12n
0273
(K12n
0273
)
A knot diagram
1
Linearized knot diagam
3 4 11 2 10 4 12 3 6 9 7 8
Solving Sequence
7,11
12 8
1,4
3 9 2 5 6 10
c
11
c
7
c
12
c
3
c
8
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−5u
7
− 12u
6
+ 30u
5
+ 104u
4
+ 50u
3
− 36u
2
+ 4b + 4u + 12,
6u
7
+ 15u
6
− 36u
5
− 130u
4
− 62u
3
+ 54u
2
+ 4a − 4u − 20,
u
8
+ 4u
7
− 2u
6
− 30u
5
− 44u
4
− 12u
3
+ 8u
2
− 4u − 4i
I
u
2
= h−au + b + 2a − u + 2, 2a
2
− au + 2a + u + 3, u
2
− 2i
I
u
3
= hu
2
+ 2b + 2a − 4u + 2, 4u
2
a + 2a
2
− 12au − u
2
+ 6a + 7u − 6, u
3
− 4u
2
+ 4u − 2i
I
u
4
= h2b + 2a + u + 2, 2a
2
+ 2au + 2a + u + 3, u
2
− 2i
I
v
1
= ha, b
2
− b + 1, v + 1i
I
v
2
= ha, b + v −1, v
2
− v + 1i
* 6 irreducible components of dim
C
= 0, with total 26 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−5u
7
−12u
6
+· · ·+4b + 12, 6u
7
+15u
6
+· · ·+4a − 20, u
8
+4u
7
+· · ·−4u − 4i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
8
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−
3
2
u
7
−
15
4
u
6
+ ··· + u + 5
5
4
u
7
+ 3u
6
+ ··· − u − 3
a
3
=
−
1
4
u
7
−
3
4
u
6
+ ··· −
9
2
u
2
+ 2
5
4
u
7
+ 3u
6
+ ··· − u − 3
a
9
=
−
9
4
u
7
−
23
4
u
6
+ ··· −
25
2
u
2
+ 6
3
4
u
7
+ 2u
6
− 4u
5
− 17u
4
−
23
2
u
3
+ 4u
2
− 2
a
2
=
5
4
u
7
+
11
4
u
6
+ ··· − u − 2
3
4
u
7
+ 2u
6
+ ··· − u − 2
a
5
=
−u
7
− 2u
6
+ 6u
5
+ 19u
4
+ 8u
3
− 9u
2
+ 3
−u
4
+ 2u
2
a
6
=
3u
7
+
31
4
u
6
+ ··· − 2u − 9
−
3
4
u
7
− 2u
6
+ ··· + u + 2
a
10
=
−
1
2
u
7
−
5
4
u
6
+ ··· −
5
2
u
2
+ 2
1
4
u
7
+ u
6
− u
5
− 8u
4
−
13
2
u
3
+ 2u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
7
− 4u
6
+ 13u
5
+ 35u
4
+ 8u
3
− 6u
2
+ 18u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 4u
7
− 142u
6
− 1344u
5
+ 923u
4
− 512u
3
+ 178u
2
− 36u + 1
c
2
, c
4
, c
10
u
8
− 2u
6
+ 40u
5
− 73u
4
+ 56u
3
− 18u
2
+ 1
c
3
, c
5
, c
9
u
8
− 6u
5
− u
4
− 6u
3
− 2u
2
− 2u − 1
c
6
u
8
− 6u
7
+ 22u
6
− 102u
5
+ 297u
4
− 492u
3
+ 402u
2
− 108u − 19
c
7
, c
11
, c
12
u
8
+ 4u
7
− 2u
6
− 30u
5
− 44u
4
− 12u
3
+ 8u
2
− 4u − 4
c
8
u
8
− 16u
7
+ 72u
6
− 4u
5
− 371u
4
− 426u
3
+ 442u
2
+ 68u − 97
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 300y
7
+ ··· − 940y + 1
c
2
, c
4
, c
10
y
8
− 4y
7
− 142y
6
− 1344y
5
+ 923y
4
− 512y
3
+ 178y
2
− 36y + 1
c
3
, c
5
, c
9
y
8
− 2y
6
