12n
0550
(K12n
0550
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 9 11 2 1 5 7 10 4
Solving Sequence
2,8
7 3
4,10
5 11 1 6 9 12
c
7
c
2
c
3
c
4
c
10
c
1
c
6
c
9
c
12
c
5
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
16
− 3u
15
+ ··· + b + 3, −3u
17
+ 7u
16
+ ··· + 2a + 1, u
18
− 3u
17
+ ··· − 5u + 2i
I
u
2
= h−u
10
− 2u
8
− 3u
6
− u
5
− 2u
4
− u
3
− u
2
+ b − u − 1,
u
11
+ u
10
+ 2u
9
+ 2u
8
+ 3u
7
+ 3u
6
+ 2u
5
+ u
4
+ u
3
+ a + u, u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1i
I
u
3
= h−u
4
a − 2u
2
a + u
3
+ au + b − a + u − 1, u
3
a + 2u
2
a + 3u
3
+ a
2
+ 2au + 3u
2
+ 2a + u + 1,
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 40 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
16
−3u
15
+· · ·+b+3, −3u
17
+7u
16
+· · ·+2a+1, u
18
−3u
17
+· · ·−5u+2i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
3
2
u
17
−
7
2
u
16
+ ··· +
7
2
u −
1
2
−u
16
+ 3u
15
+ ··· + 4u − 3
a
5
=
−
1
2
u
17
+
3
2
u
16
+ ··· −
3
2
u +
1
2
u
17
− 2u
16
+ ··· + 3u − 1
a
11
=
5
2
u
17
−
11
2
u
16
+ ··· +
11
2
u −
3
2
−2u
16
+ 5u
15
+ ··· + 7u − 5
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
−
1
2
u
17
+
3
2
u
16
+ ··· −
1
2
u +
1
2
2u
17
− 3u
16
+ ··· + 2u − 1
a
9
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
12
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
17
+ 6u
16
−12u
15
+ 22u
14
−24u
13
+ 40u
12
−34u
11
+ 46u
10
−
32u
9
+ 26u
8
− 22u
7
+ 6u
6
+ 2u
5
− 10u
4
+ 16u
3
+ 2u
2
+ 2u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 9u
17
+ ··· + 3u + 4
c
2
, c
7
u
18
+ 3u
17
+ ··· + 5u + 2
c
3
u
18
− 3u
17
+ ··· − 123u + 34
c
4
, c
5
, c
6
c
9
, c
10
u
18
+ 17u
16
+ ··· + 6u
2
+ 1
c
8
u
18
+ 15u
17
+ ··· + 1169u + 266
c
11
u
18
+ 34u
17
+ ··· + 12u + 1
c
12
u
18
+ u
17
+ ··· + 265u + 160
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
+ y
17
+ ··· + 191y + 16
c
2
, c
7
y
18
+ 9y
17
+ ··· + 3y + 4
c
3
y
18
− 7y
17
+ ··· − 3909y + 1156
c
4
, c
5
, c
6
c
9
, c
10
y
18
+ 34y
17
+ ··· + 12y + 1
c
8
y
18
− 3y
17
+ ··· + 155491y + 70756
c
11
y
18
− 126y
17
+ ··· + 16y + 1
c
12
y
18
+ 85y
17
+ ··· − 550545y + 25600
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.900621 + 0.268664I
a = −0.729720 + 0.016572I
b = −0.20830 − 2.38570I
−17.8823 − 5.8715I 1.17869 + 1.87367I
u = 0.900621 − 0.268664I
a = −0.729720 − 0.016572I
b = −0.20830 + 2.38570I
−17.8823 + 5.8715I 1.17869 − 1.87367I
u = 0.297755 + 1.056830I
a = 0.151471 − 1.118820I
b = 0.648988 + 0.730249I
−3.32385 + 0.53851I −1.31320 + 1.23447I
u = 0.297755 − 1.056830I
a = 0.151471 + 1.118820I
b = 0.648988 − 0.730249I
−3.32385 − 0.53851I −1.31320 − 1.23447I
u = −0.750892 + 0.811110I
a = −1.272950 + 0.346188I
b = 0.598966 − 0.460694I
−14.5019 − 2.7975I 1.40873 + 2.65752I
u = −0.750892 − 0.811110I
a = −1.272950 − 0.346188I
