12n
0456
(K12n
0456
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 11 5 4 12 6 9 10
Solving Sequence
5,8
4
9,11
12 3 7 6 2 1 10
c
4
c
8
c
11
c
3
c
7
c
6
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h6263646729160u
18
+ 14140809413971u
17
+ ··· + 89100980340036b − 30737815843748,
18590835030469u
18
+ 42781634933956u
17
+ ··· + 89100980340036a + 3563580716326,
u
19
+ 2u
18
+ ··· − 4u + 4i
I
u
2
= h−2u
4
+ 3u
3
− 8u
2
+ 3b + 7u − 5, −2u
4
+ 3u
3
− 8u
2
+ 3a + 7u − 5, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= hau + 3b − 4a − 2u − 1, 2a
2
+ 3au − 2a + u − 3, u
2
+ 2i
I
v
1
= ha, b − v + 2, v
2
− 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 30 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
= h6.26 × 10
12
u
18
+ 1.41 × 10
13
u
17
+ · · · + 8.91 × 10
13
b − 3.07 × 10
13
, 1.86 ×
10
13
u
18
+4.28×10
13
u
17
+· · ·+8.91×10
13
a+3.56×10
12
, u
19
+2u
18
+· · ·−4u+4i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
9
=
−u
u
3
+ u
a
11
=
−0.208649u
18
− 0.480148u
17
+ ··· + 3.87234u − 0.0399949
−0.0702983u
18
− 0.158705u
17
+ ··· + 2.22000u + 0.344977
a
12
=
−0.244992u
18
− 0.555084u
17
+ ··· + 4.64401u + 0.313303
−0.0108279u
18
− 0.00284769u
17
+ ··· + 1.31196u − 0.0173194
a
3
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
u
u
a
6
=
−0.224318u
18
− 0.601824u
17
+ ··· + 0.750211u + 2.25690
0.0287859u
18
+ 0.0720971u
17
+ ··· + 0.834764u + 0.0468869
a
2
=
−0.274839u
18
− 0.716004u
17
+ ··· + 1.86949u + 2.91655
0.135398u
18
+ 0.314325u
17
+ ··· − 1.55333u − 1.32495
a
1
=
−0.328521u
18
− 0.867710u
17
+ ··· + 1.09177u + 2.92776
0.181013u
18
+ 0.450207u
17
+ ··· − 1.64773u − 1.56042
a
10
=
0.265765u
18
+ 0.718930u
17
+ ··· + 0.293501u − 3.20338
−0.180775u
18
− 0.461139u
17
+ ··· + 2.62586u + 1.11854
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
71570433112501
66825735255027
u
18
+
150381276390913
66825735255027
u
17
+ ··· −
2412887014500530
66825735255027
u −
508458695878388
66825735255027
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 26u
18
+ ··· + 17485u + 361
c
2
, c
5
u
19
+ 4u
18
+ ··· − 105u − 19
c
3
, c
4
, c
7
c
8
u
19
− 2u
18
+ ··· − 4u − 4
c
6
, c
10
u
19
+ 2u
18
+ ··· − 384u − 288
c
9
, c
11
, c
12
u
19
+ 9u
18
+ ··· − 48u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− 58y
18
+ ··· + 218537949y −130321
c
2
, c
5
y
19
− 26y
18
+ ··· + 17485y −361
c
3
, c
4
, c
7
c
8
y
19
+ 16y
18
+ ··· + 336y −16
c
6
, c
10
y
19
+ 24y
18
+ ··· + 1101312y −82944
c
9
, c
11
, c
12
y
19
− 5y
18
+ ··· + 3042y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.074260 + 0.917453I
a = −0.994466 − 0.538689I
b = −0.317235 + 0.013665I
1.53865 − 1.34534I 0.43995 + 3.44940I
u = −0.074260 − 0.917453I
a = −0.994466 + 0.538689I
b = −0.317235 − 0.013665I
1.53865 + 1.34534I 0.43995 − 3.44940I
u = 0.682490 + 0.464817I
a = −0.506827 + 0.094261I
b = 0.670269 − 1.079920I
−1.35153 − 0.43293I −1.33817 + 2.53322I
u = 0.682490 − 0.464817I
a = −0.506827 − 0.094261I
b = 0.670269 + 1.079920I
−1.35153 + 0.43293I −1.33817 − 2.53322I
u = 0.017361 + 1.304010I
