12n
0037
(K12n
0037
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 11 12 4 6 7 10 9
Solving Sequence
6,11 2,7
5 3 1 10 12 8 9 4
c
6
c
5
c
2
c
1
c
10
c
11
c
7
c
9
c
4
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
28
− 2u
27
+ ··· + 2b − 3u, 2u
28
− 4u
27
+ ··· + 2a − 2, u
30
− 3u
29
+ ··· + 4u − 1i
I
u
2
= h−2u
4
a − 4u
3
a − 2u
4
+ 3u
2
a − 4u
3
− 8au + 3u
2
+ 19b − 7a − 8u − 7,
u
3
a − u
2
a − 2u
3
+ a
2
+ au + 2u
2
− u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 40 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
= hu
28
−2u
27
+· · ·+2b−3u, 2u
28
−4u
27
+· · ·+2a−2, u
30
−3u
29
+· · ·+4u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
−u
28
+ 2u
27
+ ··· +
3
2
u + 1
−
1
2
u
28
+ u
27
+ ··· +
5
2
u
2
+
3
2
u
a
7
=
1
−u
2
a
5
=
−
1
2
u
28
+ 2u
27
+ ··· +
3
2
u
3
+ 4u
−
1
2
u
28
+ 2u
27
+ ··· +
17
2
u
2
−
3
2
u
a
3
=
−3u
28
+ 6u
27
+ ··· +
17
2
u
2
+
5
2
u
−
3
2
u
28
+ 3u
27
+ ··· +
17
2
u
2
−
3
2
u
a
1
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
5
+ u
3
+ u
a
8
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
9
=
u
3
u
3
+ u
a
4
=
−
3
2
u
28
+ 3u
27
+ ··· +
3
2
u
3
+ 4u
−
3
2
u
28
+ 3u
27
+ ··· +
17
2
u
2
−
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
29
−
25
2
u
28
+ ··· −
59
2
u +
7
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
30
+ 22u
29
+ ··· + 23u + 1
c
2
, c
5
u
30
+ 6u
29
+ ··· + 5u + 1
c
3
u
30
− 6u
29
+ ··· + 5u + 1
c
4
, c
8
u
30
− u
29
+ ··· − 2048u − 1024
c
6
, c
10
u
30
− 3u
29
+ ··· + 4u − 1
c
7
, c
9
u
30
+ 3u
29
+ ··· + 2u − 1
c
11
u
30
+ 19u
29
+ ··· + 8u + 1
c
12
u
30
− 13u
29
+ ··· − 29592u + 1669
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
30
− 22y
29
+ ··· − 241y + 1
c
2
, c
5
y
30
+ 22y
29
+ ··· + 23y + 1
c
3
y
30
− 66y
29
+ ··· + 199y + 1
c
4
, c
8
y
30
− 55y
29
+ ··· − 4194304y + 1048576
c
6
, c
10
y
30
+ 19y
29
+ ··· + 8y + 1
c
7
, c
9
y
30
− 45y
29
+ ··· + 8y + 1
c
11
y
30
− 13y
29
+ ··· + 36y + 1
c
12
y
30
− 145y
29
+ ··· − 1268048336y + 2785561
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.976696 + 0.048976I
a = 0.12463 + 2.08766I
b = −0.56420 + 1.41173I
−16.9810 − 6.1121I −7.59589 + 2.67108I
u = 0.976696 − 0.048976I
a = 0.12463 − 2.08766I
b = −0.56420 − 1.41173I
−16.9810 + 6.1121I −7.59589 − 2.67108I
u = 0.465293 + 0.912238I
a = 1.93554 − 1.46111I
b = 0.008467 − 0.943478I
−1.62988 + 2.08354I −8.16060 − 3.45084I
u = 0.465293 − 0.912238I
a = 1.93554 + 1.46111I
b = 0.008467 + 0.943478I
−1.62988 − 2.08354I −8.16060 + 3.45084I
u = 0.960868
a = −0.707740
b = −1.15384
−12.5512 −5.29830
u = −0.166593 + 0.933326I
a = −0.92254 − 2.23571I
b = 0.617718 − 0.920078I
−1.07529 − 3.17191I −11.28317 + 2.37568I
u = −0.166593 − 0.933326I
a = −0.92254 + 2.23571I
b = 0.617718 + 0.920078I
−1.07529 + 3.17191I −11.28317 − 2.37568I
u = 0.231283 + 1.116780I
a = −0.01646 + 3.64757I
b = 0.357122 + 1.153660I
−3.77750 + 3.57782I −10.98746 − 4.01828I
u = 0.231283 − 1.116780I
a = −0.01646 − 3.64757I
b = 0.357122 − 1.153660I
−3.77750 − 3.57782I −10.98746 + 4.01828I
u = −0.828254 + 0.182083I
a = 0.61725 − 1.74485I
