12a
0787
(K12a
0787
)
A knot diagram
1
Linearized knot diagam
3 8 11 10 12 9 2 7 1 4 6 5
Solving Sequence
5,10
4 11
1,3
2 9 12 6 7 8
c
4
c
10
c
3
c
1
c
9
c
12
c
5
c
6
c
8
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
27
+ u
26
+ ··· + 16a + 1, u
28
+ 17u
26
+ ··· − u + 1i
I
u
2
= h−2521899524065u
37
− 1276991146481u
36
+ ··· + 13233245626882b + 40729676197632,
− 9138522337506u
37
− 7893613959922u
36
+ ··· + 13233245626882a + 87046035891719,
u
38
+ u
37
+ ··· − 7u + 2i
I
u
3
= hb + u, a
4
+ a
3
+ 3a
2
+ 2a + 1, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 74 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
= hb − u, u
27
+ u
26
+ · · · + 16a + 1, u
28
+ 17u
26
+ · · · − u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
11
=
u
u
3
+ u
a
1
=
−
1
16
u
27
−
1
16
u
26
+ ··· + 3u −
1
16
u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−
1
16
u
27
−
1
16
u
26
+ ··· + 2u −
1
16
−
1
16
u
27
−
1
16
u
26
+ ··· + u −
1
16
a
9
=
−
1
8
u
25
−
1
8
u
24
+ ··· −
87
8
u
3
−
1
8
−
1
16
u
27
−
1
16
u
26
+ ··· + u −
1
16
a
12
=
−
1
16
u
27
−
1
16
u
26
+ ··· + 2u −
1
16
u
a
6
=
1
16
u
27
−
1
16
u
26
+ ··· +
1
8
u +
15
16
−u
2
a
7
=
−0.687500u
27
+ 0.437500u
26
+ ··· − 1.62500u + 1.56250
−
1
16
u
27
−
1
16
u
26
+ ··· −
1
4
u +
1
16
a
8
=
−0.187500u
27
+ 0.187500u
26
+ ··· + 0.375000u + 1.31250
1
4
u
27
+
3
8
u
26
+ ··· +
7
8
u +
5
8
(ii) Obstruction class = −1
(iii) Cusp Shapes =
11
4
u
27
−
1
4
u
26
+ ··· −
9
2
u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
u
28
+ 7u
27
+ ··· + 9u + 4
c
2
, c
7
u
28
+ 3u
27
+ ··· − u + 2
c
3
, c
4
, c
5
c
10
, c
11
, c
12
u
28
+ 17u
26
+ ··· + u + 1
c
9
u
28
− 21u
27
+ ··· − 24293u + 2642
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
y
28
+ 29y
27
+ ··· + 383y + 16
c
2
, c
7
y
28
− 7y
27
+ ··· − 9y + 4
c
3
, c
4
, c
5
c
10
, c
11
, c
12
y
28
+ 34y
27
+ ··· + 5y + 1
c
9
y
28
+ y
27
+ ··· + 72951y + 6980164
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.668490 + 0.320146I
a = 1.70913 + 0.46510I
b = 0.668490 + 0.320146I
6.28423 + 6.57769I 6.41221 − 7.42471I
u = 0.668490 − 0.320146I
a = 1.70913 − 0.46510I
b = 0.668490 − 0.320146I
6.28423 − 6.57769I 6.41221 + 7.42471I
u = −0.675753 + 0.283618I
a = −1.69364 + 0.41228I
b = −0.675753 + 0.283618I
6.57383 − 0.38638I 7.28743 + 2.11062I
u = −0.675753 − 0.283618I
a = −1.69364 − 0.41228I
b = −0.675753 − 0.283618I
6.57383 + 0.38638I 7.28743 − 2.11062I
u = 0.010002 + 1.361850I
a = 0.08812 − 1.82299I
b = 0.010002 + 1.361850I
0.98241 + 3.28877I −2.28562 − 2.49826I
u = 0.010002 − 1.361850I
a = 0.08812 + 1.82299I
b = 0.010002 − 1.361850I
