12a
1276
(K12a
1276
)
A knot diagram
1
Linearized knot diagam
5 8 9 10 12 1 11 3 4 2 7 6
Solving Sequence
1,7
6 12 5 2 11 8 3 10 4 9
c
6
c
12
c
5
c
1
c
11
c
7
c
2
c
10
c
4
c
9
c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
36
− 14u
34
+ ··· − 2u + 1i
I
u
2
= hu + 1i
* 2 irreducible components of dim
C
= 0, with total 37 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
36
− 14u
34
+ · · · − 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
6
=
1
−u
2
a
12
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
11
=
−u
3
+ 2u
−u
3
+ u
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
3
=
u
19
− 8u
17
+ 26u
15
− 40u
13
+ 19u
11
+ 24u
9
− 30u
7
+ 9u
3
u
19
− 7u
17
+ 20u
15
− 27u
13
+ 11u
11
+ 13u
9
− 14u
7
+ 3u
3
+ u
a
10
=
u
15
− 6u
13
+ 14u
11
− 14u
9
+ 2u
7
+ 6u
5
− 4u
3
+ 2u
−u
17
+ 7u
15
− 19u
13
+ 22u
11
− 3u
9
− 14u
7
+ 6u
5
+ 2u
3
+ u
a
4
=
u
28
− 11u
26
+ ··· + u
2
+ 1
−u
30
+ 12u
28
+ ··· + 8u
4
− u
2
a
9
=
−u
32
+ 13u
30
+ ··· + 2u
2
+ 1
−u
32
+ 12u
30
+ ··· − 8u
6
− 10u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
35
− 56u
33
+ 4u
32
+ 352u
31
− 52u
30
− 1276u
29
+ 300u
28
+ 2804u
27
− 980u
26
−
3340u
25
+ 1872u
24
+ 400u
23
− 1720u
22
+ 5108u
21
− 588u
20
− 6980u
19
+ 3336u
18
+
1544u
17
− 2900u
16
+ 4732u
15
− 552u
14
− 4032u
13
+ 2344u
12
− 448u
11
− 840u
10
+
1592u
9
− 544u
8
− 184u
7
+ 268u
6
− 216u
5
+ 48u
4
− 24u
3
+ 12u
2
+ 20u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u
36
+ 3u
35
+ ··· − 30u − 7
c
2
, c
3
, c
4
c
8
, c
9
u
36
− 24u
34
+ ··· + 3u
2
+ 1
c
5
, c
6
, c
12
u
36
− 14u
34
+ ··· + 2u + 1
c
10
u
36
− 6u
35
+ ··· + 272u − 304
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
y
36
+ 39y
35
+ ··· − 326y + 49
c
2
, c
3
, c
4
c
8
, c
9
y
36
− 48y
35
+ ··· + 6y + 1
c
5
, c
6
, c
12
y
36
− 28y
35
+ ··· + 6y + 1
c
10
y
36
− 24y
35
+ ··· + 92000y + 92416
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.050369 + 0.890388I
−18.9058 − 5.9322I 9.90521 + 2.84532I
u = 0.050369 − 0.890388I
−18.9058 + 5.9322I 9.90521 − 2.84532I
u = −0.038390 + 0.874277I
10.50350 + 4.44528I 9.29426 − 3.93187I
u = −0.038390 − 0.874277I
10.50350 − 4.44528I 9.29426 + 3.93187I
u = 0.015851 + 0.853937I
6.10418 − 1.70405I 5.05889 + 3.68915I
u = 0.015851 − 0.853937I
6.10418 + 1.70405I 5.05889 − 3.68915I
u = 0.777405 + 0.185259I
11.28280 + 0.04041I 5.42723 + 0.89925I
u = 0.777405 − 0.185259I
11.28280 − 0.04041I 5.42723 − 0.89925I
u = 1.252890 + 0.044961I
−2.90916 − 0.08413I −0.68947 − 1.50895I
u = 1.252890 − 0.044961I
−2.90916 + 0.08413I −0.68947 + 1.50895I
u = −1.281930 + 0.128297I
