12n
0398
(K12n
0398
)
A knot diagram
1
Linearized knot diagam
3 5 12 10 2 12 3 5 4 9 6 8
Solving Sequence
5,10 4,12
3 2 6 1 9 8 7 11
c
4
c
3
c
2
c
5
c
1
c
9
c
8
c
7
c
11
c
6
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2.97014 × 10
17
u
25
− 5.76869 × 10
17
u
24
+ ··· + 3.13067 × 10
19
b − 2.63141 × 10
17
,
1.19380 × 10
20
u
25
− 2.53498 × 10
19
u
24
+ ··· + 5.94827 × 10
20
a − 2.01222 × 10
21
, u
26
+ u
25
+ ··· − 28u − 19i
I
u
2
= h−u
15
+ 4u
13
− u
12
− 10u
11
+ 3u
10
+ 16u
9
− 7u
8
− 19u
7
+ 10u
6
+ 13u
5
− 10u
4
− 5u
3
+ 5u
2
+ b − 1,
− 2u
15
+ u
14
+ 6u
13
− 4u
12
− 13u
11
+ 9u
10
+ 16u
9
− 15u
8
− 14u
7
+ 17u
6
+ 2u
5
− 11u
4
+ 3u
3
+ 3u
2
+ a − 2u,
u
16
− 4u
14
+ 10u
12
− 16u
10
+ u
9
+ 19u
8
− 3u
7
− 15u
6
+ 4u
5
+ 8u
4
− 3u
3
− 3u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 42 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
=
h2.97×10
17
u
25
−5.77×10
17
u
24
+· · ·+3.13×10
19
b−2.63×10
17
, 1.19×10
20
u
25
−
2.53 × 10
19
u
24
+ · · · + 5.95 × 10
20
a − 2.01 × 10
21
, u
26
+ u
25
+ · · · − 28u − 19i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
12
=
−0.200697u
25
+ 0.0426171u
24
+ ··· + 2.66846u + 3.38286
−0.00948723u
25
+ 0.0184264u
24
+ ··· + 0.751262u + 0.00840526
a
3
=
−0.384928u
25
+ 0.0732483u
24
+ ··· + 4.20798u + 6.50815
−0.102054u
25
+ 0.0407047u
24
+ ··· + 1.29638u + 1.21236
a
2
=
−0.282874u
25
+ 0.0325436u
24
+ ··· + 2.91160u + 5.29579
−0.102054u
25
+ 0.0407047u
24
+ ··· + 1.29638u + 1.21236
a
6
=
−0.00386158u
25
− 0.0616679u
24
+ ··· + 0.342331u + 1.96390
0.0590853u
25
− 0.0329754u
24
+ ··· − 1.46399u − 0.594028
a
1
=
−0.326961u
25
+ 0.111690u
24
+ ··· + 4.32043u + 4.70360
−0.120257u
25
+ 0.0334802u
24
+ ··· + 1.90402u + 1.63890
a
9
=
−u
−u
3
+ u
a
8
=
−u
3
−u
3
+ u
a
7
=
0.0487672u
25
+ 0.135155u
24
+ ··· + 0.267367u − 3.78355
−0.000670823u
25
+ 0.0342194u
24
+ ··· + 0.985487u − 1.00612
a
11
=
u
3
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
41971186605596834432
31306709017582656563
u
25
−
5084666455514062662
31306709017582656563
u
24
+ ··· +
725337053465471662059
31306709017582656563
u +
1192252859670655966978
31306709017582656563
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 4u
25
+ ··· − 5008u + 841
c
2
, c
5
u
26
+ 2u
25
+ ··· − 152u + 29
c
3
u
26
− 3u
25
+ ··· − 4u + 1
c
4
, c
9
u
26
+ u
25
+ ··· − 28u − 19
c
6
, c
11
u
26
+ 3u
25
+ ··· + 240u − 56
c
7
u
26
+ u
25
+ ··· − 9u + 1
c
8
u
26
+ 3u
25
+ ··· + 1729u + 2888
c
10
u
26
− 19u
25
+ ··· − 2570u + 361
c
12
u
26
− u
25
+ ··· + 305u − 278
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
+ 52y
25
+ ··· − 89120532y + 707281
c
2
, c
5
y
26
+ 4y
25
+ ··· − 5008y + 841
c
3
y
26
+ 21y
25
+ ··· + 86y + 1
c
4
, c
9
y
26
− 19y
25
+ ··· − 2570y + 361
c
6
, c
11
y
26
− 33y
25
+ ··· + 3552y + 3136
c
7
y
26
+ 59y
25
+ ··· − 63y + 1
c
8
y
26
+ 49y
25
+ ··· − 13634609y + 8340544
c
10
y
26
− 15y
25
+ ··· − 18094y + 130321
c
12
y
26
− 35y
25
+ ··· − 343225y + 77284
