12n
0748
(K12n
0748
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 2 11 4 12 5 6 8 9
Solving Sequence
4,9 2,5
6 10 1 12 8 7 3 11
c
4
c
5
c
9
c
1
c
12
c
8
c
7
c
3
c
11
c
2
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h5.55332 × 10
75
u
50
+ 2.14999 × 10
75
u
49
+ ··· + 1.63199 × 10
76
b − 9.50449 × 10
76
,
− 1.00920 × 10
77
u
50
− 6.14195 × 10
76
u
49
+ ··· + 4.89597 × 10
76
a + 2.60239 × 10
78
, u
51
+ u
50
+ ··· + 9u − 9i
I
u
2
= hu
11
+ 3u
10
− 6u
9
− 5u
8
+ 19u
7
− 14u
6
− 34u
5
+ 34u
4
+ 25u
3
− 24u
2
+ 5b + u + 8,
− 19u
11
+ 3u
10
+ 89u
9
− 75u
8
− 136u
7
+ 321u
6
+ 41u
5
− 531u
4
+ 80u
3
+ 341u
2
+ 5a − 64u − 47,
u
12
− 5u
10
+ 3u
9
+ 9u
8
− 16u
7
− 7u
6
+ 31u
5
+ 3u
4
− 24u
3
− 2u
2
+ 5u + 1i
* 2 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
= h5.55×10
75
u
50
+2.15×10
75
u
49
+· · ·+1.63×10
76
b−9.50×10
76
, −1.01×
10
77
u
50
−6.14×10
76
u
49
+· · ·+4.90×10
76
a+2.60×10
78
, u
51
+u
50
+· · ·+9u−9i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
2
=
2.06129u
50
+ 1.25449u
49
+ ··· + 156.424u − 53.1538
−0.340279u
50
− 0.131740u
49
+ ··· − 5.95216u + 5.82387
a
5
=
1
−u
2
a
6
=
−3.63256u
50
− 2.37017u
49
+ ··· − 340.652u + 80.1409
−0.228512u
50
− 0.176770u
49
+ ··· − 38.9007u + 9.50612
a
10
=
u
−u
3
+ u
a
1
=
1.72101u
50
+ 1.12275u
49
+ ··· + 150.472u − 47.3299
−0.340279u
50
− 0.131740u
49
+ ··· − 5.95216u + 5.82387
a
12
=
1.72101u
50
+ 1.12275u
49
+ ··· + 150.472u − 47.3299
−0.570525u
50
− 0.285333u
49
+ ··· − 26.8255u + 11.2082
a
8
=
3.13695u
50
+ 1.91795u
49
+ ··· + 298.052u − 76.5104
0.639941u
50
+ 0.374469u
49
+ ··· + 59.0371u − 16.4493
a
7
=
2.49701u
50
+ 1.54348u
49
+ ··· + 239.015u − 60.0611
0.639941u
50
+ 0.374469u
49
+ ··· + 59.0371u − 16.4493
a
3
=
−3.39441u
50
− 1.99256u
49
+ ··· − 330.150u + 85.1843
−1.49048u
50
− 1.03131u
49
+ ··· − 138.506u + 38.2645
a
11
=
−2.02479u
50
− 1.32882u
49
+ ··· − 189.222u + 38.3733
−1.71640u
50
− 1.07355u
49
+ ··· − 148.388u + 42.1932
(ii) Obstruction class = −1
(iii) Cusp Shapes = 23.5676u
50
+ 14.4075u
49
+ ··· + 1956.28u − 544.212
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
51
− 2u
50
+ ··· − 2798u − 211
c
2
, c
5
u
51
+ 2u
50
+ ··· − 10u + 1
c
3
, c
7
u
51
− 2u
50
+ ··· − 110u − 11
c
4
, c
9
u
51
+ u
50
+ ··· + 9u − 9
c
6
, c
10
u
51
− 2u
50
+ ··· − 17u + 1
c
8
, c
11
, c
12
u
51
− 18u
49
+ ··· + 13u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