b = 0.598966 + 0.460694I
−14.5019 + 2.7975I 1.40873 − 2.65752I
u = −0.482590 + 1.071550I
a = 0.264991 − 0.374329I
b = −0.027718 + 0.293336I
−0.95248 − 3.40681I 4.19440 + 2.15670I
u = −0.482590 − 1.071550I
a = 0.264991 + 0.374329I
b = −0.027718 − 0.293336I
−0.95248 + 3.40681I 4.19440 − 2.15670I
u = 0.541968 + 1.089520I
a = −0.848196 + 0.994100I
b = −0.205313 − 1.329710I
−1.65747 + 6.51690I 2.73570 − 9.24962I
u = 0.541968 − 1.089520I
a = −0.848196 − 0.994100I
b = −0.205313 + 1.329710I
−1.65747 − 6.51690I 2.73570 + 9.24962I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.643322 + 0.355418I
a = 0.486969 + 0.304277I
b = −0.228450 + 0.806417I
0.44494 − 1.87013I 6.06834 + 5.46541I
u = 0.643322 − 0.355418I
a = 0.486969 − 0.304277I
b = −0.228450 − 0.806417I
0.44494 + 1.87013I 6.06834 − 5.46541I
u = 0.266121 + 1.276470I
a = −0.55947 + 2.38028I
b = −1.03140 − 1.90393I
16.5141 − 2.0953I −3.38488 − 0.11711I
u = 0.266121 − 1.276470I
a = −0.55947 − 2.38028I
b = −1.03140 + 1.90393I
16.5141 + 2.0953I −3.38488 + 0.11711I
u = −0.504699 + 0.453241I
a = 0.598131 + 0.161265I
b = −0.263286 + 0.046770I
0.920321 − 0.679889I 8.59008 + 5.05445I
u = −0.504699 − 0.453241I
a = 0.598131 − 0.161265I
b = −0.263286 − 0.046770I
0.920321 + 0.679889I 8.59008 − 5.05445I
u = 0.588394 + 1.196870I
a = 2.15878 − 1.76754I
b = −0.28348 + 2.92630I
18.7937 + 11.3151I −1.47787 − 5.35681I
u = 0.588394 − 1.196870I
a = 2.15878 + 1.76754I
b = −0.28348 − 2.92630I
18.7937 − 11.3151I −1.47787 + 5.35681I
6
II. I
u
2
= h−u
10
− 2u
8
+ · · · + b − 1, u
11
+ u
10
+ · · · + a + u, u
12
+ 3u
10
+
5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−u
11
− u
10
− 2u
9
− 2u
8
− 3u
7
− 3u
6
− 2u
5
− u
4
− u
3
− u
u
10
+ 2u
8
+ 3u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ u + 1
a
5
=
u
11
− u
10
+ 2u
9
− 3u
8
+ 3u
7
− 4u
6
+ 2u
5
− 2u
4
+ u
3
+ u − 1
−u
11
− 2u
9
+ u
8
− 2u
7
+ 2u
6
+ 2u
4
+ u
3
a
11
=
−2u
11
− u
10
− 4u
9
− 2u
8
− 5u
7
− 3u
6
− 2u
5
− u
4
− u
u
11
+ u
10
+ 3u
9
+ 2u
8
+ 4u
7
+ 3u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 2u + 1
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
u
11
− u
10
+ 2u
9
− 3u
8
+ 3u
7
− 4u
6
+ 2u
5
− 2u
4
+ 2u
3
+ u − 1
−u
11
− 2u
9
+ u
8
− 2u
7
+ 2u
6
+ 2u
4
− u
a
9
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
12
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
10
− 12u
8
− 16u
6
− 8u
4
− 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
c
2
, c
7
, c
8
u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1
c
3
u
12
− u
10
+ 5u
8
+ 6u
4
− 3u
2
+ 1
c
4
, c
5
, c
6
c
9
, c
10
(u
2
+ 1)
6
c
11
(u + 1)
12
c
12
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
2
, c
7
, c
8
(y
6
+ 3y
5
+ 5y
4
+ 4y
3
+ 2y
2
+ y + 1)
2
c
3
(y
6
− y
5
+ 5y
4
+ 6y
2
− 3y + 1)
2
c
4
, c
5
, c
6
c