a = 1.72921 − 0.99084I
b = 1.41667 − 1.52874I
4.96292 + 0.91602I 7.77670 − 1.28958I
u = 0.017361 − 1.304010I
a = 1.72921 + 0.99084I
b = 1.41667 + 1.52874I
4.96292 − 0.91602I 7.77670 + 1.28958I
u = −1.302620 + 0.369270I
a = 0.245431 − 0.163795I
b = 0.25437 + 1.92922I
−11.99340 + 5.23790I −1.08128 − 2.47083I
u = −1.302620 − 0.369270I
a = 0.245431 + 0.163795I
b = 0.25437 − 1.92922I
−11.99340 − 5.23790I −1.08128 + 2.47083I
u = 0.537549
a = −2.91204
b = 0.537401
6.81454 −5.63570
u = 0.58466 + 1.34467I
a = 0.940809 − 0.609725I
b = −0.099480 − 1.008040I
1.61779 − 4.70658I 1.34919 + 3.92901I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.58466 − 1.34467I
a = 0.940809 + 0.609725I
b = −0.099480 + 1.008040I
1.61779 + 4.70658I 1.34919 − 3.92901I
u = −0.88316 + 1.34583I
a = 0.905620 + 0.537079I
b = −0.13821 + 1.90748I
−9.13960 + 2.40553I −0.343243 − 1.215158I
u = −0.88316 − 1.34583I
a = 0.905620 − 0.537079I
b = −0.13821 − 1.90748I
−9.13960 − 2.40553I −0.343243 + 1.215158I
u = 0.360199
a = −0.304748
b = 0.772322
−1.03641 −12.4460
u = 0.13765 + 1.63790I
a = −0.009641 − 0.326348I
b = −0.387996 + 0.243197I
13.18610 − 2.66860I 6.26035 − 0.13482I
u = 0.13765 − 1.63790I
a = −0.009641 + 0.326348I
b = −0.387996 − 0.243197I
13.18610 + 2.66860I 6.26035 + 0.13482I
u = −0.47543 + 1.60768I
a = −1.18254 − 0.93225I
b = −0.40816 − 2.02733I
−5.61029 + 11.60150I 1.76287 − 4.79752I
u = −0.47543 − 1.60768I
a = −1.18254 + 0.93225I
b = −0.40816 + 2.02733I
−5.61029 − 11.60150I 1.76287 + 4.79752I
u = −0.271135
a = −2.37175
b = −0.623508
1.22081 10.2060
6
II. I
u
2
= h−2u
4
+ 3u
3
− 8u
2
+ 3b + 7u − 5, −2u
4
+ 3u
3
− 8u
2
+ 3a + 7u −
5, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
9
=
−u
u
3
+ u
a
11
=
2
3
u
4
− u
3
+
8
3
u
2
−
7
3
u +
5
3
2
3
u
4
− u
3
+
8
3
u
2
−
7
3
u +
5
3
a
12
=
2
3
u
4
− u
3
+
8
3
u
2
−
4
3
u +
5
3
2
3
u
4
− 2u
3
+
8
3
u
2
−
10
3
u +
5
3
a
3
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
u
u
a
6
=
u
u
a
2
=
u
3
+ 2u
−u
4
+ u
3
− 3u
2
+ 2u − 1
a
1
=
u
−u
3
− u
a
10
=
2
3
u
4
− u
3
+
8
3
u
2
−
7
3
u +
5
3
2
3
u
4
− u
3
+
8
3
u
2
−
7
3
u +
5
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
58
9
u
4
+
13
3
u
3
−
211
9
u
2
+
128
9
u −
115
9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
2
u
5
− u
4
+ u
2
+ u − 1
c
5
u
5
+ u
4
− u
2
+ u + 1
c
6
, c
10
u
5
c
7
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
(u + 1)
5
c
11
, c
12
(u − 1)
5
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
2
, c
5
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
6
, c
10
y
5
c
9
, c
11
, c
12
(y −1)
5
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = −0.046507 − 0.815869I
b = −0.046507 − 0.815869I
3.46474 − 2.21397I 2.99716 + 4.40290I
u = 0.233677 − 0.885557I
a = −0.046507 + 0.815869I
b = −0.046507 + 0.815869I
3.46474 + 2.21397I 2.99716 − 4.40290I
u = 0.416284
a = 1.10533
b = 1.10533
0.762751 −10.8010
u = 0.05818 + 1.69128I
a = −0.172825 + 0.649395I
b = −0.172825 + 0.649395I
12.60320 − 3.33174I 0.51443 + 5.79761I
u = 0.05818 − 1.69128I
a = −0.172825 − 0.649395I