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.046430 + 0.230334I
a = 0.904468 + 0.995486I
b = −0.297194 + 0.524133I
−2.11030 − 0.82786I 9.83528 − 0.62006I
u = −1.046430 − 0.230334I
a = 0.904468 − 0.995486I
b = −0.297194 − 0.524133I
−2.11030 + 0.82786I 9.83528 + 0.62006I
u = −0.979713 + 0.526985I
a = 0.47135 − 1.47204I
b = 0.26056 − 1.46043I
−6.29591 − 2.05047I 2.31530 + 2.39529I
u = −0.979713 − 0.526985I
a = 0.47135 + 1.47204I
b = 0.26056 + 1.46043I
−6.29591 + 2.05047I 2.31530 − 2.39529I
u = 1.068060 + 0.381172I
a = 0.25798 + 1.48795I
b = −1.25101 + 1.49141I
3.21406 + 4.68481I 6.78577 − 4.01254I
u = 1.068060 − 0.381172I
a = 0.25798 − 1.48795I
b = −1.25101 − 1.49141I
3.21406 − 4.68481I 6.78577 + 4.01254I
u = −0.720226 + 0.420524I
a = 0.109243 + 0.577294I
b = −0.211859 − 0.068837I
−1.01337 − 1.74582I 2.64941 + 5.86100I
u = −0.720226 − 0.420524I
a = 0.109243 − 0.577294I
b = −0.211859 + 0.068837I
−1.01337 + 1.74582I 2.64941 − 5.86100I
u = −1.125590 + 0.414878I
a = −0.60347 + 2.11030I
b = 0.78890 + 1.53070I
3.15440 − 5.95143I 5.27054 + 8.12697I
u = −1.125590 − 0.414878I
a = −0.60347 − 2.11030I
b = 0.78890 − 1.53070I
3.15440 + 5.95143I 5.27054 − 8.12697I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.576799 + 0.539363I
a = 1.73236 − 0.76798I
b = 0.917879 + 0.763236I
1.72229 − 1.05497I 5.11673 − 2.05662I
u = 0.576799 − 0.539363I
a = 1.73236 + 0.76798I
b = 0.917879 − 0.763236I
1.72229 + 1.05497I 5.11673 + 2.05662I
u = 0.937850 + 0.875162I
a = −0.383439 − 0.046876I
b = 0.058830 − 0.513138I
−9.63476 + 3.24540I 11.28144 − 4.93616I
u = 0.937850 − 0.875162I
a = −0.383439 + 0.046876I
b = 0.058830 + 0.513138I
−9.63476 − 3.24540I 11.28144 + 4.93616I
u = −0.126374 + 1.299430I
a = 0.240279 − 0.296936I
b = 0.23603 − 1.59331I
9.18772 + 5.01541I 6.86924 − 2.09286I
u = −0.126374 − 1.299430I
a = 0.240279 + 0.296936I
b = 0.23603 + 1.59331I
9.18772 − 5.01541I 6.86924 + 2.09286I
u = −0.260661 + 0.600635I
a = −0.008179 − 0.294986I
b = −0.523677 + 1.128050I
0.64435 + 2.04528I 2.89894 − 3.38693I
u = −0.260661 − 0.600635I
a = −0.008179 + 0.294986I
b = −0.523677 − 1.128050I
0.64435 − 2.04528I 2.89894 + 3.38693I
u = 0.619771
a = 0.580980
b = 0.438499
0.786533 13.5010
u = 1.44898 + 0.19387I
a = 0.44142 + 1.70204I
b = −0.04602 + 1.63288I
6.07568 + 0.67636I 9.59349 − 0.33542I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.44898 − 0.19387I
a = 0.44142 − 1.70204I
b = −0.04602 − 1.63288I
6.07568 − 0.67636I 9.59349 + 0.33542I
u = −1.51509
a = −0.841913
b = 0.310960
9.26344 9.84080
u = −1.41163 + 0.67471I
a = 0.68636 − 1.52049I
b = −0.44218 − 1.83701I
13.2089 − 11.9429I 8.01117 + 5.02358I
u = −1.41163 − 0.67471I
a = 0.68636 + 1.52049I
b = −0.44218 + 1.83701I
13.2089 + 11.9429I 8.01117 − 5.02358I
u = 1.58660 + 0.55037I
a = −0.82317 − 1.20478I
b = 0.13502 − 1.49909I
14.6555 + 1.7140I 9.20177 − 0.71366I
u = 1.58660 − 0.55037I
a = −0.82317 + 1.20478I
b = 0.13502 + 1.49909I
14.6555 − 1.7140I 9.20177 + 0.71366I
7
II.
