12n
0679
(K12n
0679
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 10 3 11 12 5 7 8 9
Solving Sequence
7,10
11 8
3,12
6 5 2 4 1 9
c
10
c
7
c
11
c
6
c
5
c
2
c
4
c
1
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−272506u
18
− 964931u
17
+ ··· + 166246b − 454705,
202199u
18
+ 697603u
17
+ ··· + 166246a + 690296, u
19
+ 4u
18
+ ··· + 6u + 1i
I
u
2
= hu
2
+ b + u − 2, a, u
3
+ u
2
− 2u − 1i
I
u
3
= hb − 1, a + 1, u
2
− u − 1i
I
u
4
= hb + u + 1, a + u − 2, u
2
− u − 1i
* 4 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−2.73 × 10
5
u
18
− 9.65 × 10
5
u
17
+ · · · + 1.66 × 10
5
b − 4.55 × 10
5
, 2.02 ×
10
5
u
18
+6.98×10
5
u
17
+· · · +1.66×10
5
a+6.90×10
5
, u
19
+4u
18
+· · · +6u +1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
8
=
−u
−u
3
+ u
a
3
=
−1.21626u
18
− 4.19621u
17
+ ··· − 17.4998u − 4.15226
1.63917u
18
+ 5.80424u
17
+ ··· + 13.5564u + 2.73513
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
−1.42025u
18
− 4.28958u
17
+ ··· − 12.9625u − 2.68015
0.636515u
18
+ 1.48579u
17
+ ··· + 2.46235u − 0.167595
a
5
=
−0.783736u
18
− 2.80379u
17
+ ··· − 10.5002u − 2.84774
0.636515u
18
+ 1.48579u
17
+ ··· + 2.46235u − 0.167595
a
2
=
−0.783736u
18
− 2.80379u
17
+ ··· − 10.5002u − 2.84774
0.912202u
18
+ 3.77581u
17
+ ··· + 10.4811u + 2.39994
a
4
=
−1.21626u
18
− 4.19621u
17
+ ··· − 17.4998u − 4.15226
0.949954u
18
+ 3.82918u
17
+ ··· + 10.7596u + 2.06629
a
1
=
−u
4
+ 3u
2
− 1
−u
6
+ 4u
4
− 3u
2
a
9
=
u
3
− 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
6730797
166246
u
18
+
11675065
83123
u
17
+ ··· +
25540453
83123
u +
10182639
166246
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
19
− 6u
18
+ ··· + 15u + 1
c
3
, c
6
u
19
+ 3u
18
+ ··· + 4u + 8
c
5
, c
9
u
19
− 2u
18
+ ··· + 32u + 16
c
7
, c
8
, c
10
c
11
, c
12
u
19
+ 4u
18
+ ··· + 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
19
− 8y
18
+ ··· + 227y −1
c
3
, c
6
y
19
+ 15y
18
+ ··· + 2448y −64
c
5
, c
9
y
19
+ 20y
18
+ ··· + 6272y −256
c
7
, c
8
, c
10
c
11
, c
12
y
19
− 22y
18
+ ··· − 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.371183 + 0.912603I
a = −1.80989 − 0.68295I
b = 1.53400 + 0.07023I
6.24887 + 1.03604I −13.20409 − 0.16139I
u = 0.371183 − 0.912603I
a = −1.80989 + 0.68295I
b = 1.53400 − 0.07023I
6.24887 − 1.03604I −13.20409 + 0.16139I
u = 0.754085 + 0.801793I
a = 1.59489 + 0.69061I
b = −1.59706 + 0.29496I
5.09846 − 6.69074I −15.1331 + 4.7970I
u = 0.754085 − 0.801793I
a = 1.59489 − 0.69061I
b = −1.59706 − 0.29496I
5.09846 + 6.69074I −15.1331 − 4.7970I
u = 0.788633
a = −1.87771
b = 0.117506
−10.1632 −31.8240
u = −1.296690 + 0.113032I
a = −0.844571 − 0.789505I
b = 1.171320 + 0.089530I
−4.61857 + 1.71767I −18.2481 − 1.2911I
u = −1.296690 − 0.113032I
a = −0.844571 + 0.789505I
b = 1.171320 − 0.089530I
−4.61857 − 1.71767I −18.2481 + 1.2911I
u = −0.541707
a = 0.445654
b = −3.62686
−2.44677 −98.2800
u = −1.44566 + 0.37064I
a = 0.833635 − 0.884796I
b = −1.51545 − 0.17161I
0.51136 + 3.59146I −15.7311 − 1.9367I
u = −1.44566 − 0.37064I
a = 0.833635 + 0.884796I
b = −1.51545 + 0.17161I
0.51136 − 3.59146I −15.7311 + 1.9367I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.49235 + 0.21013I
a = 0.018135 − 0.695055I
b = 0.526738 − 0.141756I
−6.67966 − 1.72802I −16.2689 + 1.8813I
