12n
0403
(K12n
0403
)
A knot diagram
1
Linearized knot diagam
3 6 10 11 12 2 12 3 1 4 8 10
Solving Sequence
3,10
4 11
1,5
9 8 12 6 2 7
c
3
c
10
c
4
c
9
c
8
c
12
c
5
c
2
c
6
c
1
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
− 3u
3
− 2u
2
+ b + 1, u
3
+ 2u
2
+ a − u − 1, u
5
+ 5u
4
+ 7u
3
+ u
2
− 2u − 1i
I
u
2
= h−u
4
+ u
3
+ 2u
2
+ b − 2u + 1, u
3
+ a − 3u − 1, u
5
+ u
4
− 3u
3
− 3u
2
− 1i
I
u
3
= h−u
2
a + au + u
2
+ 2b − a − 3u + 1, u
2
a + a
2
− au − 2a + u, u
3
− 2u
2
− 1i
I
u
4
= hb − 1, 4a − u − 3, u
2
− u − 4i
I
u
5
= h−au + b − a + 1, a
2
− a − u + 2, u
2
− u − 1i
* 5 irreducible components of dim
C
= 0, with total 22 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−u
4
−3u
3
−2u
2
+b+1, u
3
+2u
2
+a−u−1, u
5
+5u
4
+7u
3
+u
2
−2u−1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
11
=
u
−u
3
+ u
a
1
=
−u
3
− 2u
2
+ u + 1
u
4
+ 3u
3
+ 2u
2
− 1
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
u
2
−u
4
− 4u
3
− 2u
2
+ 2u + 1
a
8
=
u
4
+ 4u
3
+ 3u
2
− 2u − 1
−u
4
− 4u
3
− 2u
2
+ 2u + 1
a
12
=
−u
3
− 2u
2
+ u + 1
−2u
4
− 5u
3
+ 2u
a
6
=
1
−u
4
− 4u
3
− 4u
2
+ u + 1
a
2
=
−u
4
− 4u
3
− 4u
2
+ u + 2
u
4
+ 3u
3
+ 2u
2
− 1
a
7
=
−u
3
− 2u
2
+ u + 1
−2u
4
− 6u
3
− 3u
2
+ u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
4
− 22u
3
− 12u
2
+ 18u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ u
4
+ 7u
3
+ 8u
2
+ u − 1
c
2
, c
6
, c
7
c
11
u
5
− 3u
4
+ 5u
3
− 4u
2
+ 3u − 1
c
3
, c
4
, c
10
u
5
+ 5u
4
+ 7u
3
+ u
2
− 2u − 1
c
5
, c
9
, c
12
u
5
− u
4
+ 12u
3
+ 3u
2
− u − 1
c
8
u
5
− 14u
4
+ 51u
3
− 6u
2
+ 4u − 13
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
+ 13y
4
+ 35y
3
− 48y
2
+ 17y −1
c
2
, c
6
, c
7
c
11
y
5
+ y
4
+ 7y
3
+ 8y
2
+ y −1
c
3
, c
4
, c
10
y
5
− 11y
4
+ 35y
3
− 19y
2
+ 6y −1
c
5
, c
9
, c
12
y
5
+ 23y
4
+ 148y
3
− 35y
2
+ 7y −1
c
8
y
5
− 94y
4
+ 2441y
3
+ 8y
2
− 140y −169
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.483921 + 0.312340I
a = 0.214528 + 0.727972I
b = −0.714557 − 0.120312I
−1.93405 − 1.28592I −1.53646 + 5.58816I
u = −0.483921 − 0.312340I
a = 0.214528 − 0.727972I
b = −0.714557 + 0.120312I
−1.93405 + 1.28592I −1.53646 − 5.58816I
u = 0.563096
a = 0.750397
b = 0.270326
0.922645 10.8000
u = −2.29763 + 0.27249I
a = −0.08973 − 1.51845I
b = 0.07939 − 2.65310I
−13.3317 − 8.5417I 7.63666 + 3.64244I
u = −2.29763 − 0.27249I
a = −0.08973 + 1.51845I
b = 0.07939 + 2.65310I
−13.3317 + 8.5417I 7.63666 − 3.64244I
5
II.