− 40y
5
− 73y
4
− 56y
3
− 18y
2
+ 1
c
6
y
8
+ 8y
7
+ ··· − 26940y + 361
c
7
, c
11
, c
12
y
8
− 20y
7
+ 156y
6
− 612y
5
+ 1208y
4
− 1072y
3
+ 320y
2
− 80y + 16
c
8
y
8
− 112y
7
+ ··· − 90372y + 9409
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.551137
a = 0.212410
b = 0.424769
−0.800674 −12.5950
u = −1.44764 + 0.06689I
a = −0.812880 − 0.861044I
b = −0.292648 + 0.756441I
−4.25535 + 2.66770I −4.04426 − 2.00132I
u = −1.44764 − 0.06689I
a = −0.812880 + 0.861044I
b = −0.292648 − 0.756441I
−4.25535 − 2.66770I −4.04426 + 2.00132I
u = 0.377266 + 0.364501I
a = 1.01991 − 1.65688I
b = 0.110704 + 0.754072I
1.63452 − 1.28115I 0.91619 + 5.71849I
u = 0.377266 − 0.364501I
a = 1.01991 + 1.65688I
b = 0.110704 − 0.754072I
1.63452 + 1.28115I 0.91619 − 5.71849I
u = −2.04666 + 0.56570I
a = 0.01556 + 1.77149I
b = 0.98351 − 1.43901I
13.6112 + 10.0113I −6.52481 − 3.89603I
u = −2.04666 − 0.56570I
a = 0.01556 − 1.77149I
b = 0.98351 + 1.43901I
13.6112 − 10.0113I −6.52481 + 3.89603I
u = 2.78520
a = 1.34242
b = −2.02789
−11.3105 −8.09900
5
II. I
u
2
= h−au + b + 2a − u + 2, 2a
2
− au + 2a + u + 3, u
2
− 2i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
2
a
8
=
−u
−u
a
1
=
−1
0
a
4
=
a
au − 2a + u − 2
a
3
=
au − a + u − 2
au − 2a + u − 2
a
9
=
−a −
1
2
u − 2
au − 2a + u − 1
a
2
=
2au − 3a +
3
2
u − 4
au − 2a + u − 1
a
5
=
1
0
a
6
=
au − a +
3
2
u + 1
−au + 2a − u + 1
a
10
=
a − u
au − 2a + u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8au + 16a − 8u + 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
(u
2
− u + 1)
2
c
2
, c
5
(u
2
+ u + 1)
2
c
6
u
4
+ 2u
3
+ 5u
2
+ 10u + 7
c
7
, c
11
, c
12
(u
2
− 2)
2
c
8
u
4
− 2u
3
+ 5u
2
− 10u + 7
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
(y
2
+ y + 1)
2
c
6
, c
8
y
4
+ 6y
3
− y
2
− 30y + 49
c
7
, c
11
, c
12
(y −2)
4
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.14645 + 1.47840I
b = −0.500000 − 0.866025I
−4.93480 − 4.05977I −8.00000 + 6.92820I
u = 1.41421
a = −0.14645 − 1.47840I
b = −0.500000 + 0.866025I
−4.93480 + 4.05977I −8.00000 − 6.92820I
u = −1.41421
a = −0.853553 + 0.253653I
b = −0.500000 − 0.866025I
−4.93480 − 4.05977I −8.00000 + 6.92820I
u = −1.41421
a = −0.853553 − 0.253653I
b = −0.500000 + 0.866025I
−4.93480 + 4.05977I −8.00000 − 6.92820I
9
III.