b = −0.172825 − 0.649395I
12.60320 + 3.33174I 0.51443 − 5.79761I
10
III. I
u
3
= hau + 3b − 4a − 2u − 1, 2a
2
+ 3au − 2a + u − 3, u
2
+ 2i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
2
a
9
=
−u
−u
a
11
=
a
−
1
3
au +
4
3
a +
2
3
u +
1
3
a
12
=
2
3
au +
1
3
a −
4
3
u −
2
3
1
3
au +
2
3
a −
2
3
u −
1
3
a
3
=
−1
0
a
7
=
u
u
a
6
=
−
1
3
au +
1
3
a +
7
6
u +
4
3
1
a
2
=
−
1
3
au +
1
3
a +
7
6
u +
1
3
1
a
1
=
−
1
3
au +
1
3
a +
7
6
u +
4
3
1
a
10
=
−au + a +
3
2
u + 2
−
2
3
au +
2
3
a +
1
3
u +
5
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
7
c
8
(u
2
+ 2)
2
c
6
, c
11
, c
12
(u
2
+ u − 1)
2
c
9
, c
10
(u
2
− u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
7
c
8
(y + 2)
4
c
6
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = −0.618034 − 0.270091I
b = −0.618034 + 0.874032I
12.1725 4.00000
u = 1.414210I
a = 1.61803 − 1.85123I
b = 1.61803 − 2.28825I
4.27683 4.00000
u = − 1.414210I
a = −0.618034 + 0.270091I
b = −0.618034 − 0.874032I
12.1725 4.00000
u = − 1.414210I
a = 1.61803 + 1.85123I
b = 1.61803 + 2.28825I
4.27683 4.00000
14
IV. I
v
1
= ha, b − v + 2, v
2
− 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
v
0
a
4
=
1
0
a
9
=
v
0
a
11
=
0
v − 2
a
12
=
2v − 1
v − 2
a
3
=
1
0
a
7
=
v
0
a
6
=
v
1
a
2
=
−v + 1
−1
a
1
=
−v
−1
a
10
=
−2v + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 14
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
7
c
8
u
2
c
5
(u + 1)
2
c
6
, c
9
u
2
− u − 1
c
10
, c
11
, c
12
u
2
+ u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
2
c
3
, c
4
, c
7
c
8
y
2
c
6
, c
9
, c
10
c
11
, c
12
y
2
− 3y + 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.381966
a = 0
b = −1.61803
−0.657974 14.0000
v = 2.61803
a = 0
b = 0.618034
7.23771 14.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
· (u
19
+ 26u
18
+ ··· + 17485u + 361)
c
2
((u − 1)
2
)(u + 1)
4
(u
5
− u
4
+ ··· + u − 1)(u
19
+ 4u
18
+ ··· − 105u − 19)
c
3
, c
4
u
2
(u
2
+ 2)
2
(u
5
− u
4
+ ··· + 3u − 1)(u
19
− 2u
18
+ ··· − 4u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
5
+ u
4
+ ··· + u + 1)(u
19
+ 4u
18
+ ··· − 105u − 19)
c
6
u
5
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
19
+ 2u
18
+ ··· − 384u − 288)
c
7
, c
8
u
2
(u
2
+ 2)
2
(u
5
+ u
4
+ ··· + 3u + 1)(u
19
− 2u
18
+ ··· − 4u − 4)
c
9
((u + 1)
5
)(u
2
− u − 1)
3
(u
19
+ 9u
18
+ ··· − 48u + 9)
c
10
u
5
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
19
+ 2u
18
+ ··· − 384u − 288)
c
11
, c
12
((u − 1)
5
)(u
2
+ u − 1)
3
(u
19
+ 9u
18
+ ··· − 48u + 9)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
6
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
19
− 58y
18
+ ··· + 218537949y − 130321)
c
2
, c
5
(y − 1)
6
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
· (y
19
− 26y
18
+ ··· + 17485y − 361)
c
3
, c
4
, c
7
c
8
y
2
(y + 2)
4
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
19
+ 16y
18
+ ··· + 336y − 16)
c
6
, c
10
y
5
(y
2
− 3y + 1)
3
(y
19
+ 24y
18
+ ··· + 1101312y − 82944)
c
9
, c
11
, c
12
((y − 1)
5
)(y
2
− 3y + 1)
3
(y
19
− 5y
18
+ ··· + 3042y − 81)
20