51
+ 70y
50
+ ··· + 9533684y −44521
c
2
, c
5
y
51
− 10y
50
+ ··· + 26y −1
c
3
, c
7
y
51
− 40y
50
+ ··· + 12914y −121
c
4
, c
9
y
51
− 51y
50
+ ··· + 6489y −81
c
6
, c
10
y
51
− 8y
50
+ ··· + 181y −1
c
8
, c
11
, c
12
y
51
− 36y
50
+ ··· + 279y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.966312 + 0.028011I
a = −0.536822 − 0.279028I
b = 0.882354 − 0.296412I
3.13672 − 0.62690I 0
u = 0.966312 − 0.028011I
a = −0.536822 + 0.279028I
b = 0.882354 + 0.296412I
3.13672 + 0.62690I 0
u = −0.752807 + 0.539036I
a = −0.872796 + 0.719742I
b = 0.459364 − 0.125960I
2.93432 − 5.19831I 0. + 6.04454I
u = −0.752807 − 0.539036I
a = −0.872796 − 0.719742I
b = 0.459364 + 0.125960I
2.93432 + 5.19831I 0. − 6.04454I
u = 1.08285
a = −0.166192
b = −1.36678
−8.81786 0
u = −0.237795 + 0.803808I
a = 0.997700 + 0.304588I
b = −0.237661 + 0.548165I
−1.59127 + 1.64954I −10.79264 − 4.97589I
u = −0.237795 − 0.803808I
a = 0.997700 − 0.304588I
b = −0.237661 − 0.548165I
−1.59127 − 1.64954I −10.79264 + 4.97589I
u = −1.229440 + 0.096772I
a = −0.45337 + 1.83496I
b = 0.96203 − 1.87253I
3.25985 − 4.40630I 0
u = −1.229440 − 0.096772I
a = −0.45337 − 1.83496I
b = 0.96203 + 1.87253I
3.25985 + 4.40630I 0
u = −1.23712
a = 1.45818
b = −0.238614
−3.66174 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.205562 + 0.732617I
a = 0.314612 − 0.527449I
b = −0.548118 − 0.948146I
−3.85278 − 3.00372I −11.66965 + 6.00585I
u = −0.205562 − 0.732617I
a = 0.314612 + 0.527449I
b = −0.548118 + 0.948146I
−3.85278 + 3.00372I −11.66965 − 6.00585I
u = 0.572928 + 1.104770I
a = −0.206310 − 0.298977I
b = 0.320739 − 0.725671I
−0.44013 + 9.46101I 0
u = 0.572928 − 1.104770I
a = −0.206310 + 0.298977I
b = 0.320739 + 0.725671I
−0.44013 − 9.46101I 0
u = −1.223810 + 0.312977I
a = −0.32338 + 1.52316I
b = −0.38353 − 2.16692I
1.53528 − 5.83376I 0
u = −1.223810 − 0.312977I
a = −0.32338 − 1.52316I
b = −0.38353 + 2.16692I
1.53528 + 5.83376I 0
u = 1.304010 + 0.126559I
a = −0.351078 − 0.800505I
b = 0.73472 + 1.28447I
3.37052 + 0.99994I 0
u = 1.304010 − 0.126559I
a = −0.351078 + 0.800505I
b = 0.73472 − 1.28447I
3.37052 − 0.99994I 0
u = −1.289750 + 0.452317I
a = 0.550918 − 0.357255I
b = 0.193129 + 0.814567I
−1.04630 − 1.13129I 0
u = −1.289750 − 0.452317I
a = 0.550918 + 0.357255I
b = 0.193129 − 0.814567I
−1.04630 + 1.13129I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.131606 + 0.597948I