I
u
3
= hu
2
+2b+2a−4u+2, 4u
2
a+2a
2
−12au−u
2
+6a+7u−6, u
3
−4u
2
+4u−2i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
8
=
−u
−4u
2
+ 5u − 2
a
1
=
−u
2
+ 1
−10u
2
+ 14u − 8
a
4
=
a
−
1
2
u
2
− a + 2u − 1
a
3
=
−
1
2
u
2
+ 2u − 1
−
1
2
u
2
− a + 2u − 1
a
9
=
au + u
2
− a − 2u + 2
−u
2
a + 2au −
3
2
u
2
− a + 3u − 1
a
2
=
3
2
u
2
a − 3au + u
2
+ a +
1
2
u + 1
−2u
2
a + 4au −
5
2
u
2
− 3a + 4u − 2
a
5
=
−4u
2
a + 8au + 5u
2
− 4a − 10u + 7
−10u
2
+ 14u − 8
a
6
=
2u
2
a − 5au +
3
2
u
2
+ 4a − u − 1
−2u
2
a + 4au −
3
2
u
2
− 3a + 2u + 1
a
10
=
−
3
2
u
2
a + au − u
2
− a +
5
2
u
u
2
a −
7
2
u
2
− a + 7u − 4
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
2
− 4u − 4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 43u
5
+ 630u
4
+ 2111u
3
+ 5110u
2
+ 5291u + 2401
c
2
, c
4
, c
10
u
6
+ u
5
+ 22u
4
− 19u
3
+ 54u
2
+ u + 49
c
3
, c
5
, c
9
u
6
+ 3u
5
+ 4u
4
− u
3
− u + 7
c
6
u
6
+ 8u
5
+ 37u
4
+ 56u
3
+ 31u
2
+ 8u + 47
c
7
, c
11
, c
12
(u
3
− 4u
2
+ 4u − 2)
2
c
8
u
6
+ 16u
5
+ 75u
4
− 40u
3
− 49u
2
+ 12u + 149
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 589y
5
+ ··· − 3456461y + 5764801
c
2
, c
4
, c
10
y
6
+ 43y
5
+ 630y
4
+ 2111y
3
+ 5110y
2
+ 5291y + 2401
c
3
, c
5
, c
9
y
6
− y
5
+ 22y
4
+ 19y
3
+ 54y
2
− y + 49
c
6
y
6
+ 10y
5
+ 535y
4
− 876y
3
+ 3543y
2
+ 2850y + 2209
c
7
, c
11
, c
12
(y
3
− 8y
2
− 4)
2
c
8
y
6
− 106y
5
+ 6807y
4
− 9036y
3
+ 25711y
2
− 14746y + 22201
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.580357 + 0.606291I
a = 0.967369 + 0.278463I
b = −0.791268 + 0.582254I
−0.96847 + 3.17729I −6.35220 − 1.72143I
u = 0.580357 + 0.606291I
a = −0.42368 + 1.95182I
b = 0.599780 − 1.091110I
−0.96847 + 3.17729I −6.35220 − 1.72143I
u = 0.580357 − 0.606291I
a = 0.967369 − 0.278463I
b = −0.791268 − 0.582254I
−0.96847 − 3.17729I −6.35220 + 1.72143I
u = 0.580357 − 0.606291I
a = −0.42368 − 1.95182I
b = 0.599780 + 1.091110I
−0.96847 − 3.17729I −6.35220 + 1.72143I
u = 2.83929
a = −1.04369 + 1.34813I
b = 1.69149 − 1.34813I
11.8065 −7.29560
u = 2.83929
a = −1.04369 − 1.34813I
b = 1.69149 + 1.34813I
11.8065 −7.29560
13
IV. I
u
4
= h2b + 2a + u + 2, 2a
2
+ 2au + 2a + u + 3, u
2
− 2i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