I
u
2
= h−u
4
+ u
3
+ 2u
2
+ b − 2u + 1, u
3
+ a − 3u − 1, u
5
+ u
4
− 3u
3
− 3u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
11
=
u
−u
3
+ u
a
1
=
−u
3
+ 3u + 1
u
4
− u
3
− 2u
2
+ 2u − 1
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
u
2
− 2
u
4
− 2u
2
− 1
a
8
=
−u
4
+ 3u
2
− 1
u
4
− 2u
2
− 1
a
12
=
−u
3
+ 3u + 1
−u
3
+ 2u
a
6
=
−1
u
4
− 2u
2
− u − 1
a
2
=
−u
4
+ 2u
2
+ u + 2
u
4
− u
3
− 2u
2
+ 2u − 1
a
7
=
u
3
− 3u − 1
2u
4
− 5u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
+ 6u
3
+ 4u
2
− 14u + 5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− u
4
+ 3u
3
+ u + 1
c
2
, c
7
u
5
− u
4
+ u
3
+ u − 1
c
3
, c
4
u
5
+ u
4
− 3u
3
− 3u
2
− 1
c
5
, c
12
u
5
− u
4
− u
2
+ u − 1
c
6
, c
11
u
5
+ u
4
+ u
3
+ u + 1
c
8
u
5
− 3u
3
+ 6u
2
− 4u + 1
c
9
u
5
+ u
4
+ u
2
+ u + 1
c
10
u
5
− u
4
− 3u
3
+ 3u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
+ 5y
4
+ 11y
3
+ 8y
2
+ y −1
c
2
, c
6
, c
7
c
11
y
5
+ y
4
+ 3y
3
+ y −1
c
3
, c
4
, c
10
y
5
− 7y
4
+ 15y
3
− 7y
2
− 6y −1
c
5
, c
9
, c
12
y
5
− y
4
− 3y
2
− y −1
c
8
y
5
− 6y
4
+ y
3
− 12y
2
+ 4y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.48162 + 0.12936I
a = −0.266775 − 0.461665I
b = −0.54328 − 1.49449I
6.00251 − 5.77307I 6.19041 + 5.09435I
u = −1.48162 − 0.12936I
a = −0.266775 + 0.461665I
b = −0.54328 + 1.49449I
6.00251 + 5.77307I 6.19041 − 5.09435I
u = 0.099006 + 0.496292I
a = 1.36921 + 1.59652I
b = −0.210516 + 0.857202I
0.38751 + 3.74061I 2.14222 − 7.10791I
u = 0.099006 − 0.496292I
a = 1.36921 − 1.59652I
b = −0.210516 − 0.857202I
0.38751 − 3.74061I 2.14222 + 7.10791I
u = 1.76524
a = 0.795136
b = 0.507589
6.95916 6.33470
9
III.
I
u
3
= h−u
2
a + au + u
2
+ 2b − a − 3u + 1, u
2
a + a
2
− au − 2a + u, u
3
− 2u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
11
=
u
−2u
2
+ u − 1
a
1
=
a
1
2
u
2
a −
1
2
u
2
+ ··· +
1
2
a −
1
2
a
5
=
−u
2
+ 1
2u
2
+ u + 2
a
9
=
−u
2
a + 2au − u
2
− a
−
1
2
u
2
a −
3
2
u
2
+ ··· −
1
2
a −
1
2
a
8
=
−
1
2
u
2
a +
1
2
u
2
+ ··· −
1
2
a +
1
2
−
1
2
u
2
a −
3
2
u
2
+ ··· −
1
2
a −
1
2
a
12
=
a
−
1
2
u
2
a −
1
2
u
2
+ ··· +
1
2
a −
1
2
a
6
=
1
−
1
2
u
2
a +
1
2
u
2
+ ··· +
1
2
a −
1
2
a
2
=
−
1
2
u
2
a +
1
2
u
2
+ ··· +
1
2
a +
1
2
1
2
u
2
a −
1
2
u
2
+ ··· +
1
2
a −
1
2
a
7
=
a
−u
2
a + au + u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
2
− 2u + 7
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 8u
4
− 4u
3
+ 8u
2
− 9u + 4
c
2
, c
6
, c
7
c
11
u
6
− 2u
5
+ 2u
4
+ 2u
3
− 2u
2
+ u + 2
c
3
, c
4
, c
10
(u
3
− 2u
2
− 1)
2
c
5
, c
9
, c
