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+2a4u+2, 4u
2
a+2a
2
12auu
2
+6a+7u6, u
3
4u
2
+4u2i
(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