9
, c
10
(y + 1)
12
c
11
(y −1)
12
c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.295542 + 1.002190I
a = −0.48664 − 2.49244I
b = 1.92274 + 0.99741I
−5.18047 + 0.92430I −5.71672 − 0.79423I
u = 0.295542 − 1.002190I
a = −0.48664 + 2.49244I
b = 1.92274 − 0.99741I
−5.18047 − 0.92430I −5.71672 + 0.79423I
u = −0.295542 + 1.002190I
a = −0.883753 − 0.365919I
b = 0.593678 − 0.140919I
−5.18047 − 0.92430I −5.71672 + 0.79423I
u = −0.295542 − 1.002190I
a = −0.883753 + 0.365919I
b = 0.593678 + 0.140919I
−5.18047 + 0.92430I −5.71672 − 0.79423I
u = 0.664531 + 0.428243I
a = 1.116540 − 0.158471I
b = 0.37850 + 1.59457I
−1.39926 − 0.92430I 1.71672 + 0.79423I
u = 0.664531 − 0.428243I
a = 1.116540 + 0.158471I
b = 0.37850 − 1.59457I
−1.39926 + 0.92430I 1.71672 − 0.79423I
u = −0.664531 + 0.428243I
a = 0.719425 − 0.699888I
b = −0.212587 + 0.409813I
−1.39926 + 0.92430I 1.71672 − 0.79423I
u = −0.664531 − 0.428243I
a = 0.719425 + 0.699888I
b = −0.212587 − 0.409813I
−1.39926 − 0.92430I 1.71672 + 0.79423I
u = 0.558752 + 1.073950I
a = −2.11043 + 0.88125I
b = 0.71759 − 2.27409I
−3.28987 + 5.69302I −2.00000 − 5.51057I
u = 0.558752 − 1.073950I
a = −2.11043 − 0.88125I
b = 0.71759 + 2.27409I
−3.28987 − 5.69302I −2.00000 + 5.51057I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.558752 + 1.073950I
a = 0.644857 + 0.118748I
b = −0.399916 + 0.126193I
−3.28987 − 5.69302I −2.00000 + 5.51057I
u = −0.558752 − 1.073950I
a = 0.644857 − 0.118748I
b = −0.399916 − 0.126193I
−3.28987 + 5.69302I −2.00000 − 5.51057I
11
III. I
u
3
= h−u
4
a − 2u
2
a + u
3
+ au + b − a + u − 1, u
3
a + 3u
3
+ · · · + 2a +
1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
a
u
4
a + 2u
2
a − u
3
− au + a − u + 1
a
5
=
−u
3
a + u
3
− au + 2u
2
+ a + 2u + 3
u
4
a + u
3
a + u
4
+ 2u
2
a + au + 2u
2
+ a − u
a
11
=
u
4
a + u
2
a − u
3
− au + 2a − u + 1
u
4
a + u
4
+ 3u
2
a − au + 2u
2
+ 2a + 2
a
1
=
u
3
−u
4
− u
3
− u
2
− 1
a
6
=
u
4
a + 2u
2
a − 3u
3
− au − 2u
2
− 3u − 3
2u
4
a + u
4
+ 2u
2
a + 2u
3
− au + 2a + 2u + 2
a
9
=
−2u
3
− u
2u
4
+ 2u
3
+ 2u
2
+ u + 2
a
12
=
2u
3
+ 1
−2u
4
− 2u
3
− 2u
2
− 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u
2
− 4u − 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
c
2
, c
7
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
3
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
4
, c
5
, c
6
c
9
, c
10
u
10
− u
9
+ 8u
8
− 4u
7
+ 28u
6
− 6u
5
+ 53u
4
+ 5u
3
+ 50u
2
+ 12u + 17
c
8
(u
5
− 5u
4
+ 8u
3
− 3u
2
− u − 1)
2
c
11
u
10
+ 15u
9
+ ··· + 1556u + 289
c
12
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
2
, c
7
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
3
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
4
, c
5
, c
6
c
9
, c
10
y
10
+ 15y
9
+ ··· + 1556y + 289
c
8
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1)