12
u
6
+ 3u
5
+ 16u
4
+ 22u
3
+ 34u
2
+ 11u + 1
c
8
u
6
+ 14u
5
+ 69u
4
+ 114u
3
+ 127u
2
+ 27u + 22
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
+ 16y
5
+ 80y
4
+ 120y
3
+ 56y
2
− 17y + 16
c
2
, c
6
, c
7
c
11
y
6
+ 8y
4
− 4y
3
+ 8y
2
− 9y + 4
c
3
, c
4
, c
10
(y
3
− 4y
2
− 4y −1)
2
c
5
, c
9
, c
12
y
6
+ 23y
5
+ 192y
4
+ 540y
3
+ 704y
2
− 53y + 1
c
8
y
6
− 58y
5
+ 1823y
4
+ 3818y
3
+ 13009y
2
+ 4859y + 484
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.102785 + 0.665457I
a = 0.019191 + 0.283733I
b = −0.317796 + 1.154010I
1.03690 + 2.56897I 6.77330 − 1.46771I
u = −0.102785 + 0.665457I
a = 2.31029 + 0.51852I
b = 0.544495 + 0.313702I
1.03690 + 2.56897I 6.77330 − 1.46771I
u = −0.102785 − 0.665457I
a = 0.019191 − 0.283733I
b = −0.317796 − 1.154010I
1.03690 − 2.56897I 6.77330 + 1.46771I
u = −0.102785 − 0.665457I
a = 2.31029 − 0.51852I
b = 0.544495 − 0.313702I
1.03690 − 2.56897I 6.77330 + 1.46771I
u = 2.20557
a = −0.32948 + 1.44811I
b = −0.22670 + 2.64929I
−13.5883 7.45340
u = 2.20557
a = −0.32948 − 1.44811I
b = −0.22670 − 2.64929I
−13.5883 7.45340
13
IV. I
u
4
= hb − 1, 4a − u − 3, u
2
− u − 4i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u − 4
a
11
=
u
−4u − 4
a
1
=
1
4
u +
3
4
1
a
5
=
−u − 3
7u + 12
a
9
=
5
4
u +
7
4
2u + 1
a
8
=
−
3
4
u +
3
4
2u + 1
a
12
=
1
4
u +
3
4
−2u − 3
a
6
=
1
4
u −
5
4
1
a
2
=
1
4
u −
1
4
1
a
7
=
1
2
u −
3
2
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 14
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
6
, c
7
c
11
(u + 1)
2
c
3
, c
4
, c
10
u
2
− u − 4
c
5
, c
9
, c
12
u
2
− 3u − 2
c
8
u
2
− 4u − 13
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
(y − 1)
2
c
3
, c
4
, c
10
y
2
− 9y + 16
c
5
, c
9
, c
12
y
2
− 13y + 4
c
8
y
2
− 42y + 169
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.56155
a = 0.359612
b = 1.00000
8.22467 14.0000
u = 2.56155
a = 1.39039
b = 1.00000
8.22467 14.0000
17
V. I
u
5
= h−au + b − a + 1, a
2
− a − u + 2, u
2
− u − 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u − 1
a
11
=
u
−u − 1
a
1
=
a
au + a − 1
a
5
=
−u
u
a
9
=
au − u + 1
au + a
a
8
=
−a − u + 1
au + a
a
12
=
a
−1
a
6
=
−1
au
a
2
=
−au + 1
au + a − 1
a
7
=
−a
2au + a − u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u + 6
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
12
u
4
− u
3
+ u
2
− u + 1
c
3
, c
4
(u
2
− u − 1)
2
c
6
, c
9
, c
11
u
4
+ u
3
+ u
2
+ u + 1
c
8
u
4
+ 3u
3
+ 4u
2
+ 2u + 1
c
10
(u
2
+ u − 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
c
11
, c
12
y
4
+ y
3
+ y
