12n
0013
(K12n
0013
)
A knot diagram
1
Linearized knot diagam
3 5 6 7 2 11 4 6 12 7 10 9
Solving Sequence
6,11 2,7
5 4 3 1 10 12 9 8
c
6
c
5
c
4
c
3
c
1
c
10
c
11
c
9
c
8
c
2
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
20
+ 2u
19
+ ··· + 3u
2
+ 2b, −u
20
+ 2u
19
+ ··· + 2a + 1, u
22
− 3u
21
+ ··· − u + 1i
I
u
2
= h−u
3
a − 2u
2
a − u
3
− au − 2u
2
+ 2b − a − u − 1, u
2
a + u
3
+ a
2
+ au + u
2
+ 2a + u, u
4
+ u
3
+ u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 30 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−u
20
+2u
19
+· · ·+3u
2
+2b, −u
20
+2u
19
+· · ·+2a+1, u
22
−3u
21
+· · ·−u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
1
2
u
20
− u
19
+ ··· +
3
2
u −
1
2
1
2
u
20
− u
19
+ ··· +
3
2
u
3
−
3
2
u
2
a
7
=
1
u
2
a
5
=
−u
19
+
5
2
u
18
+ ··· −
5
2
u +
5
2
−u
21
+
5
2
u
20
+ ··· +
5
2
u
2
− u
a
4
=
3
2
u
18
− 2u
17
+ ··· −
3
2
u +
5
2
3
2
u
20
− 3u
19
+ ··· +
5
2
u
2
− u
a
3
=
3
2
u
20
− 3u
19
+ ··· −
5
2
u +
5
2
3
2
u
20
− 3u
19
+ ··· +
5
2
u
2
− u
a
1
=
−u
7
− 2u
3
−u
9
− u
7
− 3u
5
− 2u
3
− u
a
10
=
u
u
3
+ u
a
12
=
u
3
u
5
+ u
3
+ u
a
9
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
a
8
=
−u
7
− 2u
3
u
7
+ u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
7
2
u
21
− 8u
20
+
27
2
u
19
−
19
2
u
18
+ 32u
17
−
99
2
u
16
+
163
2
u
15
− 45u
14
+
197
2
u
13
− 102u
12
+
365
2
u
11
−
157
2
u
10
+
253
2
u
9
−58u
8
+ 151u
7
−40u
6
+ 53u
5
+
59
2
u
4
+
23
2
u
3
+
35
2
u
2
+
21
2
u +
17
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 17u
21
+ ··· + 31u + 1
c
2
, c
5
u
22
+ 5u
21
+ ··· + 7u + 1
c
3
u
22
− 5u
21
+ ··· + 5u + 1
c
4
, c
7
u
22
+ u
21
+ ··· − 640u + 256
c
6
, c
10
u
22
+ 3u
21
+ ··· + u + 1
c
8
u
22
+ 3u
21
+ ··· − 2455u + 2425
c
9
, c
11
, c
12
u
22
− 3u
21
+ ··· − 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
− 19y
21
+ ··· − 29y + 1
c
2
, c
5
y
22
+ 17y
21
+ ··· + 31y + 1
c
3
y
22
− 55y
21
+ ··· + 143y + 1
c
4
, c
7
y
22
+ 45y
21
+ ··· + 344064y + 65536
c
6
, c
10
y
22
+ 3y
21
+ ··· + 11y + 1
c
8
y
22
+ 135y
21
+ ··· + 316362175y + 5880625
c
9
, c
11
, c
12
y
22
+ 35y
21
+ ··· + 11y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.443267 + 0.917989I
a = 1.79705 + 0.42952I
b = −0.004921 + 1.060270I
−1.64918 − 2.09688I 0.35789 + 3.47675I
u = 0.443267 − 0.917989I
a = 1.79705 − 0.42952I
b = −0.004921 − 1.060270I
−1.64918 + 2.09688I 0.35789 − 3.47675I
u = 0.720168 + 0.521314I
