12a
0878
(K12a
0878
)
A knot diagram
1
Linearized knot diagam
4 6 7 10 2 3 11 12 1 5 8 9
Solving Sequence
2,5
6 3
7,10
11 8 4 1 9 12
c
5
c
2
c
6
c
10
c
7
c
4
c
1
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h5u
35
+ 7u
34
+ ··· + b − 4, −2u
35
− u
34
+ ··· + 2a + 13, u
36
+ 3u
35
+ ··· + 6u − 1i
I
u
2
= hb, a − u + 2, u
2
− u − 1i
I
u
3
= hb − 1, −u
3
+ a + 2u + 1, u
4
− u
3
− 2u
2
+ 2u − 1i
I
u
4
= hb − 1, a, u + 1i
I
u
5
= hb, a − 1, u
2
− u − 1i
* 5 irreducible components of dim
C
= 0, with total 45 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
= h5u
35
+7u
34
+· · ·+b−4, −2u
35
−u
34
+· · ·+2a+13, u
36
+3u
35
+· · ·+6u−1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
u
35
+
1
2
u
34
+ ··· +
27
2
u −
13
2
−5u
35
− 7u
34
+ ··· − 23u + 4
a
11
=
−4u
35
−
13
2
u
34
+ ··· −
19
2
u −
5
2
−5u
35
− 7u
34
+ ··· − 23u + 4
a
8
=
−
1
2
u
34
− u
33
+ ··· +
11
2
u +
3
2
−u
9
+ 5u
7
− 7u
5
+ 2u
3
− u
a
4
=
u
3
− 2u
u
5
− 3u
3
+ u
a
1
=
u
7
− 4u
5
+ 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
9
=
1
2
u
35
− u
34
+ ··· +
21
2
u − 6
−
1
2
u
35
−
1
2
u
34
+ ··· − u +
1
2
a
12
=
−0.500000u
35
+ 10.5000u
33
+ ··· − 16.5000u
2
− 9.50000u
−
1
2
u
35
−
1
2
u
34
+ ··· − u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
15
2
u
35
+ 13u
34
+ ··· +
137
2
u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
36
− 7u
35
+ ··· − 204u − 7
c
2
, c
3
, c
5
c
6
u
36
+ 3u
35
+ ··· + 6u − 1
c
4
, c
10
u
36
+ 4u
35
+ ··· + 80u + 16
c
7
, c
8
, c
9
c
11
, c
12
u
36
− 3u
35
+ ··· − 12u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
36
+ 19y
35
+ ··· − 24116y + 49
c
2
, c
3
, c
5
c
6
y
36
− 41y
35
+ ··· − 56y + 1
c
4
, c
10
y
36
− 20y
35
+ ··· − 3200y + 256
c
7
, c
8
, c
9
c
11
, c
12
y
36
− 49y
35
+ ··· + 24y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.680450 + 0.624888I
a = 1.58464 + 1.12498I
b = −1.37928 + 0.58427I
15.3018 − 8.1184I 7.62644 + 5.89402I
u = 0.680450 − 0.624888I
a = 1.58464 − 1.12498I
b = −1.37928 − 0.58427I
15.3018 + 8.1184I 7.62644 − 5.89402I
u = 0.634805 + 0.572486I
a = −1.68198 − 1.14140I
b = 1.250780 − 0.467984I
5.35265 − 6.44473I 7.06333 + 7.49999I
u = 0.634805 − 0.572486I
a = −1.68198 + 1.14140I
b = 1.250780 + 0.467984I
5.35265 + 6.44473I 7.06333 − 7.49999I
u = 0.286743 + 0.718481I
a = −1.61383 − 0.66499I
b = 1.41214 + 0.42853I
16.4706 + 3.6258I 9.89093 − 0.70806I
u = 0.286743 − 0.718481I
a = −1.61383 + 0.66499I
b = 1.41214 − 0.42853I
16.4706 − 3.6258I 9.89093 + 0.70806I
u = −1.216880 + 0.225154I
a = 0.176584 − 0.371182I
b = −1.374430 + 0.160487I
11.69880 − 0.20082I 6.33515 + 0.I
u = −1.216880 − 0.225154I
a = 0.176584 + 0.371182I
b = −1.374430 − 0.160487I
11.69880 + 0.20082I 6.33515 + 0.I
u = 0.556839 + 0.497955I
a = 1.87783 + 1.11937I
b = −1.079090 + 0.293810I
1.19518 − 3.43864I 3.32627 + 7.55199I
u = 0.556839 − 0.497955I
a = 1.87783 − 1.11937I
b = −1.079090 − 0.293810I
1.19518 + 3.43864I 3.32627 − 7.55199I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.488097 + 0.539433I