b = −0.176405 − 1.246890I
−5.31113 + 2.21238I −8.26070 − 1.34538I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.828254 − 0.182083I
a = 0.61725 + 1.74485I
b = −0.176405 + 1.246890I
−5.31113 − 2.21238I −8.26070 + 1.34538I
u = 0.247708 + 0.775830I
a = 0.759182 + 0.262675I
b = 0.064907 + 0.268421I
−0.445314 + 1.227680I −5.13147 − 5.35598I
u = 0.247708 − 0.775830I
a = 0.759182 − 0.262675I
b = 0.064907 − 0.268421I
−0.445314 − 1.227680I −5.13147 + 5.35598I
u = −0.431100 + 1.154270I
a = −0.376542 + 0.218469I
b = −0.568898 − 0.159015I
−4.80479 − 4.01525I −8.38777 + 4.38030I
u = −0.431100 − 1.154270I
a = −0.376542 − 0.218469I
b = −0.568898 + 0.159015I
−4.80479 + 4.01525I −8.38777 − 4.38030I
u = −0.538187 + 1.166640I
a = 1.43637 + 2.76799I
b = −0.271108 + 1.248230I
−8.20594 − 7.19126I −10.97609 + 5.35204I
u = −0.538187 − 1.166640I
a = 1.43637 − 2.76799I
b = −0.271108 − 1.248230I
−8.20594 + 7.19126I −10.97609 − 5.35204I
u = −0.033664 + 0.692526I
a = 1.161840 − 0.009465I
b = 0.453127 + 0.691055I
−0.304353 + 1.377760I −5.31625 − 4.96434I
u = −0.033664 − 0.692526I
a = 1.161840 + 0.009465I
b = 0.453127 − 0.691055I
−0.304353 − 1.377760I −5.31625 + 4.96434I
u = −0.343267 + 1.272440I
a = −0.45988 − 3.28730I
b = −0.116378 − 1.358180I
−9.80427 − 1.78849I −12.19268 + 1.56602I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.343267 − 1.272440I
a = −0.45988 + 3.28730I
b = −0.116378 + 1.358180I
−9.80427 + 1.78849I −12.19268 − 1.56602I
u = −0.662170
a = 0.443818
b = −0.426253
−1.60545 −5.56060
u = 0.486497 + 1.299810I
a = −1.68061 − 0.72887I
b = −1.177550 + 0.042833I
−16.5686 + 5.1451I −8.39302 − 2.77801I
u = 0.486497 − 1.299810I
a = −1.68061 + 0.72887I
b = −1.177550 − 0.042833I
−16.5686 − 5.1451I −8.39302 + 2.77801I
u = 0.517727 + 1.292950I
a = 0.64113 − 3.50293I
b = −0.59774 − 1.40210I
18.6656 + 11.4427I −10.35634 − 5.55493I
u = 0.517727 − 1.292950I
a = 0.64113 + 3.50293I
b = −0.59774 + 1.40210I
18.6656 − 11.4427I −10.35634 + 5.55493I
u = 0.459310 + 1.324670I
a = −1.26343 + 2.96678I
b = −0.55077 + 1.45009I
18.1990 − 1.0261I −10.88309 + 0.I
u = 0.459310 − 1.324670I
a = −1.26343 − 2.96678I
b = −0.55077 − 1.45009I
18.1990 + 1.0261I −10.88309 + 0.I
u = 0.307204 + 0.297461I
a = 1.17548 + 0.93277I
b = 0.311753 + 0.846408I
−0.35666 + 1.51654I −2.64602 − 3.80074I
u = 0.307204 − 0.297461I
a = 1.17548 − 0.93277I
b = 0.311753 − 0.846408I
−0.35666 − 1.51654I −2.64602 + 3.80074I
7
II. I
u
2
= h−2u
4
a − 2u
4
+ · · · − 7a − 7, u
3
a − u
2
a − 2u
3
+ a
2
+ au + 2u
2
− u +
2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
a
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a + 0.368421
a
7
=
1
−u
2
a
5
=
−0.105263au
4
− 0.105263u
4
+ ··· + 0.631579a − 0.368421
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
a
3
=
u
3
− u
2
+ a + u − 1
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
a
1
=
−1
0
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
4
− u
3
+ u
2
+ 1
a
8
=
u
3
u
3
+ u
a
9
=
u
3
u
3
+ u
a
4
=
−0.105263au
4
− 0.105263u
4
+ ··· + 0.631579a − 0.368421
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