a = −0.12085 − 1.89524I
b = 0.210578 − 1.177030I
−3.15354 − 2.22003I −2.57059 + 3.13171I
u = 0.720168 − 0.521314I
a = −0.12085 + 1.89524I
b = 0.210578 + 1.177030I
−3.15354 + 2.22003I −2.57059 − 3.13171I
u = −0.786228 + 0.864892I
a = 0.309770 − 0.406841I
b = −0.672095 + 0.089076I
−5.42259 + 2.92304I 0.66405 − 3.09728I
u = −0.786228 − 0.864892I
a = 0.309770 + 0.406841I
b = −0.672095 − 0.089076I
−5.42259 − 2.92304I 0.66405 + 3.09728I
u = −0.948373 + 0.755313I
a = −0.16577 + 1.46179I
b = −0.211609 + 1.390430I
−10.31720 − 0.20205I −2.87081 − 0.56297I
u = −0.948373 − 0.755313I
a = −0.16577 − 1.46179I
b = −0.211609 − 1.390430I
−10.31720 + 0.20205I −2.87081 + 0.56297I
u = −0.763942 + 1.021840I
a = 1.53053 − 1.26261I
b = −0.296827 − 1.316550I
−9.36150 + 6.49304I −1.80593 − 4.67801I
u = −0.763942 − 1.021840I
a = 1.53053 + 1.26261I
b = −0.296827 + 1.316550I
−9.36150 − 6.49304I −1.80593 + 4.67801I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.269873 + 0.669231I
a = 0.754251 + 0.193264I
b = −0.084971 − 0.208905I
0.305023 − 1.133800I 3.87062 + 6.16556I
u = 0.269873 − 0.669231I
a = 0.754251 − 0.193264I
b = −0.084971 + 0.208905I
0.305023 + 1.133800I 3.87062 − 6.16556I
u = 0.967508 + 0.974980I
a = −0.224218 + 0.619641I
b = −1.215960 − 0.024945I
−18.0328 − 3.5472I 0.60128 + 2.10334I
u = 0.967508 − 0.974980I
a = −0.224218 − 0.619641I
b = −1.215960 + 0.024945I
−18.0328 + 3.5472I 0.60128 − 2.10334I
u = 1.005100 + 0.939078I
a = −0.378437 − 1.203500I
b = −0.58715 − 1.46073I
16.7892 + 2.8754I −1.69600 − 0.52262I
u = 1.005100 − 0.939078I
a = −0.378437 + 1.203500I
b = −0.58715 + 1.46073I
16.7892 − 2.8754I −1.69600 + 0.52262I
u = 0.948146 + 1.018020I
a = 1.14718 + 1.69204I
b = −0.61183 + 1.42911I
17.0662 − 10.0252I −1.33592 + 4.78932I
u = 0.948146 − 1.018020I
a = 1.14718 − 1.69204I
b = −0.61183 − 1.42911I
17.0662 + 10.0252I −1.33592 − 4.78932I
u = −0.036441 + 0.595658I
a = 0.47769 + 1.42665I
b = 0.429450 − 0.716106I
0.68417 − 1.38791I 7.27307 + 5.07376I
u = −0.036441 − 0.595658I
a = 0.47769 − 1.42665I
b = 0.429450 + 0.716106I
0.68417 + 1.38791I 7.27307 − 5.07376I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.319079 + 0.434625I
a = −2.12719 + 1.76082I
b = 0.545330 + 0.947805I
−0.06729 + 2.75299I 1.012349 − 0.159946I
u = −0.319079 − 0.434625I
a = −2.12719 − 1.76082I
b = 0.545330 − 0.947805I
−0.06729 − 2.75299I 1.012349 + 0.159946I
7
II.
