12n
0251
(K12n
0251
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 11 10 3 5 7 6 8
Solving Sequence
6,11
7 12
3,5
2 1 4 10 8 9
c
6
c
11
c
5
c
2
c
1
c
4
c
10
c
7
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ 2u
10
+ 8u
9
+ 13u
8
+ 22u
7
+ 27u
6
+ 24u
5
+ 19u
4
+ 9u
3
+ 4u
2
+ b − 1,
u
13
+ 2u
12
+ 10u
11
+ 16u
10
+ 37u
9
+ 46u
8
+ 62u
7
+ 57u
6
+ 46u
5
+ 30u
4
+ 12u
3
+ 7u
2
+ a − u,
u
14
+ 2u
13
+ 11u
12
+ 18u
11
+ 46u
10
+ 60u
9
+ 91u
8
+ 90u
7
+ 86u
6
+ 60u
5
+ 34u
4
+ 17u
3
+ 2u
2
− 1i
I
u
2
= hb − u + 1, u
4
− u
3
+ 4u
2
+ a − 2u + 2, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
* 2 irreducible components of dim
C
= 0, with total 19 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
11
+2u
10
+· · ·+b−1, u
13
+2u
12
+· · ·+a−u, u
14
+2u
13
+· · ·+2u
2
−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
u
a
3
=
−u
13
− 2u
12
+ ··· − 7u
2
+ u
−u
11
− 2u
10
+ ··· − 4u
2
+ 1
a
5
=
u
2
+ 1
−u
2
a
2
=
−u
13
− 3u
12
+ ··· − 9u
2
+ 1
u
12
+ 5u
10
− 2u
9
+ 3u
8
− 11u
7
− 14u
6
− 18u
5
− 16u
4
− 8u
3
− 5u
2
+ 1
a
1
=
−u
7
− 4u
5
− 4u
3
− 2u
−u
9
− 5u
7
− 7u
5
− 2u
3
+ u
a
4
=
u
13
+ 2u
12
+ ··· + u − 1
−u
11
− 4u
9
+ 2u
8
+ 9u
6
+ 12u
5
+ 11u
4
+ 6u
3
+ 4u
2
+ u − 1
a
10
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
7
+ 4u
5
+ 4u
3
+ 2u
−u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= u
13
+2u
12
+11u
11
+18u
10
+45u
9
+56u
8
+80u
7
+65u
6
+50u
5
+12u
4
−7u
3
−11u
2
−9u−14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 26u
13
+ ··· + 14u + 1
c
2
, c
4
u
14
− 6u
13
+ ··· − 2u − 1
c
3
, c
8
u
14
+ u
13
+ ··· + 64u + 32
c
5
, c
6
, c
7
c
10
, c
11
u
14
− 2u
13
+ ··· + 2u
2
− 1
c
9
, c
12
u
14
− 2u
13
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 86y
13
+ ··· − 730y + 1
c
2
, c
4
y
14
− 26y
13
+ ··· − 14y + 1
c
3
, c
8
y
14
− 33y
13
+ ··· − 1536y + 1024
c
5
, c
6
, c
7
c
10
, c
11
y
14
+ 18y
13
+ ··· − 4y + 1
c
9
, c
12
y
14
− 30y
13
+ ··· − 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.550724 + 0.891947I
a = −0.096240 + 0.175738I
b = −1.07587 − 1.79627I
−15.0245 + 4.4031I −12.54526 − 3.39165I
u = −0.550724 − 0.891947I
a = −0.096240 − 0.175738I
b = −1.07587 + 1.79627I
−15.0245 − 4.4031I −12.54526 + 3.39165I
u = 0.190452 + 0.810025I
a = 0.013565 − 0.546935I
b = −0.396657 + 0.339392I
1.71814 − 1.64819I −5.73834 + 4.69390I
u = 0.190452 − 0.810025I
a = 0.013565 + 0.546935I
b = −0.396657 − 0.339392I
1.71814 + 1.64819I −5.73834 − 4.69390I
u = −0.772289
a = −2.27398
b = −0.485231
−17.7180 −15.7100
u = −0.241199 + 0.492313I
a = 0.340540 + 1.345040I
b = 1.022190 + 0.391429I
−1.48613 + 0.97077I −11.76317 − 1.95166I
u = −0.241199 − 0.492313I
a = 0.340540 − 1.345040I
b = 1.022190 − 0.391429I
−1.48613 − 0.97077I −11.76317 + 1.95166I
u = −0.04571 + 1.57188I
a = −1.87359 − 0.58564I
b = 2.33538 + 1.40783I