3
19
u
4
a −
13
19
u
3
a −
16
19
u
4
+
5
19
u
2
a +
82
19
u
3
−
7
19
au −
90
19
u
2
−
37
19
a +
69
19
u −
170
19
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
7
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
9
, c
12
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
10
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
11
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
7
, c
9
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = 0.523653 + 0.423720I
b = 0.500000 + 0.866025I
−0.329100 + 0.499304I −5.91654 + 2.81652I
u = −0.339110 + 0.822375I
a = −1.39487 − 1.53138I
b = 0.500000 − 0.866025I
−0.32910 − 3.56046I −1.60756 + 7.85087I
u = −0.339110 − 0.822375I
a = 0.523653 − 0.423720I
b = 0.500000 − 0.866025I
−0.329100 − 0.499304I −5.91654 − 2.81652I
u = −0.339110 − 0.822375I
a = −1.39487 + 1.53138I
b = 0.500000 + 0.866025I
−0.32910 + 3.56046I −1.60756 − 7.85087I
u = 0.766826
a = −0.314857 + 1.186700I
b = 0.500000 + 0.866025I
−2.40108 + 2.02988I −6.55976 − 2.76390I
u = 0.766826
a = −0.314857 − 1.186700I
b = 0.500000 − 0.866025I
−2.40108 − 2.02988I −6.55976 + 2.76390I
u = 0.455697 + 1.200150I
a = 0.85051 − 1.45588I
b = 0.500000 − 0.866025I
−5.87256 + 2.37095I −10.62344 − 1.09779I
u = 0.455697 + 1.200150I
a = −0.66443 + 2.33052I
b = 0.500000 + 0.866025I
−5.87256 + 6.43072I −9.29269 − 5.42389I
u = 0.455697 − 1.200150I
a = 0.85051 + 1.45588I
b = 0.500000 + 0.866025I
−5.87256 − 2.37095I −10.62344 + 1.09779I
u = 0.455697 − 1.200150I
a = −0.66443 − 2.33052I
b = 0.500000 − 0.866025I
−5.87256 − 6.43072I −9.29269 + 5.42389I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
30
+ 22u
29
+ ··· + 23u + 1)
c
2
((u
2
+ u + 1)
5
)(u
30
+ 6u
29
+ ··· + 5u + 1)
c
3
((u
2
− u + 1)
5
)(u
30
− 6u
29
+ ··· + 5u + 1)
c
4
, c
8
u
10
(u
30
− u
29
+ ··· − 2048u − 1024)
c
5
((u
2
− u + 1)
5
)(u
30
+ 6u
29
+ ··· + 5u + 1)
c
6
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
30
− 3u
29
+ ··· + 4u − 1)
c
7
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
30
+ 3u
29
+ ··· + 2u − 1)
c
9
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
30
+ 3u
29
+ ··· + 2u − 1)
c
10
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
30
− 3u
29
+ ··· + 4u − 1)
c
11
((u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
)(u
30
+ 19u
29
+ ··· + 8u + 1)
c
12
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
30
− 13u
29
+ ··· − 29592u + 1669)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
30
− 22y
29
+ ··· − 241y + 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
30
+ 22y
29
+ ··· + 23y + 1)
c
3
((y
2
+ y + 1)
5
)(y
30
− 66y
29
+ ··· + 199y + 1)
c
4
, c
8
y
10
(y
30
− 55y
29
+ ··· − 4194304y + 1048576)
c
6
, c
10
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
30
+ 19y
29
+ ··· + 8y + 1)
c
7
, c
9
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
30
− 45y
29
+ ··· + 8y + 1)
c
11
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
30
− 13y
29
+ ··· + 36y + 1)
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
· (y
30
− 145y
29
+ ··· − 1268048336y + 2785561)
13