2
a
8
=
−u
−u
a
1
=
−1
0
a
4
=
a
−a −
1
2
u − 1
a
3
=
−
1
2
u − 1
−a −
1
2
u − 1
a
9
=
−au − a − u
a +
1
2
u + 1
a
2
=
−
1
2
au − a − u −
5
2
−a −
1
2
u
a
5
=
1
0
a
6
=
au + 2a +
3
2
u + 1
−a −
1
2
u − 1
a
10
=
−
1
2
au + a + u +
5
2
a +
1
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
(u
2
− u + 1)
2
c
2
, c
5
(u
2
+ u + 1)
2
c
6
u
4
− 4u
3
+ 8u
2
− 8u + 7
c
7
, c
11
, c
12
(u
2
− 2)
2
c
8
u
4
+ 4u
3
+ 8u
2
+ 8u + 7
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
(y
2
+ y + 1)
2
c
6
, c
8
y
4
+ 14y
2
+ 48y + 49
c
7
, c
11
, c
12
(y −2)
4
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −1.20711 + 0.86603I
b = −0.500000 − 0.866025I
−4.93480 −8.00000
u = 1.41421
a = −1.20711 − 0.86603I
b = −0.500000 + 0.866025I
−4.93480 −8.00000
u = −1.41421
a = 0.207107 + 0.866025I
b = −0.500000 − 0.866025I
−4.93480 −8.00000
u = −1.41421
a = 0.207107 − 0.866025I
b = −0.500000 + 0.866025I
−4.93480 −8.00000
17
V. I
v
1
= ha, b
2
− b + 1, v + 1i
(i) Arc colorings
a
7
=
−1
0
a
11
=
1
0
a
12
=
1
0
a
8
=
−1
0
a
1
=
1
0
a
4
=
0
b
a
3
=
b
b
a
9
=
−b
−b + 1
a
2
=
b
b − 1
a
5
=
−1
0
a
6
=
−1
−b + 1
a
10
=
0
−b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8b − 4
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
2
− u + 1
c
2
, c
3
, c
6
c
8
, c
9
u
2
+ u + 1
c
7
, c
11
, c
12
u
2
19
(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
8
, c
9
, c
10
y
2
+ y + 1
c
7
, c
11
, c
12
y
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 0.500000 + 0.866025I
− 4.05977I 0. + 6.92820I
v = −1.00000
a = 0
b = 0.500000 − 0.866025I
4.05977I 0. − 6.92820I
21
VI. I
v
2
= ha, b + v − 1, v
2
− v + 1i
(i) Arc colorings
a
7
=
v
0
a
11
=
1
0
a
12
=
1
0
a
8
=
v
0
a
1
=
1
0
a
4
=
0
−v + 1
a
3
=
−v + 1
−v + 1
a
9
=
1
−v + 1
a
2
=
−v + 1
−v
a
5
=
−1
0
a
6
=
v
−v + 1
a
10
=
−v + 2
−v
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
2
− u + 1
c
2
, c
3
, c
9
u
2
+ u + 1
c
6
, c
8
(u − 1)
2
c
7
, c
11
, c
12
u
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
c
10
y
2
+ y + 1
c
6
, c
8
(y − 1)
2
c
7
, c
11
, c
12
y
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
2
√
−1(vol +
√
−1CS) Cusp shape
v = 0.500000 + 0.866025I
a = 0
b = 0.500000 − 0.866025I
0 −6.00000