0.98241 − 3.28877I −2.28562 + 2.49826I
u = 0.479155 + 0.323957I
a = 1.41870 + 0.64071I
b = 0.479155 + 0.323957I
−0.70246 + 3.29244I 1.36299 − 9.65780I
u = 0.479155 − 0.323957I
a = 1.41870 − 0.64071I
b = 0.479155 − 0.323957I
−0.70246 − 3.29244I 1.36299 + 9.65780I
u = 0.018495 + 0.555616I
a = 0.11963 + 2.15929I
b = 0.018495 + 0.555616I
5.02040 − 3.10627I 5.07926 + 2.30848I
u = 0.018495 − 0.555616I
a = 0.11963 − 2.15929I
b = 0.018495 − 0.555616I
5.02040 + 3.10627I 5.07926 − 2.30848I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.12806 + 1.49665I
a = 0.635356 − 0.790802I
b = 0.12806 + 1.49665I
−8.20165 + 3.12776I −1.78341 − 2.86928I
u = 0.12806 − 1.49665I
a = 0.635356 + 0.790802I
b = 0.12806 − 1.49665I
−8.20165 − 3.12776I −1.78341 + 2.86928I
u = −0.467343 + 0.123789I
a = −1.256740 + 0.253657I
b = −0.467343 + 0.123789I
0.942412 − 0.321823I 9.93783 + 1.87531I
u = −0.467343 − 0.123789I
a = −1.256740 − 0.253657I
b = −0.467343 − 0.123789I
0.942412 + 0.321823I 9.93783 − 1.87531I
u = 0.33592 + 1.48906I
a = 1.343430 − 0.294813I
b = 0.33592 + 1.48906I
−4.85757 + 8.12866I −0.65039 − 3.19597I
u = 0.33592 − 1.48906I
a = 1.343430 + 0.294813I
b = 0.33592 − 1.48906I
−4.85757 − 8.12866I −0.65039 + 3.19597I
u = 0.24720 + 1.51029I
a = 1.054950 − 0.471434I
b = 0.24720 + 1.51029I
−10.29860 + 5.81281I −1.13573 − 2.99582I
u = 0.24720 − 1.51029I
a = 1.054950 + 0.471434I
b = 0.24720 − 1.51029I
−10.29860 − 5.81281I −1.13573 + 2.99582I
u = −0.34687 + 1.50237I
a = −1.331920 − 0.237656I
b = −0.34687 + 1.50237I
−5.4858 − 14.4431I −1.57414 + 7.85260I
u = −0.34687 − 1.50237I
a = −1.331920 + 0.237656I
b = −0.34687 − 1.50237I
−5.4858 + 14.4431I −1.57414 − 7.85260I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.09545 + 1.55729I
a = −0.408600 − 0.576483I
b = −0.09545 + 1.55729I
−9.25929 + 2.16393I −3.88367 − 2.85288I
u = −0.09545 − 1.55729I
a = −0.408600 + 0.576483I
b = −0.09545 − 1.55729I
−9.25929 − 2.16393I −3.88367 + 2.85288I
u = −0.28719 + 1.53991I
a = −1.099080 − 0.292514I
b = −0.28719 + 1.53991I
−13.1202 − 9.6253I −6.14913 + 7.05980I
u = −0.28719 − 1.53991I
a = −1.099080 + 0.292514I
b = −0.28719 − 1.53991I
−13.1202 + 9.6253I −6.14913 − 7.05980I
u = −0.21010 + 1.56283I
a = −0.823988 − 0.384208I
b = −0.21010 + 1.56283I
−14.2962 − 3.2369I −8.20460 + 0.I
u = −0.21010 − 1.56283I
a = −0.823988 + 0.384208I
b = −0.21010 − 1.56283I
−14.2962 + 3.2369I −8.20460 + 0.I
u = 0.195378 + 0.368416I
a = 0.744647 + 1.097280I
b = 0.195378 + 0.368416I
−1.28464 − 0.85587I −2.41301 + 0.23823I
u = 0.195378 − 0.368416I
a = 0.744647 − 1.097280I
b = 0.195378 − 0.368416I
−1.28464 + 0.85587I −2.41301 − 0.23823I
7
II.
