12a
0838
(K12a
0838
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 1 3 6 7 8
Solving Sequence
7,12
8
1,4
2 9 3 11 6 10 5
c
7
c
12
c
1
c
8
c
3
c
11
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
22
− 17u
20
+ ··· + b + 1, u
22
+ u
21
+ ··· + a + 2, u
23
+ 2u
22
+ ··· − 12u
2
+ 1i
I
u
2
= h−u
2
+ b + 1, −u
2
+ a + 2, u
3
− u
2
− 2u + 1i
* 2 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
= hu
22
−17u
20
+· · ·+b+1, u
22
+u
21
+· · ·+a+2, u
23
+2u
22
+· · ·−12u
2
+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
3
+ u
a
4
=
−u
22
− u
21
+ ··· + 11u − 2
−u
22
+ 17u
20
+ ··· + 2u − 1
a
2
=
u
21
− 16u
19
+ ··· − 11u + 2
−u
22
+ 16u
20
+ ··· + 9u
2
− u
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
u
22
− u
21
+ ··· + 10u − 1
3u
22
− 48u
20
+ ··· + u + 1
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
10
=
−u
3
+ 2u
−u
3
+ u
a
5
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
22
+ 8u
21
− 81u
20
− 126u
19
+ 556u
18
+ 835u
17
− 2107u
16
−
3020u
15
+ 4822u
14
+ 6443u
13
− 6906u
12
− 8110u
11
+ 6388u
10
+ 5547u
9
− 4175u
8
−
1464u
7
+ 2179u
6
− 280u
5
− 757u
4
+ 196u
3
+ 80u
2
− 23u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
23
− 4u
22
+ ··· + 5u − 1
c
3
, c
9
u
23
− u
22
+ ··· − 28u − 8
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
u
23
− 2u
22
+ ··· + 12u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
23
− 26y
22
+ ··· + 57y −1
c
3
, c
9
y
23
+ 21y
22
+ ··· + 592y −64
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
23
− 36y
22
+ ··· + 24y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.899656 + 0.367597I
a = −0.129750 − 0.316248I
b = 1.37701 + 0.64432I
−10.62640 + 4.79693I −18.4963 − 4.5941I
u = −0.899656 − 0.367597I
a = −0.129750 + 0.316248I
b = 1.37701 − 0.64432I
−10.62640 − 4.79693I −18.4963 + 4.5941I
u = 0.869158
a = −0.611937
b = 1.66429
−5.48753 −17.2890
u = −0.821958 + 0.135064I
a = −0.253247 + 1.039930I
b = −0.445233 + 0.309895I
−3.58507 + 2.11349I −17.2477 − 5.0037I
u = −0.821958 − 0.135064I
a = −0.253247 − 1.039930I
b = −0.445233 − 0.309895I
−3.58507 − 2.11349I −17.2477 + 5.0037I
u = −1.29295
a = −0.815349
b = −0.215176
−6.99093 −10.8460
u = 0.292199 + 0.547469I
a = −0.70419 + 1.55774I
b = −0.871306 + 0.010708I
−6.92984 − 1.76193I −14.7690 + 3.3456I
u = 0.292199 − 0.547469I
a = −0.70419 − 1.55774I
b = −0.871306 − 0.010708I
−6.92984 + 1.76193I −14.7690 − 3.3456I
u = 1.43893 + 0.05540I
a = −0.573918 + 0.627821I
b = −0.269240 + 1.142030I
−11.26780 − 2.80601I −17.5990 + 3.0357I
u = 1.43893 − 0.05540I
a = −0.573918 − 0.627821I
b = −0.269240 − 1.142030I
−11.26780 + 2.80601I −17.5990 − 3.0357I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.45875
a = 2.49389
b = 1.79311
−13.4340 −18.2600
u = 0.523970
a = 0.174078
b = −0.424666
−0.880680 −10.8440
u = 1.46882 + 0.17141I
a = 1.75548 − 0.96316I
b = 1.371010 − 0.251551I
−18.5944 − 6.8465I −19.2959 + 3.6692I
u = 1.46882 − 0.17141I
a = 1.75548 + 0.96316I
b = 1.371010 + 0.251551I
−18.5944 + 6.8465I −19.2959 − 3.6692I
u = 0.198961 + 0.259775I
a = 1.162630 + 0.050871I
b = 0.098000 − 0.372398I
−0.444837 − 0.821194I −9.62649 + 8.14856I
u = 0.198961 − 0.259775I
a = 1.162630 − 0.050871I
b = 0.098000 + 0.372398I
−0.444837 + 0.821194I −9.62649 − 8.14856I
u = −0.236506
a = −3.87954
b = −0.871608
−1.99765 0.552820
u = 1.81292
a = 1.34454
b = 2.75785
−18.5335 −9.99140
u = −1.85429 + 0.01361I
a = 1.048680 + 0.206116I
b = 2.22063 + 0.96437I
15.7173 + 3.1639I −17.6160 − 2.4156I
u = −1.85429 − 0.01361I
a = 1.048680 − 0.206116I
b = 2.22063 − 0.96437I
15.7173 − 3.1639I −17.6160 + 2.4156I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.85896
a = −3.64519
b = −8.09850
13.4167 −18.4390
u = −1.86140 + 0.04356I
a = −2.83593 − 1.10657I
b = −6.28352 − 2.42136I
8.27161 + 7.98934I −19.2912 − 3.1659I
u = −1.86140 − 0.04356I
a = −2.83593 + 1.10657I
b = −6.28352 + 2.42136I
8.27161 − 7.98934I −19.2912 + 3.1659I
7
II. I
u
2
= h−u
2
+ b + 1, −u
2
+ a + 2, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
2
− u + 1
a
4
=
u
2
− 2
u
2
− 1
a
2
=
u
2
− u − 2
−u
a
9
=
−u
2
+ 1
−u
2
− u + 1
a
3
=
u
2
− 2
u
2
− 1
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
10
=
−u
2
+ 1
−u
2
− u + 1
a
5
=
u
u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− u − 23
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
9
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
c
8
u
3
− u
2
− 2u + 1
c
10
, c
11
, c
12
u
3
+ u
2
− 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
9
y
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = −0.445042
b = 0.554958
−7.98968 −20.1980
u = 0.445042
a = −1.80194
b = −0.801938
−2.34991 −23.2470
u = 1.80194
a = 1.24698
b = 2.24698
−19.2692 −21.5550
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
23
− 4u
22
+ ··· + 5u − 1)
c
3
, c
9
u
3
(u
23
− u
22
+ ··· − 28u − 8)
c
4
((u + 1)
3
)(u
23
− 4u
22
+ ··· + 5u − 1)
c
5
, c
6
, c
7
c
8
(u
3
− u
2
− 2u + 1)(u
23
− 2u
22
+ ··· + 12u
2
− 1)
c
10
, c
11
, c
12
(u
3
+ u
2
− 2u − 1)(u
23
− 2u
22
+ ··· + 12u
2
− 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
3
)(y
23
− 26y
22
+ ··· + 57y −1)
c
3
, c
9
y
3
(y
23
+ 21y
22
+ ··· + 592y −64)
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
(y
3
− 5y
2
+ 6y −1)(y
23
− 36y
22
+ ··· + 24y −1)
13