I
u
2
= h−u
15
+4u
13
+· · ·+b−1, −2u
15
+u
14
+· · ·+a−2u, u
16
−4u
14
+· · ·+u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
12
=
2u
15
− u
14
+ ··· − 3u
2
+ 2u
u
15
− 4u
13
+ ··· − 5u
2
+ 1
a
3
=
2u
15
− 3u
14
+ ··· + 2u + 4
−u
13
+ 3u
11
− 7u
9
+ 9u
7
− u
6
− 9u
5
+ 2u
4
+ 4u
3
− 2u
2
− u + 1
a
2
=
2u
15
− 3u
14
+ ··· + 3u + 3
−u
13
+ 3u
11
− 7u
9
+ 9u
7
− u
6
− 9u
5
+ 2u
4
+ 4u
3
− 2u
2
− u + 1
a
6
=
−u
15
+ 4u
13
− 9u
11
+ 13u
9
− u
8
− 12u
7
+ 3u
6
+ 6u
5
− 3u
4
+ u
3
+ u
2
+ 1
−u
15
+ 4u
13
+ ··· + 3u
2
+ 2u
a
1
=
3u
15
− u
14
+ ··· − 4u
2
+ u
u
15
− 4u
13
+ ··· − u + 1
a
9
=
−u
−u
3
+ u
a
8
=
−u
3
−u
3
+ u
a
7
=
−u
15
− u
14
+ ··· + 2u
2
+ 4u
−u
14
+ 4u
12
+ ··· + 3u + 2
a
11
=
u
3
u
5
− u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shap es = −8u
15
+ 5u
14
+ 32u
13
− 19u
12
− 77u
11
+ 45u
10
+ 119u
9
− 78u
8
−
125u
7
+ 100u
6
+ 79u
5
− 84u
4
− 20u
3
+ 38u
2
+ 2u − 4
8
(iv) u-Polynomials at the component
9
Crossings u-Polynomials at each crossing
c
1
u
16
− 15u
15
+ ··· − 19u + 1
c
2
u
16
+ u
15
+ ··· + u + 1
c
3
u
16
+ 4u
15
+ ··· + u + 1
c
4
u
16
− 4u
14
+ ··· + u + 1
c
5
u
16
− u
15
+ ··· − u + 1
c
6
u
16
+ 2u
15
+ ··· + 2u
2
+ 1
c
7
u
16
+ 15u
14
+ ··· − 8u + 1
c
8
u
16
− 4u
14
+ ··· − u + 1
c
9
u
16
− 4u
14
+ ··· − u + 1
c
10
u
16
− 8u
15
+ ··· − 7u + 1
c
11
u
16
− 2u
15
+ ··· + 2u
2
+ 1
c
12
u
16
+ 2u
14
+ ··· − 2u + 1
10
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 13y
15
+ ··· − 49y + 1
c
2
, c
5
y
16
+ 15y
15
+ ··· + 19y + 1
c
3
y
16
− 8y
15
+ ··· + y + 1
c
4
, c
9
y
16
− 8y
15
+ ··· − 7y + 1
c
6
, c
11
y
16
+ 2y
15
+ ··· + 4y + 1
c
7
y
16
+ 30y
15
+ ··· + 24y + 1
c
8
y
16
− 8y
15
+ ··· − 3y + 1
c
10
y
16
+ 8y
15
+ ··· + 13y + 1
c
12
y
16
+ 4y
15
+ ··· + 2y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.928459 + 0.210985I
a = −0.08578 + 1.94449I
b = 0.44492 + 1.40305I
−5.69032 + 0.91508I 6.42574 + 1.46044I
u = 0.928459 − 0.210985I
a = −0.08578 − 1.94449I
b = 0.44492 − 1.40305I
−5.69032 − 0.91508I 6.42574 − 1.46044I
u = −0.891284 + 0.284250I
a = 1.049080 + 0.856325I
b = −0.424111 − 0.038339I
−2.75842 − 1.23687I −0.93586 + 5.66129I
u = −0.891284 − 0.284250I
a = 1.049080 − 0.856325I
b = −0.424111 + 0.038339I
−2.75842 + 1.23687I −0.93586 − 5.66129I
u = 0.879726 + 0.695218I
a = −0.783339 − 1.034040I
b = −0.251596 − 1.308430I
−5.37416 + 2.67548I 8.13212 − 4.78244I
u = 0.879726 − 0.695218I
a = −0.783339 + 1.034040I
b = −0.251596 + 1.308430I
−5.37416 − 2.67548I 8.13212 + 4.78244I
u = 0.502884 + 0.656202I
a = 0.921548 + 0.037821I
b = −0.330221 + 1.286020I
1.24395 + 2.85601I 7.24981 − 2.90050I
u = 0.502884 − 0.656202I