I
u
2
= h−2.52×10
12
u
37
−1.28×10
12
u
36
+· · ·+1.32×10
13
b+4.07×10
13
, −9.14×
10
12
u
37
−7.89×10
12
u
36
+· · ·+1.32×10
13
a+8.70×10
13
, u
38
+u
37
+· · ·−7u+2i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
11
=
u
u
3
+ u
a
1
=
0.690573u
37
+ 0.596499u
36
+ ··· + 27.4552u − 6.57783
0.190573u
37
+ 0.0964987u
36
+ ··· + 7.95523u − 3.07783
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
0.504054u
37
+ 0.688185u
36
+ ··· + 16.6062u − 3.16265
0.385043u
37
+ 0.407527u
36
+ ··· + 10.1840u − 3.31197
a
9
=
1.06524u
37
+ 0.963635u
36
+ ··· + 35.3708u − 9.46751
0.374672u
37
+ 0.367136u
36
+ ··· + 8.91556u − 2.88968
a
12
=
1
2
u
37
+
1
2
u
36
+ ··· +
39
2
u −
7
2
0.190573u
37
+ 0.0964987u
36
+ ··· + 7.95523u − 3.07783
a
6
=
1.53891u
37
+ 1.72949u
36
+ ··· + 25.2431u − 1.81717
−0.0940743u
37
+ 0.0900246u
36
+ ··· − 1.74382u + 0.618854
a
7
=
−0.540566u
37
− 0.819412u
36
+ ··· − 5.62383u − 4.68707
−0.203220u
37
− 0.490174u
36
+ ··· − 4.02159u − 2.26098
a
8
=
−0.215007u
37
+ 0.108358u
36
+ ··· − 9.07233u + 2.38121
−0.203697u
37
− 0.475149u
36
+ ··· − 4.71463u − 1.10982
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
2578336863836
6616622813441
u
37
−
917470771354
6616622813441
u
36
+ ··· +
231718964327524
6616622813441
u −
47642846027518
6616622813441
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
(u
19
+ 5u
18
+ ··· + 2u + 1)
2
c
2
, c
7
(u
19
− u
18
+ ··· + u
2
− 1)
2
c
3
, c
4
, c
5
c
10
, c
11
, c
12
u
38
− u
37
+ ··· + 7u + 2
c
9
(u
19
+ 7u
18
+ ··· + 2u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
(y
19
+ 19y
18
+ ··· + 10y −1)
2
c
2
, c
7
(y
19
− 5y
18
+ ··· + 2y −1)
2
c
3
, c
4
, c
5
c
10
, c
11
, c
12
y
38
+ 31y
37
+ ··· + 107y + 4
c
9
(y
19
+ 11y
18
+ ··· + 42y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.745217 + 0.660881I
a = 0.786897 − 0.760848I
b = 0.03575 − 1.42698I
−6.91199 + 0.16816I −6.16829 − 0.91431I
u = −0.745217 − 0.660881I
a = 0.786897 + 0.760848I
b = 0.03575 + 1.42698I
−6.91199 − 0.16816I −6.16829 + 0.91431I
u = −0.909367 + 0.387387I
a = 1.17340 − 1.04324I
b = 0.24265 − 1.43973I
0.60648 − 9.88550I 1.13872 + 7.31129I
u = −0.909367 − 0.387387I
a = 1.17340 + 1.04324I
b = 0.24265 + 1.43973I
0.60648 + 9.88550I 1.13872 − 7.31129I
u = −0.835357 + 0.509822I
a = 1.020060 − 0.906332I
b = 0.14784 − 1.43865I
−6.41945 − 5.52702I −4.42794 + 7.00248I
u = −0.835357 − 0.509822I
a = 1.020060 + 0.906332I
b = 0.14784 + 1.43865I
−6.41945 + 5.52702I −4.42794 − 7.00248I
u = 0.881694 + 0.365628I
a = −1.21257 − 1.01487I
b = −0.24482 − 1.41618I
1.12421 + 3.71612I 2.19900 − 2.45937I
u = 0.881694 − 0.365628I
a = −1.21257 + 1.01487I
b = −0.24482 + 1.41618I
1.12421 − 3.71612I 2.19900 + 2.45937I
u = 0.190982 + 1.053290I
a = 0.0243158 − 0.1341050I
b = 0.190982 − 1.053290I
−4.19724 −7.47222 + 0.I
u = 0.190982 − 1.053290I
a = 0.0243158 + 0.1341050I
b = 0.190982 + 1.053290I
−4.19724 −7.47222 + 0.I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618900 + 0.885728I