a = 0.755619 + 0.012297I
b = −0.236259 + 0.347071I
−0.317906 + 1.195420I −3.90544 − 5.43225I
u = 0.131606 − 0.597948I
a = 0.755619 − 0.012297I
b = −0.236259 − 0.347071I
−0.317906 − 1.195420I −3.90544 + 5.43225I
u = 1.385030 + 0.283021I
a = 0.04231 − 1.92490I
b = −0.77123 + 2.30254I
1.21275 + 6.65866I 0
u = 1.385030 − 0.283021I
a = 0.04231 + 1.92490I
b = −0.77123 − 2.30254I
1.21275 − 6.65866I 0
u = −1.46412 + 0.01784I
a = −0.244361 − 1.204180I
b = −0.79728 + 1.66527I
5.76286 − 1.13846I 0
u = −1.46412 − 0.01784I
a = −0.244361 + 1.204180I
b = −0.79728 − 1.66527I
5.76286 + 1.13846I 0
u = 0.514876
a = 3.02685
b = 0.317888
−10.7307 10.2750
u = 1.48372 + 0.08688I
a = 0.167132 − 1.009410I
b = 0.37784 + 1.38305I
4.37936 + 0.98278I 0
u = 1.48372 − 0.08688I
a = 0.167132 + 1.009410I
b = 0.37784 − 1.38305I
4.37936 − 0.98278I 0
u = 1.49336 + 0.11013I
a = −0.41151 − 1.40552I
b = −0.51248 + 1.86936I
7.35814 + 6.04698I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.49336 − 0.11013I
a = −0.41151 + 1.40552I
b = −0.51248 − 1.86936I
7.35814 − 6.04698I 0
u = −0.491947
a = 1.66651
b = −0.0705730
−1.36137 −6.25410
u = −0.399561 + 0.244629I
a = −2.11272 − 0.74059I
b = 0.615884 − 1.246730I
1.00696 − 4.56920I −3.90257 + 5.25320I
u = −0.399561 − 0.244629I
a = −2.11272 + 0.74059I
b = 0.615884 + 1.246730I
1.00696 + 4.56920I −3.90257 − 5.25320I
u = −1.50785 + 0.38384I
a = −0.017099 + 1.147920I
b = −0.35140 − 1.57334I
4.58888 − 5.41893I 0
u = −1.50785 − 0.38384I
a = −0.017099 − 1.147920I
b = −0.35140 + 1.57334I
4.58888 + 5.41893I 0
u = 1.57518 + 0.19913I
a = −0.200647 + 1.350350I
b = 0.06483 − 2.13765I
10.54910 + 8.09027I 0
u = 1.57518 − 0.19913I
a = −0.200647 − 1.350350I
b = 0.06483 + 2.13765I
10.54910 − 8.09027I 0
u = −0.407564
a = 1.19912
b = −1.34717
−2.57791 9.00240
u = −1.61819 + 0.08048I
a = −0.265175 − 1.291110I
b = 0.19764 + 2.01735I
11.95830 − 0.22760I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61819 − 0.08048I
a = −0.265175 + 1.291110I
b = 0.19764 − 2.01735I
11.95830 + 0.22760I 0
u = −1.58168 + 0.38198I
a = −0.17902 − 1.40951I
b = 0.91256 + 2.00149I
6.4507 − 14.7902I 0
u = −1.58168 − 0.38198I
a = −0.17902 + 1.40951I
b = 0.91256 − 2.00149I
6.4507 + 14.7902I 0
u = −0.364492
a = −2.15420
b = −1.45022
−6.63015 −21.7480
u = 1.63613 + 0.31897I
a = −0.126349 + 1.180690I
b = 0.85837 − 1.71259I
7.91149 + 6.61409I 0
u = 1.63613 − 0.31897I