2
c
11
y
10
− y
9
+ ··· − 3940y + 83521
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.55867 − 1.51591I
b = 0.549371 − 0.042056I
−3.61897 + 1.53058I 1.48489 − 4.43065I
u = 0.339110 + 0.822375I
a = −1.46526 − 0.97188I
b = 1.65698 + 0.38291I
−3.61897 + 1.53058I 1.48489 − 4.43065I
u = 0.339110 − 0.822375I
a = 0.55867 + 1.51591I
b = 0.549371 + 0.042056I
−3.61897 − 1.53058I 1.48489 + 4.43065I
u = 0.339110 − 0.822375I
a = −1.46526 + 0.97188I
b = 1.65698 − 0.38291I
−3.61897 − 1.53058I 1.48489 + 4.43065I
u = −0.766826
a = −0.595741 + 0.538146I
b = 0.25856 + 1.76977I
−5.69095 0.518860
u = −0.766826
a = −0.595741 − 0.538146I
b = 0.25856 − 1.76977I
−5.69095 0.518860
u = −0.455697 + 1.200150I
a = −1.45542 − 1.61863I
b = −0.53813 + 1.89796I
−9.16243 − 4.40083I −2.74431 + 3.49859I
u = −0.455697 + 1.200150I
a = 0.95775 + 2.38693I
b = 0.57321 − 2.51986I
−9.16243 − 4.40083I −2.74431 + 3.49859I
u = −0.455697 − 1.200150I
a = −1.45542 + 1.61863I
b = −0.53813 − 1.89796I
−9.16243 + 4.40083I −2.74431 − 3.49859I
u = −0.455697 − 1.200150I
a = 0.95775 − 2.38693I
b = 0.57321 + 2.51986I
−9.16243 + 4.40083I −2.74431 − 3.49859I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
· (u
18
+ 9u
17
+ ··· + 3u + 4)
c
2
, c
7
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
(u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1)
· (u
18
+ 3u
17
+ ··· + 5u + 2)
c
3
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
(u
12
− u
10
+ 5u
8
+ 6u
4
− 3u
2
+ 1)
· (u
18
− 3u
17
+ ··· − 123u + 34)
c
4
, c
5
, c
6
c
9
, c
10
(u
2
+ 1)
6
· (u
10
− u
9
+ 8u
8
− 4u
7
+ 28u
6
− 6u
5
+ 53u
4
+ 5u
3
+ 50u
2
+ 12u + 17)
· (u
18
+ 17u
16
+ ··· + 6u
2
+ 1)
c
8
((u
5
− 5u
4
+ 8u
3
− 3u
2
− u − 1)
2
)(u
12
+ 3u
10
+ ··· + u
2
+ 1)
· (u
18
+ 15u
17
+ ··· + 1169u + 266)
c
11
((u + 1)
12
)(u
10
+ 15u
9
+ ··· + 1556u + 289)
· (u
18
+ 34u
17
+ ··· + 12u + 1)
c
12
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
· (u
18
+ u
17
+ ··· + 265u + 160)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
18
+ y
17
+ ··· + 191y + 16)
c
2
, c
7
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
(y
6
+ 3y
5
+ 5y
4
+ 4y
3
+ 2y
2
+ y + 1)
2
· (y
18
+ 9y
17
+ ··· + 3y + 4)
c
3
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
(y
6
− y
5
+ 5y
4
+ 6y
2
− 3y + 1)
2
· (y
18
− 7y
17
+ ··· − 3909y + 1156)
c
4
, c
5
, c
6
c
9
, c
10
((y + 1)
12
)(y
10
+ 15y
9
+ ··· + 1556y + 289)
· (y
18
+ 34y
17
+ ··· + 12y + 1)
c
8
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1)
2
· (y
6
+ 3y
5
+ 5y
4
+ 4y
3
+ 2y
2
+ y + 1)
2
· (y
18
− 3y
17
+ ··· + 155491y + 70756)
c
11
((y −1)
12
)(y
10
− y
9
+ ··· − 3940y + 83521)
· (y
18
− 126y
17
+ ··· + 16y + 1)
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
18
+ 85y
17
+ ··· − 550545y + 25600)
17