a = −0.755389 − 0.156628I
b = −0.139231 + 1.154510I
11.32400 + 1.86563I 6.84113 − 3.39356I
u = −0.488097 − 0.539433I
a = −0.755389 + 0.156628I
b = −0.139231 − 1.154510I
11.32400 − 1.86563I 6.84113 + 3.39356I
u = 0.302987 + 0.626688I
a = 1.73596 + 0.65197I
b = −1.228520 − 0.302969I
6.32076 + 2.37657I 9.77017 − 1.34852I
u = 0.302987 − 0.626688I
a = 1.73596 − 0.65197I
b = −1.228520 + 0.302969I
6.32076 − 2.37657I 9.77017 + 1.34852I
u = −0.482616 + 0.407041I
a = 0.667048 − 0.013183I
b = 0.156607 − 0.892042I
1.87081 + 1.46712I 6.30394 − 4.92073I
u = −0.482616 − 0.407041I
a = 0.667048 + 0.013183I
b = 0.156607 + 0.892042I
1.87081 − 1.46712I 6.30394 + 4.92073I
u = −0.603920 + 0.151230I
a = −0.257154 + 0.105987I
b = −0.288863 + 0.427112I
−1.107760 + 0.363241I −6.95847 − 1.67967I
u = −0.603920 − 0.151230I
a = −0.257154 − 0.105987I
b = −0.288863 − 0.427112I
−1.107760 − 0.363241I −6.95847 + 1.67967I
u = −1.38180
a = −0.715291
b = 1.22298
1.39348 6.13670
u = −1.52012 + 0.10884I
a = 1.040470 − 0.777211I
b = −1.121760 − 0.323243I
−4.81513 + 1.90456I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.52012 − 0.10884I
a = 1.040470 + 0.777211I
b = −1.121760 + 0.323243I
−4.81513 − 1.90456I 0
u = 1.52376 + 0.14580I
a = 0.388481 + 0.623220I
b = 0.395969 + 1.220420I
4.64948 − 4.27120I 0
u = 1.52376 − 0.14580I
a = 0.388481 − 0.623220I
b = 0.395969 − 1.220420I
4.64948 + 4.27120I 0
u = 1.54036 + 0.09785I
a = −0.291616 − 0.548274I
b = −0.355163 − 1.046370I
−4.96649 − 3.17823I 0
u = 1.54036 − 0.09785I
a = −0.291616 + 0.548274I
b = −0.355163 + 1.046370I
−4.96649 + 3.17823I 0
u = −1.55388 + 0.14251I
a = −0.903024 + 1.018170I
b = 1.164680 + 0.488851I
−5.88929 + 5.74406I 0
u = −1.55388 − 0.14251I
a = −0.903024 − 1.018170I
b = 1.164680 − 0.488851I
−5.88929 − 5.74406I 0
u = 0.439340
a = 3.99822
b = −0.448670
8.22478 19.8310
u = 1.57896 + 0.04089I
a = 0.168039 + 0.376790I
b = 0.269429 + 0.728694I
−8.61991 − 1.07838I 0
u = 1.57896 − 0.04089I
a = 0.168039 − 0.376790I
b = 0.269429 − 0.728694I
−8.61991 + 1.07838I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.57622 + 0.17454I
a = 0.742839 − 1.118400I
b = −1.254650 − 0.614758I
−2.04882 + 9.20350I 0
u = −1.57622 − 0.17454I
a = 0.742839 + 1.118400I
b = −1.254650 + 0.614758I
−2.04882 − 9.20350I 0
u = −1.59552 + 0.19703I
a = −0.638410 + 1.172730I
b = 1.32886 + 0.71326I
7.68400 + 11.19530I 0
u = −1.59552 − 0.19703I
a = −0.638410 − 1.172730I
b = 1.32886 − 0.71326I
7.68400 − 11.19530I 0
u = 1.65310
a = −0.316454
b = −0.667605
−7.37086 0
u = 0.154049
a = −3.44747
b = 0.378377
0.766693 13.5400
8
II. I
u
2
= hb, a − u + 2, u
2
− u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u + 1
a
3
=
−u
−u − 1
a
7
=
−u
−u
a
10
=
u − 2
0
a
11
=
u − 2
0
a
8
=
−3u + 3
−u
a
4
=
1
0
a
1
=
u
u
a
9
=
u − 3
−1
a
12
=
−2u + 4
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
11
, c
12
u
2
+ u − 1
c
4
, c
10
u
2
c
5
, c
6
, c
7
c
8
, c
9
u
2
− u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