2
+ y + 1
c
3
, c
4
, c
10
(y
2
− 3y + 1)
2
c
8
y
4
− y
3
+ 6y
2
+ 4y + 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.50000 + 1.53884I
b = −0.809017 + 0.587785I
−0.657974 5.38200
u = −0.618034
a = 0.50000 − 1.53884I
b = −0.809017 − 0.587785I
−0.657974 5.38200
u = 1.61803
a = 0.500000 + 0.363271I
b = 0.309017 + 0.951057I
7.23771 7.61800
u = 1.61803
a = 0.500000 − 0.363271I
b = 0.309017 − 0.951057I
7.23771 7.61800
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
(u
4
− u
3
+ u
2
− u + 1)(u
5
− u
4
+ 3u
3
+ u + 1)
· (u
5
+ u
4
+ 7u
3
+ 8u
2
+ u − 1)(u
6
+ 8u
4
− 4u
3
+ 8u
2
− 9u + 4)
c
2
, c
7
(u + 1)
2
(u
4
− u
3
+ u
2
− u + 1)(u
5
− 3u
4
+ 5u
3
− 4u
2
+ 3u − 1)
· (u
5
− u
4
+ u
3
+ u − 1)(u
6
− 2u
5
+ 2u
4
+ 2u
3
− 2u
2
+ u + 2)
c
3
, c
4
(u
2
− u − 4)(u
2
− u − 1)
2
(u
3
− 2u
2
− 1)
2
(u
5
+ u
4
− 3u
3
− 3u
2
− 1)
· (u
5
+ 5u
4
+ 7u
3
+ u
2
− 2u − 1)
c
5
, c
12
(u
2
− 3u − 2)(u
4
− u
3
+ u
2
− u + 1)(u
5
− u
4
− u
2
+ u − 1)
· (u
5
− u
4
+ 12u
3
+ 3u
2
− u − 1)(u
6
+ 3u
5
+ ··· + 11u + 1)
c
6
, c
11
(u + 1)
2
(u
4
+ u
3
+ u
2
+ u + 1)(u
5
− 3u
4
+ 5u
3
− 4u
2
+ 3u − 1)
· (u
5
+ u
4
+ u
3
+ u + 1)(u
6
− 2u
5
+ 2u
4
+ 2u
3
− 2u
2
+ u + 2)
c
8
(u
2
− 4u − 13)(u
4
+ 3u
3
+ 4u
2
+ 2u + 1)(u
5
− 3u
3
+ 6u
2
− 4u + 1)
· (u
5
− 14u
4
+ 51u
3
− 6u
2
+ 4u − 13)
· (u
6
+ 14u
5
+ 69u
4
+ 114u
3
+ 127u
2
+ 27u + 22)
c
9
(u
2
− 3u − 2)(u
4
+ u
3
+ u
2
+ u + 1)(u
5
− u
4
+ 12u
3
+ 3u
2
− u − 1)
· (u
5
+ u
4
+ u
2
+ u + 1)(u
6
+ 3u
5
+ 16u
4
+ 22u
3
+ 34u
2
+ 11u + 1)
c
10
(u
2
− u − 4)(u
2
+ u − 1)
2
(u
3
− 2u
2
− 1)
2
(u
5
− u
4
− 3u
3
+ 3u
2
+ 1)
· (u
5
+ 5u
4
+ 7u
3
+ u
2
− 2u − 1)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
2
(y
4
+ y
3
+ y
2
+ y + 1)(y
5
+ 5y
4
+ 11y
3
+ 8y
2
+ y − 1)
· (y
5
+ 13y
4
+ 35y
3
− 48y
2
+ 17y − 1)
· (y
6
+ 16y
5
+ 80y
4
+ 120y
3
+ 56y
2
− 17y + 16)
c
2
, c
6
, c
7
c
11
(y − 1)
2
(y
4
+ y
3
+ y
2
+ y + 1)(y
5
+ y
4
+ 3y
3
+ y − 1)
· (y
5
+ y
4
+ 7y
3
+ 8y
2
+ y − 1)(y
6
+ 8y
4
− 4y
3
+ 8y
2
− 9y + 4)
c
3
, c
4
, c
10
(y
2
− 9y + 16)(y
2
− 3y + 1)
2
(y
3
− 4y
2
− 4y − 1)
2
· (y
5
− 11y
4
+ 35y
3
− 19y
2
+ 6y − 1)(y
5
− 7y
4
+ 15y
3
− 7y
2
− 6y − 1)
c
5
, c
9
, c
12
(y
2
− 13y + 4)(y
4
+ y
3
+ y
2
+ y + 1)(y
5
− y
4
− 3y
2
− y − 1)
· (y
5
+ 23y
4
+ 148y
3
− 35y
2
+ 7y − 1)
· (y
6
+ 23y
5
+ 192y
4
+ 540y
3
+ 704y
2
− 53y + 1)
c
8
(y
2
− 42y + 169)(y
4
− y
3
+ 6y
2
+ 4y + 1)
· (y
5
− 94y
4
+ 2441y
3
+ 8y
2
− 140y − 169)
· (y
5
− 6y
4
+ y
3
− 12y
2
+ 4y − 1)
· (y
6
− 58y
5
+ 1823y
4
+ 3818y
3
+ 13009y
2
+ 4859y + 484)
23