u = 1.49235 − 0.21013I
a = 0.018135 + 0.695055I
b = 0.526738 + 0.141756I
−6.67966 + 1.72802I −16.2689 − 1.8813I
u = 1.60645
a = −0.333194
b = 3.67997
−10.0473 −54.2350
u = −0.369925
a = −0.699220
b = −0.403627
−0.654259 −14.8880
u = −1.65257 + 0.26840I
a = −0.793286 + 0.765963I
b = 1.67968 + 0.52712I
−2.92778 + 10.79900I −18.2574 − 5.0475I
u = −1.65257 − 0.26840I
a = −0.793286 − 0.765963I
b = 1.67968 − 0.52712I
−2.92778 − 10.79900I −18.2574 + 5.0475I
u = −0.082620 + 0.268767I
a = −0.49970 − 2.54134I
b = −0.546098 + 0.125693I
−0.761541 − 0.128012I −12.14415 − 0.42322I
u = −0.082620 − 0.268767I
a = −0.49970 + 2.54134I
b = −0.546098 − 0.125693I
−0.761541 + 0.128012I −12.14415 + 0.42322I
u = −1.76360
a = 0.466063
b = −0.273248
19.6998 −27.7990
6
II. I
u
2
= hu
2
+ b + u − 2, a, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
8
=
−u
u
2
− u − 1
a
3
=
0
−u
2
− u + 2
a
12
=
−u
2
+ 1
−u
2
+ u + 1
a
6
=
0
u
a
5
=
u
u
a
2
=
−u
−u
2
− 2u + 2
a
4
=
0
−u
2
− u + 2
a
1
=
−u
−u
a
9
=
−u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
− 7u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
u
3
− u
2
− 2u + 1
c
9
, c
10
, c
11
c
12
u
3
+ u
2
− 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = 0
b = −0.801938
−7.98968 −19.1690
u = −0.445042
a = 0
b = 2.24698
−2.34991 3.53080
u = −1.80194
a = 0
b = 0.554958
−19.2692 −11.3620
10
III. I
u
3
= hb − 1, a + 1, u
2
− u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u + 1
a
8
=
−u
−u − 1
a
3
=
−1
1
a
12
=
−u
−u
a
6
=
u
0
a
5
=
u
0
a
2
=
u
1
a
4
=
−1
−u
a
1
=
0
−u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −19
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
u
2
+ u − 1
c
4
, c
6
, c
10
c
11
, c
12
u
2
− u − 1
c
5
, c
9
u
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.00000
b = 1.00000
−1.97392 −19.0000
u = 1.61803
a = −1.00000
b = 1.00000
−17.7653 −19.0000
14
IV. I
u
4
= hb + u + 1, a + u − 2, u
2
− u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u + 1
a
8
=
−u
−u − 1
a
3
=
−u + 2
−u − 1
a
12
=
−u
−u
a
6
=
2u − 3
0
a
5
=
2u − 3
0
a
2
=
2u − 3
−u − 1
a
4
=
−u + 2
−u
a
1
=
0
−u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
u
2
+ u − 1
c
4
, c
6
, c
10
c
11
, c
12
u
2
− u − 1
c
5
, c
9
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 2.61803
b = −0.381966
−9.86960 −4.00000
u = 1.61803
a = 0.381966
b = −2.61803
−9.86960 −4.00000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
2
+ u − 1)
2
(u
19
− 6u
18
+ ··· + 15u + 1)
c
3
u
3
(u
2
+ u − 1)
2
(u
19
+ 3u
18
+ ··· + 4u + 8)
c
4
((u + 1)
3
)(u
2
− u − 1)
2
(u
19
− 6u
18
+ ··· + 15u + 1)
c
5
u
4
(u
3
− u
2
− 2u + 1)(u
19
− 2u
18
+ ··· + 32u + 16)
c
6
u
3
(u
2
− u − 1)
2
(u
19
+ 3u
18
+ ··· + 4u + 8)
c
7
, c
8
((u
2
+ u − 1)
2
)(u
3
− u
2
− 2u + 1)(u
19
+ 4u
18
+ ··· + 6u + 1)
c
9
u
4
(u
3
+ u
2
− 2u − 1)(u
19
− 2u
18
+ ··· + 32u + 16)
c
10
, c
11
, c
12
((u
2
− u − 1)
2
)(u
3
+ u
2
− 2u − 1)(u
19
+ 4u
18
+ ··· + 6u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
3
)(y
2
− 3y + 1)
2
(y
19
− 8y
18
+ ··· + 227y −1)
c
3
, c
6
y
3
(y
2
− 3y + 1)
2
(y
19
+ 15y
18
+ ··· + 2448y −64)
c
5
, c
9
y
4
(y
3
− 5y
2
+ 6y −1)(y
19
+ 20y
18
+ ··· + 6272y −256)
c
7
, c
8
, c
10
c
11
, c
12
((y
2
− 3y + 1)
2
)(y
3
− 5y
2
+ 6y −1)(y
19
− 22y
18
+ ··· − 6y −1)
20