a = −0.126349 − 1.180690I
b = 0.85837 + 1.71259I
7.91149 − 6.61409I 0
u = 0.169849 + 0.057002I
a = −7.63550 − 0.35036I
b = 0.860446 − 0.474071I
0.100763 + 0.877365I −4.31516 − 3.77329I
u = 0.169849 − 0.057002I
a = −7.63550 + 0.35036I
b = 0.860446 + 0.474071I
0.100763 − 0.877365I −4.31516 + 3.77329I
u = 1.87867
a = 0.431311
b = −0.853122
1.07210 0
u = −0.19520 + 1.90391I
a = 0.0437236 + 0.0504691I
b = −0.097631 + 0.463019I
−0.098440 − 0.706098I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.19520 − 1.90391I
a = 0.0437236 − 0.0504691I
b = −0.097631 − 0.463019I
−0.098440 + 0.706098I 0
10
II. I
u
2
=
hu
11
+3u
10
+· · ·+5b+8, −19u
11
+3u
10
+· · ·+5a−47, u
12
−5u
10
+· · ·+5u+1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
2
=
19
5
u
11
−
3
5
u
10
+ ··· +
64
5
u +
47
5
−
1
5
u
11
−
3
5
u
10
+ ··· −
1
5
u −
8
5
a
5
=
1
−u
2
a
6
=
−
12
5
u
11
+
9
5
u
10
+ ··· −
97
5
u −
51
5
−
2
5
u
11
−
6
5
u
10
+ ··· +
33
5
u +
14
5
a
10
=
u
−u
3
+ u
a
1
=
18
5
u
11
−
6
5
u
10
+ ··· +
63
5
u +
39
5
−
1
5
u
11
−
3
5
u
10
+ ··· −
1
5
u −
8
5
a
12
=
18
5
u
11
−
6
5
u
10
+ ··· +
63
5
u +
39
5
−
8
5
u
11
+
1
5
u
10
+ ··· −
13
5
u −
14
5
a
8
=
11
5
u
11
−
2
5
u
10
+ ··· +
101
5
u +
33
5
−
2
5
u
11
−
6
5
u
10
+ ··· +
33
5
u +
9
5
a
7
=
13
5
u
11
+
4
5
u
10
+ ··· +
68
5
u +
24
5
−
2
5
u
11
−
6
5
u
10
+ ··· +
33
5
u +
9
5
a
3
=
−
17
5
u
11
−
6
5
u
10
+ ··· +
18
5
u −
1
5
3
5
u
11
+
4
5
u
10
+ ··· −
7
5
u −
1
5
a
11
=
−
22
5
u
11
+
9
5
u
10
+ ··· −
97
5
u −
31
5
−
3
5
u
11
+
1
5
u
10
+ ··· −
13
5
u −
14
5
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −
1
5
u
11
−
33
5
u
10
−
9
5
u
9
+ 27u
8
−
74
5
u
7
−
216
5
u
6
+
439
5
u
5
+
176
5
u
4
−135u
3
−
71
5
u
2
+
294
5
u −
43
5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− u
11
+ ··· + 6u − 1
c
2
u
12
+ 3u
11
+ ··· + 2u − 1
c
3
u
12
− u
11
+ ··· + 15u
2
− 1
c
4
u
12
− 5u
10
+ ··· + 5u + 1
c
5
u
12
− 3u
11
+ ··· − 2u − 1
c
6
u
12
− u
11
+ ··· + u − 1
c
7
u
12
+ u
11
+ ··· + 15u
2
− 1
c
8
u
12
+ u
11
+ ··· − 5u + 1
c
9
u
12
− 5u
10
+ ··· − 5u + 1
c
10
u
12
+ u
11
+ ··· − u − 1
c
11
, c
12
u
12
− u
11
+ ··· + 5u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 7y
11
+ ··· − 4y + 1
c
2
, c
5
y
12
− 13y
11
+ ··· − 6y + 1
c
3
, c
7
y
12
− 11y
11
+ ··· − 30y + 1
c
4
, c
9
y
12
− 10y
11
+ ··· − 29y + 1
c
6
, c
10
y
12
− 7y
11
+ ··· − 13y + 1
c
8
, c
11
, c
12
y
12
− 15y
11
