0.899803 − 0.590281I
b = 0.164458 − 1.085950I
−2.47930 − 0.06362I 6.51040 + 0.07327I
u = 0.975256 − 0.756613I
a = 0.899803 + 0.590281I
b = 0.164458 + 1.085950I
−2.47930 + 0.06362I 6.51040 − 0.07327I
u = −0.652424 + 0.390106I
a = −0.956679 − 0.190508I
b = −1.47282 − 0.31238I
1.09474 + 0.91656I 7.73888 + 4.66446I
u = −0.652424 − 0.390106I
a = −0.956679 + 0.190508I
b = −1.47282 + 0.31238I
1.09474 − 0.91656I 7.73888 − 4.66446I
16
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.926976 + 0.934269I
a = 0.304797 + 1.251840I
b = −0.150225 + 0.737433I
−9.18549 − 3.41839I 12.55837 + 4.87301I
u = −0.926976 − 0.934269I
a = 0.304797 − 1.251840I
b = −0.150225 − 0.737433I
−9.18549 + 3.41839I 12.55837 − 4.87301I
u = −1.54033
a = 0.0892248
b = 0.322461
8.03261 65.8590
u = −0.425538
a = 2.96764
b = −0.961500
3.24533 7.32810
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
24
− 19u
23
+ ··· − 13u + 1)
· (u
42
+ 64u
41
+ ··· − 29237998u + 1857769)
c
2
(u
24
+ u
23
+ ··· − u − 1)(u
42
+ 2u
41
+ ··· − 2362u − 1363)
c
3
(u
24
− 7u
22
+ ··· − 2u + 1)(u
42
+ u
41
+ ··· − 7u − 11)
c
4
(u
24
+ 4u
23
+ ··· + u + 1)(u
42
+ 3u
41
+ ··· + 10u + 1)
c
5
(u
24
− u
23
+ ··· + u − 1)(u
42
+ 2u
41
+ ··· − 2362u − 1363)
c
6
(u
24
− 4u
22
+ ··· − 2u − 1)(u
42
− u
41
+ ··· + 237u − 367)
c
7
(u
24
− 4u
23
+ ··· − u + 1)(u
42
+ 3u
41
+ ··· + 10u + 1)
c
8
(u
24
+ 3u
23
+ ··· + 3u − 1)(u
42
+ u
40
+ ··· − 458u + 59)
c
9
(u
24
− 7u
22
+ ··· + 2u + 1)(u
42
+ u
41
+ ··· − 7u − 11)
c
10
(u
24
− 14u
23
+ ··· − 18u + 1)(u
42
− 9u
41
+ ··· − 1765u + 121)
c
11
(u
24
− 4u
22
+ ··· + 2u − 1)(u
42
− u
41
+ ··· + 237u − 367)
c
12
(u
24
+ u
23
+ ··· + 46u − 103)(u
42
+ 6u
41
+ ··· + 144391u − 29237)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
24
− 37y
23
+ ··· + 17y + 1)
· (y
42
− 180y
41
+ ··· + 171645687681114y + 3451305657361)
c
2
, c
5
(y
24
+ 19y
23
+ ··· + 13y + 1)
· (y
42
+ 64y
41
+ ··· − 29237998y + 1857769)
c
3
, c
9
(y
24
− 14y
23
+ ··· − 18y + 1)(y
42
− 9y
41
+ ··· − 1765y + 121)
c
4
, c
7
(y
24
+ 8y
23
+ ··· + 17y + 1)(y
42
+ 37y
41
+ ··· + 386y + 1)
c
6
, c
11
(y
24
− 8y
23
+ ··· − 16y + 1)(y
42
+ 9y
41
+ ··· − 699887y + 134689)
c
8
(y
24
− 11y
23
+ ··· − 19y + 1)(y
42
+ 2y
41
+ ··· − 127282y + 3481)
c
10
(y
24
+ 6y
23
+ ··· − 26y + 1)(y
42
+ 63y
41
+ ··· − 706841y + 14641)
c
12
(y
24
+ 5y
23
+ ··· − 49084y + 10609)
· (y
42
+ 66y
41
+ ··· − 9618536811y + 854802169)
19