I
u
2
= h−u
3
a−u
3
+· · ·−a−1, u
2
a+u
3
+a
2
+au +u
2
+2a +u, u
4
+u
3
+u
2
+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
a
1
2
u
3
a +
1
2
u
3
+ ··· +
1
2
a +
1
2
a
7
=
1
u
2
a
5
=
−
1
2
u
3
a −
1
2
u
3
+ ··· +
1
2
a +
3
2
1
2
u
3
a +
1
2
u
3
+ ··· +
1
2
a −
1
2
a
4
=
−
1
2
u
3
a −
1
2
u
3
+ ··· +
1
2
a +
3
2
1
2
u
3
a +
1
2
u
3
+ ··· +
1
2
a −
1
2
a
3
=
u
2
+ a + u + 1
1
2
u
3
a +
1
2
u
3
+ ··· +
1
2
a −
1
2
a
1
=
−1
0
a
10
=
u
u
3
+ u
a
12
=
u
3
u
3
+ u
2
+ 1
a
9
=
u
2
+ 1
u
2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
3
a − 3u
2
a − 2u
3
− 3au − 3a + u + 2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
7
u
8
c
6
(u
4
+ u
3
+ u
2
+ 1)
2
c
8
, c
11
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
10
(u
4
− u
3
+ u
2
+ 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)
4
c
4
, c
7
y
8
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = 0.084432 − 0.576081I
b = 0.500000 + 0.866025I
0.211005 + 0.614778I 1.10064 + 1.99408I
u = 0.351808 + 0.720342I
a = −2.04112 − 0.65111I
b = 0.500000 − 0.866025I
0.21101 − 3.44499I 5.86133 + 9.77094I
u = 0.351808 − 0.720342I
a = 0.084432 + 0.576081I
b = 0.500000 − 0.866025I
0.211005 − 0.614778I 1.10064 − 1.99408I
u = 0.351808 − 0.720342I
a = −2.04112 + 0.65111I
b = 0.500000 + 0.866025I
0.21101 + 3.44499I 5.86133 − 9.77094I
u = −0.851808 + 0.911292I
a = 0.033637 − 0.507913I
b = 0.500000 − 0.866025I
−6.79074 + 1.13408I −1.56110 − 0.68902I
u = −0.851808 + 0.911292I
a = −1.07695 + 1.14911I
b = 0.500000 + 0.866025I
−6.79074 + 5.19385I −0.90087 − 4.17049I
u = −0.851808 − 0.911292I
a = 0.033637 + 0.507913I
b = 0.500000 + 0.866025I
−6.79074 − 1.13408I −1.56110 + 0.68902I
u = −0.851808 − 0.911292I
a = −1.07695 − 1.14911I
b = 0.500000 − 0.866025I
−6.79074 − 5.19385I −0.90087 + 4.17049I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
22
+ 17u
21
+ ··· + 31u + 1)
c
2
((u
2
+ u + 1)
4
)(u
22
+ 5u
21
+ ··· + 7u + 1)
c
3
((u
2
− u + 1)
4
)(u
22
− 5u
21
+ ··· + 5u + 1)
c
4
, c
7
u
8
(u
22
+ u
21
+ ··· − 640u + 256)
c
5
((u
2
− u + 1)
4
)(u
22
+ 5u
21
+ ··· + 7u + 1)
c
6
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
22
+ 3u
21
+ ··· + u + 1)
c
8
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
22
+ 3u
21
+ ··· − 2455u + 2425)
c
9
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
22
− 3u
21
+ ··· − 11u + 1)
c
10
((u
4
− u
3
+ u
2
+ 1)
2
)(u
22
+ 3u
21
+ ··· + u + 1)
c
11
, c
12
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
22
− 3u
21
+ ··· − 11u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
22
− 19y
21
+ ··· − 29y + 1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
22
+ 17y
21
+ ··· + 31y + 1)
c
3
((y
2
+ y + 1)
4
)(y
22
− 55y
21
+ ··· + 143y + 1)
c
4
, c
7
y
8
(y
22
+ 45y
21
+ ··· + 344064y + 65536)
c
6
, c
10
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
22
+ 3y
21
+ ··· + 11y + 1)
c
8
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
22
+ 135y
21
+ ··· + 316362175y + 5880625)
c
9
, c
11
, c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
22
+ 35y
21
+ ··· + 11y + 1)
13