+ ··· − 27y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21822
a = 1.79757
b = −0.601851
−4.03710 −18.0780
u = −1.27309
a = 0.277397
b = 1.16135
−7.88309 −1.62320
u = −1.268290 + 0.330427I
a = −0.58188 + 1.78206I
b = −0.09657 − 2.27913I
2.50992 − 6.48574I −1.33595 + 8.54705I
u = −1.268290 − 0.330427I
a = −0.58188 − 1.78206I
b = −0.09657 + 2.27913I
2.50992 + 6.48574I −1.33595 − 8.54705I
u = 1.320490 + 0.182645I
a = −0.416916 − 1.142720I
b = 1.12535 + 1.50126I
3.03542 + 2.75700I −4.31533 − 2.69611I
u = 1.320490 − 0.182645I
a = −0.416916 + 1.142720I
b = 1.12535 − 1.50126I
3.03542 − 2.75700I −4.31533 + 2.69611I
u = 0.638711
a = −0.671197
b = −1.41524
−6.24905 0.243470
u = −1.48898
a = 0.537683
b = −0.370407
1.74863 −2.66980
u = 0.72640 + 1.30991I
a = −0.191140 + 0.258916I
b = 0.005412 + 0.445100I
−0.304228 + 0.769223I −12.9708 + 6.3200I
u = 0.72640 − 1.30991I
a = −0.191140 − 0.258916I
b = 0.005412 − 0.445100I
−0.304228 − 0.769223I −12.9708 − 6.3200I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.411452
a = −3.50522
b = −0.610386
−10.9545 −25.9780
u = −0.240615
a = 2.94363
b = −1.23188
−2.84629 −21.6500
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
− u
11
+ ··· + 6u − 1)(u
51
− 2u
50
+ ··· − 2798u − 211)
c
2
(u
12
+ 3u
11
+ ··· + 2u − 1)(u
51
+ 2u
50
+ ··· − 10u + 1)
c
3
(u
12
− u
11
+ ··· + 15u
2
− 1)(u
51
− 2u
50
+ ··· − 110u − 11)
c
4
(u
12
− 5u
10
+ ··· + 5u + 1)(u
51
+ u
50
+ ··· + 9u − 9)
c
5
(u
12
− 3u
11
+ ··· − 2u − 1)(u
51
+ 2u
50
+ ··· − 10u + 1)
c
6
(u
12
− u
11
+ ··· + u − 1)(u
51
− 2u
50
+ ··· − 17u + 1)
c
7
(u
12
+ u
11
+ ··· + 15u
2
− 1)(u
51
− 2u
50
+ ··· − 110u − 11)
c
8
(u
12
+ u
11
+ ··· − 5u + 1)(u
51
− 18u
49
+ ··· + 13u − 1)
c
9
(u
12
− 5u
10
+ ··· − 5u + 1)(u
51
+ u
50
+ ··· + 9u − 9)
c
10
(u
12
+ u
11
+ ··· − u − 1)(u
51
− 2u
50
+ ··· − 17u + 1)
c
11
, c
12
(u
12
− u
11
+ ··· + 5u + 1)(u
51
− 18u
49
+ ··· + 13u − 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
+ 7y
11
+ ··· − 4y + 1)(y
51
+ 70y
50
+ ··· + 9533684y −44521)
c
2
, c
5
(y
12
− 13y
11
+ ··· − 6y + 1)(y
51
− 10y
50
+ ··· + 26y −1)
c
3
, c
7
(y
12
− 11y
11
+ ··· − 30y + 1)(y
51
− 40y
50
+ ··· + 12914y −121)
c
4
, c
9
(y
12
− 10y
11
+ ··· − 29y + 1)(y
51
− 51y
50
+ ··· + 6489y −81)
c
6
, c
10
(y
12
− 7y
11
+ ··· − 13y + 1)(y
51
− 8y
50
+ ··· + 181y −1)
c
8
, c
11
, c
12
(y
12
− 15y
11
+ ··· − 27y + 1)(y
51
− 36y
50
+ ··· + 279y −1)
17