a = −0.403494 − 0.627123I
b = 0.126589 − 1.385740I
−0.45606 + 1.53005I −0.20605 − 2.54963I
u = 0.618900 − 0.885728I
a = −0.403494 + 0.627123I
b = 0.126589 + 1.385740I
−0.45606 − 1.53005I −0.20605 + 2.54963I
u = −0.681822 + 0.866267I
a = 0.455615 − 0.712832I
b = −0.10542 − 1.42563I
−0.85217 + 4.39903I −0.93348 − 2.80289I
u = −0.681822 − 0.866267I
a = 0.455615 + 0.712832I
b = −0.10542 + 1.42563I
−0.85217 − 4.39903I −0.93348 + 2.80289I
u = 0.704452 + 0.498527I
a = −1.061700 − 0.694204I
b = −0.115852 − 1.363560I
−3.75823 + 2.32534I 1.72826 − 3.09456I
u = 0.704452 − 0.498527I
a = −1.061700 + 0.694204I
b = −0.115852 + 1.363560I
−3.75823 − 2.32534I 1.72826 + 3.09456I
u = −0.063360 + 1.136750I
a = 0.279755 + 0.579784I
b = 0.230874 − 0.297193I
−1.87881 − 1.72326I 3.81965 + 5.18112I
u = −0.063360 − 1.136750I
a = 0.279755 − 0.579784I
b = 0.230874 + 0.297193I
−1.87881 + 1.72326I 3.81965 − 5.18112I
u = −0.068851 + 0.792352I
a = 0.19917 + 1.58842I
b = 0.090322 + 0.335817I
5.01775 − 3.11880I 5.58624 + 2.69239I
u = −0.068851 − 0.792352I
a = 0.19917 − 1.58842I
b = 0.090322 − 0.335817I
5.01775 + 3.11880I 5.58624 − 2.69239I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.115852 + 1.363560I
a = 0.766315 + 0.229589I
b = 0.704452 − 0.498527I
−3.75823 − 2.32534I 2.00000 + 3.09456I
u = −0.115852 − 1.363560I
a = 0.766315 − 0.229589I
b = 0.704452 + 0.498527I
−3.75823 + 2.32534I 2.00000 − 3.09456I
u = 0.126589 + 1.385740I
a = 0.553523 − 0.170066I
b = 0.618900 − 0.885728I
−0.45606 − 1.53005I 0. + 2.54963I
u = 0.126589 − 1.385740I
a = 0.553523 + 0.170066I
b = 0.618900 + 0.885728I
−0.45606 + 1.53005I 0. − 2.54963I
u = 0.03575 + 1.42698I
a = −0.762764 + 0.039458I
b = −0.745217 − 0.660881I
−6.91199 − 0.16816I −6.16829 + 0.I
u = 0.03575 − 1.42698I
a = −0.762764 − 0.039458I
b = −0.745217 + 0.660881I
−6.91199 + 0.16816I −6.16829 + 0.I
u = −0.10542 + 1.42563I
a = −0.630237 − 0.168639I
b = −0.681822 − 0.866267I
−0.85217 − 4.39903I 0. + 2.80289I
u = −0.10542 − 1.42563I
a = −0.630237 + 0.168639I
b = −0.681822 + 0.866267I
−0.85217 + 4.39903I 0. − 2.80289I
u = −0.24482 + 1.41618I
a = 1.000220 + 0.320005I
b = 0.881694 − 0.365628I
1.12421 − 3.71612I 0
u = −0.24482 − 1.41618I
a = 1.000220 − 0.320005I
b = 0.881694 + 0.365628I
1.12421 + 3.71612I 0
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.14784 + 1.43865I
a = −0.906043 + 0.178010I
b = −0.835357 − 0.509822I
−6.41945 + 5.52702I −4.42794 − 7.00248I
u = 0.14784 − 1.43865I
a = −0.906043 − 0.178010I
b = −0.835357 + 0.509822I
−6.41945 − 5.52702I −4.42794 + 7.00248I
u = 0.24265 + 1.43973I
a = −1.023190 + 0.288002I
b = −0.909367 − 0.387387I
0.60648 + 9.88550I 0. − 7.31129I
u = 0.24265 − 1.43973I
a = −1.023190 − 0.288002I
b = −0.909367 + 0.387387I
0.60648 − 9.88550I 0. + 7.31129I
u = 0.230874 + 0.297193I
a = −1.69352 + 0.96168I
b = −0.063360 − 1.136750I
−1.87881 + 1.72326I 3.81965 − 5.18112I
u = 0.230874 − 0.297193I
