12n
0065
(K12n
0065
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 11 10 5 12 7 6 9
Solving Sequence
6,11
7
2,12
5 3 1 10 8 4 9
c
6
c
11
c
5
c
2
c
1
c
10
c
7
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
22
+ 2u
21
+ ··· + 2b − 2u, −2u
22
+ 5u
21
+ ··· + 2a + 4, u
23
− 3u
22
+ ··· − 4u + 1i
I
u
2
= h−4u
3
a − 2u
2
a − 4u
3
− 11au − 2u
2
+ 11b − 8a − 11u − 8,
u
3
a + u
2
a − u
3
+ a
2
+ 3au − 2u
2
+ 2a − 4u − 2, u
4
+ u
3
+ 3u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 31 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
22
+2u
21
+· · ·+2b−2u, −2u
22
+5u
21
+· · ·+2a+4, u
23
−3u
22
+· · ·−4u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
2
=
u
22
−
5
2
u
21
+ ··· −
11
2
u
2
− 2
1
2
u
22
− u
21
+ ··· + 2u
2
+ u
a
12
=
u
u
a
5
=
−
1
2
u
22
+
3
2
u
21
+ ··· − 7u + 2
1
2
u
22
− u
21
+ ··· + u − 1
a
3
=
−
1
2
u
21
− 4u
19
+ ··· − 5u + 1
3
2
u
22
− 4u
21
+ ··· + 4u − 2
a
1
=
−u
9
− 4u
7
− 3u
5
+ 2u
3
− u
−u
9
− 5u
7
− 7u
5
− 2u
3
− u
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
4
=
−
3
2
u
22
+
7
2
u
21
+ ··· − 9u + 3
3
2
u
22
− 4u
21
+ ··· + 4u − 2
a
9
=
u
5
+ 2u
3
− u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
3
2
u
22
+3u
21
−21u
20
+
73
2
u
19
−124u
18
+186u
17
−400u
16
+506u
15
−
1519
2
u
14
+
1537
2
u
13
−
845u
12
+ 605u
11
−
1015
2
u
10
+ 178u
9
−
265
2
u
8
−
33
2
u
7
−
29
2
u
6
−
67
2
u
5
−
5
2
u
4
−49u
3
+
19
2
u
2
+
7
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 3u
22
+ ··· + 4u − 1
c
2
, c
5
u
23
+ 5u
22
+ ··· − 4u − 1
c
3
u
23
− 5u
22
+ ··· − 2678u − 593
c
4
, c
8
u
23
− u
22
+ ··· + 128u − 256
c
6
, c
7
, c
10
c
11
u
23
+ 3u
22
+ ··· − 4u − 1
c
9
, c
12
u
23
− u
22
+ ··· + 2u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
+ 39y
22
+ ··· + 4y −1
c
2
, c
5
y
23
+ 3y
22
+ ··· + 4y −1
c
3
y
23
+ 75y
22
+ ··· − 15091908y −351649
c
4
, c
8
y
23
− 45y
22
+ ··· − 212992y −65536
c
6
, c
7
, c
10
c
11
y
23
+ 25y
22
+ ··· + 4y −1
c
9
, c
12
y
23
+ 41y
22
+ ··· + 4y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.768656 + 0.574390I
a = 0.96411 − 1.30127I
b = −1.02013 − 1.06010I
14.5386 + 6.4049I 5.87992 − 4.56999I
u = 0.768656 − 0.574390I
a = 0.96411 + 1.30127I
b = −1.02013 + 1.06010I
14.5386 − 6.4049I 5.87992 + 4.56999I
u = 0.792664 + 0.509239I
a = −0.193691 − 0.034587I
b = −1.06482 + 1.00488I
14.7384 − 1.2262I 6.30437 − 0.37163I
u = 0.792664 − 0.509239I
a = −0.193691 + 0.034587I
b = −1.06482 − 1.00488I
14.7384 + 1.2262I 6.30437 + 0.37163I
u = −0.502322 + 0.520642I
a = 0.811150 + 0.610068I
b = −0.461371 + 0.176461I
0.70832 − 1.75933I 4.65925 + 3.45911I
u = −0.502322 − 0.520642I
a = 0.811150 − 0.610068I
b = −0.461371 − 0.176461I
0.70832 + 1.75933I 4.65925 − 3.45911I
u = −0.038925 + 1.309910I
a = 0.467966 − 0.365892I
b = 0.881690 − 0.526981I
−2.62006 − 1.64777I 3.52749 + 2.17174I
u = −0.038925 − 1.309910I
a = 0.467966 + 0.365892I
b = 0.881690 + 0.526981I
−2.62006 + 1.64777I 3.52749 − 2.17174I
u = 0.091640 + 1.402330I
a = −0.60254 + 2.18385I
b = 0.586427 + 1.117770I
−4.58949 + 3.87928I 1.77141 − 2.75540I
u = 0.091640 − 1.402330I
a = −0.60254 − 2.18385I
b = 0.586427 − 1.117770I
−4.58949 − 3.87928I 1.77141 + 2.75540I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.091228 + 0.543350I
a = 1.15824 − 1.74645I
b = 0.243917 − 0.711016I
−1.17938 − 1.49722I −2.53958 + 5.27560I
u = −0.091228 − 0.543350I
a = 1.15824 + 1.74645I
b = 0.243917 + 0.711016I
−1.17938 + 1.49722I −2.53958 − 5.27560I
u = −0.464696
a = 0.105740
b = 0.580286
1.11404 10.0250
u = −0.05085 + 1.53575I
a = 0.55421 − 2.04140I
b = 0.003120 − 0.853741I
−8.14932 − 2.13339I −3.53476 + 3.27759I
u = −0.05085 − 1.53575I
a = 0.55421 + 2.04140I
b = 0.003120 + 0.853741I
−8.14932 + 2.13339I −3.53476 − 3.27759I
u = 0.29102 + 1.52358I
a = −1.009020 + 0.577805I
b = −1.08811 + 0.92523I
8.14473 + 2.74909I 3.44638 − 0.72919I
u = 0.29102 − 1.52358I
a = −1.009020 − 0.577805I
b = −1.08811 − 0.92523I
8.14473 − 2.74909I 3.44638 + 0.72919I
u = −0.13755 + 1.55696I
a = 0.363925 + 1.080750I
b = −0.509748 + 0.419465I
−6.30985 − 4.03193I 1.37603 + 1.02672I
u = −0.13755 − 1.55696I
a = 0.363925 − 1.080750I
b = −0.509748 − 0.419465I
−6.30985 + 4.03193I 1.37603 − 1.02672I
u = 0.26608 + 1.55802I
a = 0.27719 − 2.09273I
b = −0.96060 − 1.09404I
7.54756 + 10.22760I 2.79101 − 4.77678I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.26608 − 1.55802I
a = 0.27719 + 2.09273I
b = −0.96060 + 1.09404I
7.54756 − 10.22760I 2.79101 + 4.77678I
u = 0.343175 + 0.152187I
a = −2.34441 + 0.12809I
b = 0.599485 + 0.898495I
0.46499 + 2.40467I 3.80616 − 1.75250I
u = 0.343175 − 0.152187I
a = −2.34441 − 0.12809I
b = 0.599485 − 0.898495I
0.46499 − 2.40467I 3.80616 + 1.75250I
7
II.