y
2
− 3y + 1
c
4
, c
10
y
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −2.61803
b = 0
7.89568 −5.00000
u = 1.61803
a = −0.381966
b = 0
−7.89568 −5.00000
12
III. I
u
3
= hb − 1, −u
3
+ a + 2u + 1, u
4
− u
3
− 2u
2
+ 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
3
+ 2u − 1
a
10
=
u
3
− 2u − 1
1
a
11
=
u
3
− 2u
1
a
8
=
−u
3
− u
2
+ 2u
−u
3
+ u − 1
a
4
=
u
3
− 2u
1
a
1
=
u
2
+ u − 1
u
3
− u + 1
a
9
=
−u
2
− u − 1
−u
3
− u
2
+ u
a
12
=
−u
3
− 1
−u
3
− u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 3u
3
+ 2u
2
− 2u + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
11
, c
12
u
4
− u
3
− 2u
2
+ 2u − 1
c
4
, c
10
(u − 1)
4
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
− 5y
3
− 6y
2
+ 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
11
, c
12
y
4
− 5y
3
+ 6y
2
+ 1
c
4
, c
10
(y −1)
4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.407392 + 0.476565I
a = −2.02474 − 0.82408I
b = 1.00000
1.64493 6.00000
u = 0.407392 − 0.476565I
a = −2.02474 + 0.82408I
b = 1.00000
1.64493 6.00000
u = −1.50507
a = −1.39919
b = 1.00000
1.64493 6.00000
u = 1.69028
a = 0.448678
b = 1.00000
1.64493 6.00000
16
IV. I
u
4
= hb − 1, a, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
7
=
0
1
a
10
=
0
1
a
11
=
1
1
a
8
=
1
2
a
4
=
1
1
a
1
=
−1
−2
a
9
=
−1
−1
a
12
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
u + 1
c
4
, c
10
u − 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
1.64493 6.00000
20
V. I
u
5
= hb, a − 1, u
2
− u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u + 1
a
3
=
−u
−u − 1
a
7
=
−u
−u
a
10
=
1
0
a
11
=
1
0
a
8
=
−2u
−u
a
4
=
1
0
a
1
=
u
u
a
9
=
u + 2
u + 1
a
12
=
−2u − 1
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
11
, c
12
u
2
+ u − 1
c
4
, c
10
u
2
c
5
, c
6
, c
7
c
8
, c
9
u
2
− u − 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
11
c
12
y
2
− 3y + 1
c
4
, c
10
y
2
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.00000
b = 0
0 0
u = 1.61803
a = 1.00000
b = 0
0 0
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
2
+ u − 1)
2
(u
4
− 3u
3
+ 2u
2
− 2u + 1)
· (u
36
− 7u
35
+ ··· − 204u − 7)
c
2
, c
3
(u + 1)(u
2
+ u − 1)
2
(u
4
− u
3
+ ··· + 2u − 1)(u
36
+ 3u
35
+ ··· + 6u − 1)
c
4
, c
10
u
4
(u − 1)
5
(u
36
+ 4u
35
+ ··· + 80u + 16)
c
5
, c
6
(u + 1)(u
2
− u − 1)
2
(u
4
− u
3
+ ··· + 2u − 1)(u
36
+ 3u
35
+ ··· + 6u − 1)
c
7
, c
8
, c
9
(u + 1)(u
2
− u − 1)
2
(u
4
− u
3
+ ··· + 2u − 1)(u
36
− 3u
35
+ ··· − 12u
2
− 1)
c
11
, c
12
(u + 1)(u
2
+ u − 1)
2
(u
4
− u
3
+ ··· + 2u − 1)(u
36
− 3u
35
+ ··· − 12u
2
− 1)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
2
− 3y + 1)
2
(y
4
− 5y
3
− 6y
2
+ 1)
· (y
36
+ 19y
35
+ ··· − 24116y + 49)
c
2
, c
3
, c
5
c
6
(y −1)(y
2
− 3y + 1)
2
(y
4
− 5y
3
+ 6y
2
+ 1)(y
36
− 41y
35
+ ··· − 56y + 1)
c
4
, c
10
y
4
(y −1)
5
(y
36
− 20y
35
+ ··· − 3200y + 256)
c
7
, c
8
, c
9
c
11
, c
12
(y −1)(y
2
− 3y + 1)
2
(y
4
− 5y
3
+ 6y
2
+ 1)(y
36
− 49y
35
+ ··· + 24y + 1)
26