a = −1.69352 − 0.96168I
b = −0.063360 + 1.136750I
−1.87881 − 1.72326I 3.81965 + 5.18112I
u = 0.090322 + 0.335817I
a = −0.81574 + 3.56928I
b = −0.068851 + 0.792352I
5.01775 − 3.11880I 5.58624 + 2.69239I
u = 0.090322 − 0.335817I
a = −0.81574 − 3.56928I
b = −0.068851 − 0.792352I
5.01775 + 3.11880I 5.58624 − 2.69239I
14
III. I
u
3
= hb + u, a
4
+ a
3
+ 3a
2
+ 2a + 1, u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
−1
a
11
=
u
0
a
1
=
a
−u
a
3
=
0
−1
a
2
=
a
a − u
a
9
=
−a
2
u
−a + u
a
12
=
a + u
−u
a
6
=
au
1
a
7
=
a
3
u + au
a
2
− au + 1
a
8
=
a
3
u + a
2
u + 2au + u
−a
3
+ a
2
u − 2a + u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
3
− 4a
2
− 12a − 8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
, c
7
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
3
, c
4
, c
5
c
10
, c
11
, c
12
(u
2
+ 1)
4
c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
9
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
7
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
3
, c
4
, c
5
c
10
, c
11
, c
12
(y + 1)
8
c
9
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.395123 + 0.506844I
b = − 1.000000I
−3.50087 + 1.41510I −3.82674 − 4.90874I
u = 1.000000I
a = −0.395123 − 0.506844I
b = − 1.000000I
−3.50087 − 1.41510I −3.82674 + 4.90874I
u = 1.000000I
a = −0.10488 + 1.55249I
b = − 1.000000I
3.50087 + 3.16396I −0.17326 − 2.56480I
u = 1.000000I
a = −0.10488 − 1.55249I
b = − 1.000000I
3.50087 − 3.16396I −0.17326 + 2.56480I
u = − 1.000000I
a = −0.395123 + 0.506844I
b = 1.000000I
−3.50087 + 1.41510I −3.82674 − 4.90874I
u = − 1.000000I
a = −0.395123 − 0.506844I
b = 1.000000I
−3.50087 − 1.41510I −3.82674 + 4.90874I
u = − 1.000000I
a = −0.10488 + 1.55249I
b = 1.000000I
3.50087 + 3.16396I −0.17326 − 2.56480I
u = − 1.000000I
a = −0.10488 − 1.55249I
b = 1.000000I
3.50087 − 3.16396I −0.17326 + 2.56480I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
19
+ 5u
18
+ ··· + 2u + 1)
2
· (u
28
+ 7u
27
+ ··· + 9u + 4)
c
2
, c
7
(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)(u
19
− u
18
+ ··· + u
2
− 1)
2
· (u
28
+ 3u
27
+ ··· − u + 2)
c
3
, c
4
, c
5
c
10
, c
11
, c
12
((u
2
+ 1)
4
)(u
28
+ 17u
26
+ ··· + u + 1)(u
38
− u
37
+ ··· + 7u + 2)
c
8
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
19
+ 5u
18
+ ··· + 2u + 1)
2
· (u
28
+ 7u
27
+ ··· + 9u + 4)
c
9
(u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)(u
19
+ 7u
18
+ ··· + 2u − 1)
2
· (u
28
− 21u
27
+ ··· − 24293u + 2642)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
19
+ 19y
18
+ ··· + 10y −1)
2
· (y
28
+ 29y
27
+ ··· + 383y + 16)
c
2
, c
7
((y
4
− y
3
+ 3y
2
− 2y + 1)
2
)(y
19
− 5y
18
+ ··· + 2y −1)
2
· (y
28
− 7y
27
+ ··· − 9y + 4)
c
3
, c
4
, c
5
c
10
, c
11
, c
12
((y + 1)
8
)(y
28
+ 34y
27
+ ··· + 5y + 1)(y
38
+ 31y
37
+ ··· + 107y + 4)
c
9
((y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
)(y
19
+ 11y
18
+ ··· + 42y −1)
2
· (y
28
+ y
27
+ ··· + 72951y + 6980164)
20