I
u
2
= h−4u
3
a−4u
3
+· · ·−8a−8, u
3
a−u
3
+· · ·+2a−2, u
4
+u
3
+3u
2
+2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
2
=
a
0.363636au
3
+ 0.363636u
3
+ ··· + 0.727273a + 0.727273
a
12
=
u
u
a
5
=
−0.363636au
3
+ 0.636364u
3
+ ··· + 0.272727a + 1.27273
0.363636au
3
+ 0.363636u
3
+ ··· + 0.727273a − 0.272727
a
3
=
u
3
+ u
2
+ a + 3u + 1
0.363636au
3
+ 0.363636u
3
+ ··· + 0.727273a − 0.272727
a
1
=
−1
0
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
u
3
+ u
2
+ 2u + 1
a
4
=
−0.363636au
3
+ 0.636364u
3
+ ··· + 0.272727a + 1.27273
0.363636au
3
+ 0.363636u
3
+ ··· + 0.727273a − 0.272727
a
9
=
u
2
+ 1
u
3
+ u
2
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
3
a − u
2
a + 2u
3
− 6au + 3u
2
− 3a + 7u + 7
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
8
u
8
c
6
, c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
9
(u
4
+ u
3
+ u
2
+ 1)
2
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
12
(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
8
y
8
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 0.584432 + 0.289945I
b = 0.500000 + 0.866025I
0.211005 + 0.614778I 4.65255 + 0.59814I
u = −0.395123 + 0.506844I
a = −1.54112 − 1.51713I
b = 0.500000 − 0.866025I
0.21101 − 3.44499I 1.64912 + 8.49900I
u = −0.395123 − 0.506844I
a = 0.584432 − 0.289945I
b = 0.500000 − 0.866025I
0.211005 − 0.614778I 4.65255 − 0.59814I
u = −0.395123 − 0.506844I
a = −1.54112 + 1.51713I
b = 0.500000 + 0.866025I
0.21101 + 3.44499I 1.64912 − 8.49900I
u = −0.10488 + 1.55249I
a = 0.53364 + 1.37394I
b = 0.500000 + 0.866025I
−6.79074 − 1.13408I 1.99896 − 0.39034I
u = −0.10488 + 1.55249I
a = −0.57695 − 2.01514I
b = 0.500000 − 0.866025I
−6.79074 − 5.19385I −1.80063 + 6.43123I
u = −0.10488 − 1.55249I
a = 0.53364 − 1.37394I
b = 0.500000 − 0.866025I
−6.79074 + 1.13408I 1.99896 + 0.39034I
u = −0.10488 − 1.55249I
a = −0.57695 + 2.01514I
b = 0.500000 + 0.866025I
−6.79074 + 5.19385I −1.80063 − 6.43123I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
23
+ 3u
22
+ ··· + 4u − 1)
c
2
((u
2
+ u + 1)
4
)(u
23
+ 5u
22
+ ··· − 4u − 1)
c
3
((u
2
− u + 1)
4
)(u
23
− 5u
22
+ ··· − 2678u − 593)
c
4
, c
8
u
8
(u
23
− u
22
+ ··· + 128u − 256)
c
5
((u
2
− u + 1)
4
)(u
23
+ 5u
22
+ ··· − 4u − 1)
c
6
, c
7
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
23
+ 3u
22
+ ··· − 4u − 1)
c
9
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
23
− u
22
+ ··· + 2u
2
− 1)
c
10
, c
11
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
23
+ 3u
22
+ ··· − 4u − 1)
c
12
((u
4
− u
3
+ u
2
+ 1)
2
)(u
23
− u
22
+ ··· + 2u
2
− 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
23
+ 39y
22
+ ··· + 4y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
23
+ 3y
22
+ ··· + 4y −1)
c
3
((y
2
+ y + 1)
4
)(y
23
+ 75y
22
+ ··· − 1.50919 × 10
7
y −351649)
c
4
, c
8
y
8
(y
23
− 45y
22
+ ··· − 212992y −65536)
c
6
, c
7
, c
10
c
11
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
23
+ 25y
22
+ ··· + 4y −1)
c
9
, c
12
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
23
+ 41y
22
+ ··· + 4y −1)
13