v = 0.500000 − 0.866025I
a = 0
b = 0.500000 + 0.866025I
0 −6.00000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
6
· (u
6
+ 43u
5
+ 630u
4
+ 2111u
3
+ 5110u
2
+ 5291u + 2401)
· (u
8
− 4u
7
− 142u
6
− 1344u
5
+ 923u
4
− 512u
3
+ 178u
2
− 36u + 1)
c
2
(u
2
+ u + 1)
6
(u
6
+ u
5
+ 22u
4
− 19u
3
+ 54u
2
+ u + 49)
· (u
8
− 2u
6
+ 40u
5
− 73u
4
+ 56u
3
− 18u
2
+ 1)
c
3
, c
9
(u
2
− u + 1)
4
(u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 4u
4
− u
3
− u + 7)
· (u
8
− 6u
5
− u
4
− 6u
3
− 2u
2
− 2u − 1)
c
4
, c
10
(u
2
− u + 1)
6
(u
6
+ u
5
+ 22u
4
− 19u
3
+ 54u
2
+ u + 49)
· (u
8
− 2u
6
+ 40u
5
− 73u
4
+ 56u
3
− 18u
2
+ 1)
c
5
(u
2
− u + 1)
2
(u
2
+ u + 1)
4
(u
6
+ 3u
5
+ 4u
4
− u
3
− u + 7)
· (u
8
− 6u
5
− u
4
− 6u
3
− 2u
2
− 2u − 1)
c
6
((u − 1)
2
)(u
2
+ u + 1)(u
4
− 4u
3
+ ··· − 8u + 7)(u
4
+ 2u
3
+ ··· + 10u + 7)
· (u
6
+ 8u
5
+ 37u
4
+ 56u
3
+ 31u
2
+ 8u + 47)
· (u
8
− 6u
7
+ 22u
6
− 102u
5
+ 297u
4
− 492u
3
+ 402u
2
− 108u − 19)
c
7
, c
11
, c
12
u
4
(u
2
− 2)
4
(u
3
− 4u
2
+ 4u − 2)
2
· (u
8
+ 4u
7
− 2u
6
− 30u
5
− 44u
4
− 12u
3
+ 8u
2
− 4u − 4)
c
8
((u − 1)
2
)(u
2
+ u + 1)(u
4
− 2u
3
+ ··· − 10u + 7)(u
4
+ 4u
3
+ ··· + 8u + 7)
· (u
6
+ 16u
5
+ 75u
4
− 40u
3
− 49u
2
+ 12u + 149)
· (u
8
− 16u
7
+ 72u
6
− 4u
5
− 371u
4
− 426u
3
+ 442u
2
+ 68u − 97)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
6
− 589y
5
+ ··· − 3456461y + 5764801)
· (y
8
− 300y
7
+ ··· − 940y + 1)
c
2
, c
4
, c
10
(y
2
+ y + 1)
6
· (y
6
+ 43y
5
+ 630y
4
+ 2111y
3
+ 5110y
2
+ 5291y + 2401)
· (y
8
− 4y
7
− 142y
6
− 1344y
5
+ 923y
4
− 512y
3
+ 178y
2
− 36y + 1)
c
3
, c
5
, c
9
(y
2
+ y + 1)
6
(y
6
− y
5
+ 22y
4
+ 19y
3
+ 54y
2
− y + 49)
· (y
8
− 2y
6
− 40y
5
− 73y
4
− 56y
3
− 18y
2
+ 1)
c
6
((y − 1)
2
)(y
2
+ y + 1)(y
4
+ 14y
2
+ 48y + 49)(y
4
+ 6y
3
+ ··· − 30y + 49)
· (y
6
+ 10y
5
+ 535y
4
− 876y
3
+ 3543y
2
+ 2850y + 2209)
· (y
8
+ 8y
7
+ ··· − 26940y + 361)
c
7
, c
11
, c
12
y
4
(y − 2)
8
(y
3
− 8y
2
− 4)
2
· (y
8
− 20y
7
+ 156y
6
− 612y
5
+ 1208y
4
− 1072y
3
+ 320y
2
− 80y + 16)
c
8
((y − 1)
2
)(y
2
+ y + 1)(y
4
+ 14y
2
+ 48y + 49)(y
4
+ 6y
3
+ ··· − 30y + 49)
· (y
6
− 106y
5
+ 6807y
4
− 9036y
3
+ 25711y
2
− 14746y + 22201)
· (y
8
− 112y
7
+ ··· − 90372y + 9409)
27