5.67567 + 1.86276I −10.57290 − 1.15181I
u = −0.04571 − 1.57188I
a = −1.87359 + 0.58564I
b = 2.33538 − 1.40783I
5.67567 − 1.86276I −10.57290 + 1.15181I
u = 0.05378 + 1.66919I
a = 0.834731 − 0.136645I
b = −1.099410 − 0.067263I
10.47610 − 2.59125I −5.03885 + 1.58782I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.05378 − 1.66919I
a = 0.834731 + 0.136645I
b = −1.099410 + 0.067263I
10.47610 + 2.59125I −5.03885 − 1.58782I
u = −0.16470 + 1.67887I
a = 1.88194 + 1.97217I
b = −2.16875 − 3.14542I
−6.19448 + 7.22352I −10.76531 − 2.66085I
u = −0.16470 − 1.67887I
a = 1.88194 − 1.97217I
b = −2.16875 + 3.14542I
−6.19448 − 7.22352I −10.76531 + 2.66085I
u = 0.288492
a = −0.927924
b = 0.251456
−0.575448 −17.4430
6
II. I
u
2
= hb − u + 1, u
4
− u
3
+ 4u
2
+ a − 2u + 2, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
u
a
3
=
−u
4
+ u
3
− 4u
2
+ 2u − 2
u − 1
a
5
=
u
2
+ 1
−u
2
a
2
=
−u
4
+ u
3
− 5u
2
+ 2u − 3
u
2
+ u − 1
a
1
=
−u
2
− 1
u
2
a
4
=
−u
4
+ u
3
− 4u
2
+ 2u − 2
u − 1
a
10
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
+ 3u
3
− 12u
2
+ 10u − 19
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
u
5
c
4
(u + 1)
5
c
5
, c
6
, c
7
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
9
, c
12
u
5
+ u
4
− u
2
+ u + 1
c
10
, c
11
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
6
, c
7
c
10
, c
11
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
9
, c
12
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.487744 + 0.170166I
b = −0.766323 + 0.885557I
0.17487 − 2.21397I −10.60206 + 4.05273I
u = 0.233677 − 0.885557I
a = 0.487744 − 0.170166I
b = −0.766323 − 0.885557I
0.17487 + 2.21397I −10.60206 − 4.05273I
u = 0.416284
a = −1.81849
b = −0.583716
−2.52712 −16.7900
u = 0.05818 + 1.69128I
a = 0.92150 − 1.10071I
b = −0.94182 + 1.69128I
9.31336 − 3.33174I −10.00277 + 3.46299I
u = 0.05818 − 1.69128I
a = 0.92150 + 1.10071I
b = −0.94182 − 1.69128I
9.31336 + 3.33174I −10.00277 − 3.46299I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
14
+ 26u
13
+ ··· + 14u + 1)
c
2
((u − 1)
5
)(u
14
− 6u
13
+ ··· − 2u − 1)
c
3
, c
8
u
5
(u
14
+ u
13
+ ··· + 64u + 32)
c
4
((u + 1)
5
)(u
14
− 6u
13
+ ··· − 2u − 1)
c
5
, c
6
, c
7
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
14
− 2u
13
+ ··· + 2u
2
− 1)
c
9
, c
12
(u
5
+ u
4
− u
2
+ u + 1)(u
14
− 2u
13
+ ··· − 2u − 1)
c
10
, c
11
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
14
− 2u
13
+ ··· + 2u
2
− 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
14
− 86y
13
+ ··· − 730y + 1)
c
2
, c
4
((y −1)
5
)(y
14
− 26y
13
+ ··· − 14y + 1)
c
3
, c
8
y
5
(y
14
− 33y
13
+ ··· − 1536y + 1024)
c
5
, c
6
, c
7
c
10
, c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
14
+ 18y
13
+ ··· − 4y + 1)
c
9
, c
12
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
14
− 30y
13
+ ··· − 4y + 1)
12