−4.37662 + 2.52856I −5.94778 − 5.32405I
u = −1.281930 − 0.128297I
−4.37662 − 2.52856I −5.94778 + 5.32405I
u = 1.233580 + 0.435957I
16.9206 + 1.1935I 6.78229 + 0.45952I
u = 1.233580 − 0.435957I
16.9206 − 1.1935I 6.78229 − 0.45952I
u = −1.241030 + 0.416731I
6.78682 + 0.17612I 6.07706 + 0.51655I
u = −1.241030 − 0.416731I
6.78682 − 0.17612I 6.07706 − 0.51655I
u = 1.300760 + 0.181559I
−1.26935 − 5.01630I 1.24352 + 7.13074I
u = 1.300760 − 0.181559I
−1.26935 + 5.01630I 1.24352 − 7.13074I
u = 1.260470 + 0.393875I
2.24673 − 2.77191I 1.60426 − 0.44575I
u = 1.260470 − 0.393875I
2.24673 + 2.77191I 1.60426 + 0.44575I
u = −1.34160
5.39957 −0.485750
u = −1.286130 + 0.392373I
2.05223 + 6.17629I 0.87692 − 6.64270I
u = −1.286130 − 0.392373I
2.05223 − 6.17629I 0.87692 + 6.64270I
u = −1.328350 + 0.209703I
7.96118 + 6.17101I 2.39842 − 5.27362I
u = −1.328350 − 0.209703I
7.96118 − 6.17101I 2.39842 + 5.27362I
u = 0.253274 + 0.597391I
12.89040 − 3.33854I 8.26297 + 4.57844I
u = 0.253274 − 0.597391I
12.89040 + 3.33854I 8.26297 − 4.57844I
u = 1.304210 + 0.403622I
6.31578 − 9.02541I 5.28025 + 6.79809I
u = 1.304210 − 0.403622I
6.31578 + 9.02541I 5.28025 − 6.79809I
u = −1.315470 + 0.411974I
16.3071 + 10.5958I 6.03260 − 5.51770I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.315470 − 0.411974I
16.3071 − 10.5958I 6.03260 + 5.51770I
u = −0.215467 + 0.524772I
3.39035 + 2.53006I 8.15428 − 6.42695I
u = −0.215467 − 0.524772I
3.39035 − 2.53006I 8.15428 + 6.42695I
u = −0.487225
1.85818 3.26760
u = 0.172360 + 0.345725I
0.027257 − 0.790103I 0.84818 + 8.67184I
u = 0.172360 − 0.345725I
0.027257 + 0.790103I 0.84818 − 8.67184I
6
II. I
u
2
= hu + 1i
(i) Arc colorings
a
1
=
0
−1
a
7
=
1
0
a
6
=
1
−1
a
12
=
−1
0
a
5
=
0
−1
a
2
=
0
−1
a
11
=
−1
0
a
8
=
1
0
a
3
=
−1
−1
a
10
=
−1
−1
a
4
=
1
0
a
9
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
12
u − 1
c
10
u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
y
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
, c
12
y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
1.64493 6.00000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
11
u(u
36
+ 3u
35
+ ··· − 30u − 7)
c
2
, c
3
, c
4
c
8
, c
9
(u − 1)(u
36
− 24u
34
+ ··· + 3u
2
+ 1)
c
5
, c
6
, c
12
(u − 1)(u
36
− 14u
34
+ ··· + 2u + 1)
c
10
(u + 1)(u
36
− 6u
35
+ ··· + 272u − 304)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
11
y(y
36
+ 39y
35
+ ··· − 326y + 49)
c
2
, c
3
, c
4
c
8
, c
9
(y −1)(y
36
− 48y
35
+ ··· + 6y + 1)
c
5
, c
6
, c
12
(y −1)(y
36
− 28y
35
+ ··· + 6y + 1)
c
10
(y −1)(y
36
− 24y
35
+ ··· + 92000y + 92416)
12