a = 0.921548 − 0.037821I
b = −0.330221 − 1.286020I
1.24395 − 2.85601I 7.24981 + 2.90050I
u = −1.117360 + 0.442159I
a = −0.68765 + 2.18024I
b = 0.95379 + 1.98741I
4.14272 − 5.66059I 13.1246 + 7.2150I
u = −1.117360 − 0.442159I
a = −0.68765 − 2.18024I
b = 0.95379 − 1.98741I
4.14272 + 5.66059I 13.1246 − 7.2150I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.928870 + 0.835313I
a = −0.130318 − 0.323140I
b = 0.141022 + 0.055042I
−10.05780 − 3.13031I −6.80246 + 0.44703I
u = −0.928870 − 0.835313I
a = −0.130318 + 0.323140I
b = 0.141022 − 0.055042I
−10.05780 + 3.13031I −6.80246 − 0.44703I
u = 1.144080 + 0.550514I
a = 1.18903 + 1.05584I
b = −0.11513 + 1.65347I
3.33898 + 2.00138I 10.41585 − 0.74839I
u = 1.144080 − 0.550514I
a = 1.18903 − 1.05584I
b = −0.11513 − 1.65347I
3.33898 − 2.00138I 10.41585 + 0.74839I
u = −0.517636 + 0.368558I
a = −1.97258 − 0.55483I
b = −0.91868 + 1.34697I
1.99556 + 2.04137I 8.89023 − 4.80901I
u = −0.517636 − 0.368558I
a = −1.97258 + 0.55483I
b = −0.91868 − 1.34697I
1.99556 − 2.04137I 8.89023 + 4.80901I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
16
− 15u
15
+ ··· − 19u + 1)(u
26
+ 4u
25
+ ··· − 5008u + 841)
c
2
(u
16
+ u
15
+ ··· + u + 1)(u
26
+ 2u
25
+ ··· − 152u + 29)
c
3
(u
16
+ 4u
15
+ ··· + u + 1)(u
26
− 3u
25
+ ··· − 4u + 1)
c
4
(u
16
− 4u
14
+ ··· + u + 1)(u
26
+ u
25
+ ··· − 28u − 19)
c
5
(u
16
− u
15
+ ··· − u + 1)(u
26
+ 2u
25
+ ··· − 152u + 29)
c
6
(u
16
+ 2u
15
+ ··· + 2u
2
+ 1)(u
26
+ 3u
25
+ ··· + 240u − 56)
c
7
(u
16
+ 15u
14
+ ··· − 8u + 1)(u
26
+ u
25
+ ··· − 9u + 1)
c
8
(u
16
− 4u
14
+ ··· − u + 1)(u
26
+ 3u
25
+ ··· + 1729u + 2888)
c
9
(u
16
− 4u
14
+ ··· − u + 1)(u
26
+ u
25
+ ··· − 28u − 19)
c
10
(u
16
− 8u
15
+ ··· − 7u + 1)(u
26
− 19u
25
+ ··· − 2570u + 361)
c
11
(u
16
− 2u
15
+ ··· + 2u
2
+ 1)(u
26
+ 3u
25
+ ··· + 240u − 56)
c
12
(u
16
+ 2u
14
+ ··· − 2u + 1)(u
26
− u
25
+ ··· + 305u − 278)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
16
− 13y
15
+ ··· − 49y + 1)
· (y
26
+ 52y
25
+ ··· − 89120532y + 707281)
c
2
, c
5
(y
16
+ 15y
15
+ ··· + 19y + 1)(y
26
+ 4y
25
+ ··· − 5008y + 841)
c
3
(y
16
− 8y
15
+ ··· + y + 1)(y
26
+ 21y
25
+ ··· + 86y + 1)
c
4
, c
9
(y
16
− 8y
15
+ ··· − 7y + 1)(y
26
− 19y
25
+ ··· − 2570y + 361)
c
6
, c
11
(y
16
+ 2y
15
+ ··· + 4y + 1)(y
26
− 33y
25
+ ··· + 3552y + 3136)
c
7
(y
16
+ 30y
15
+ ··· + 24y + 1)(y
26
+ 59y
25
+ ··· − 63y + 1)
c
8
(y
16
− 8y
15
+ ··· − 3y + 1)
· (y
26
+ 49y
25
+ ··· − 13634609y + 8340544)
c
10
(y
16
+ 8y
15
+ ··· + 13y + 1)(y
26
− 15y
25
+ ··· − 18094y + 130321)
c
12
(y
16
+ 4y
15
+ ··· + 2y + 1)(y
26
− 35y
25
+ ··· − 343225y + 77284)
16
