12n
0200
(K12n
0200
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 11 10 3 12 5 6 9 8
Solving Sequence
5,11 3,6
2 1 4 10 7 9 12 8
c
5
c
2
c
1
c
4
c
10
c
6
c
9
c
11
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
− u
6
− 4u
5
− 3u
4
− 4u
3
− 2u
2
+ b, u
7
+ u
6
+ 5u
5
+ 4u
4
+ 7u
3
+ 4u
2
+ a + 2u,
u
11
+ 2u
10
+ 8u
9
+ 12u
8
+ 22u
7
+ 24u
6
+ 24u
5
+ 16u
4
+ 9u
3
+ u
2
+ 2u + 1i
I
u
2
= hb + 1, −u
2
+ a + u − 2, u
3
+ 2u + 1i
I
u
3
= hb + 1, u
3
+ a + u − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 18 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
7
− u
6
− 4u
5
− 3u
4
− 4u
3
− 2u
2
+ b, u
7
+ u
6
+ 5u
5
+ 4u
4
+ 7u
3
+
4u
2
+ a + 2u, u
11
+ 2u
10
+ · · · + 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−u
7
− u
6
− 5u
5
− 4u
4
− 7u
3
− 4u
2
− 2u
u
7
+ u
6
+ 4u
5
+ 3u
4
+ 4u
3
+ 2u
2
a
6
=
1
−u
2
a
2
=
−u
5
− u
4
− 3u
3
− 2u
2
− 2u
u
7
+ u
6
+ 4u
5
+ 3u
4
+ 4u
3
+ 2u
2
a
1
=
3u
7
− u
6
+ 12u
5
− 3u
4
+ 12u
3
− 2u
2
+ 2
u
9
+ 4u
8
+ 5u
7
+ 17u
6
+ 7u
5
+ 18u
4
+ 2u
3
+ u
2
+ u
a
4
=
u
8
+ u
7
+ 4u
6
+ 4u
5
+ 3u
4
+ 3u
3
− 2u
2
− u + 1
u
8
+ 5u
6
+ u
5
+ 7u
4
+ 2u
3
+ 2u
2
+ u
a
10
=
−u
u
3
+ u
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
3
− 2u
u
3
+ u
a
12
=
u
7
+ 4u
5
+ 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
8
=
2u
10
+ 2u
9
+ 12u
8
+ 10u
7
+ 24u
6
+ 16u
5
+ 16u
4
+ 8u
3
+ u
2
+ 1
−2u
10
− 3u
9
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
10
+ 8u
9
+ 33u
8
+ 46u
7
+ 87u
6
+ 82u
5
+ 79u
4
+ 38u
3
+ 14u
2
− 8u + 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
+ 30u
10
+ ··· + 93u + 1
c
2
, c
4
u
11
− 8u
10
+ ··· + 13u − 1
c
3
, c
7
u
11
− u
10
+ ··· − 64u − 128
c
5
, c
6
, c
10
u
11
− 2u
10
+ ··· + 2u − 1
c
8
, c
11
, c
12
u
11
+ 12u
9
+ 38u
7
+ 2u
6
+ 14u
5
+ 12u
4
+ 13u
3
+ u
2
− 1
c
9
u
11
+ 2u
10
+ ··· − 15u
2
− 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 202y
10
+ ··· + 8901y −1
c
2
, c
4
y
11
− 30y
10
+ ··· + 93y −1
c
3
, c
7
y
11
+ 81y
10
+ ··· + 192512y −16384
c
5
, c
6
, c
10
y
11
+ 12y
10
+ ··· + 2y −1
c
8
, c
11
, c
12
y
11
+ 24y
10
+ ··· + 2y −1
c
9
y
11
+ 12y
10
+ ··· − 240y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.810323 + 0.554853I
a = −2.69043 − 1.72437I
b = 2.74686 + 0.14673I
15.5955 − 2.6821I 1.82264 + 2.33402I
u = −0.810323 − 0.554853I
a = −2.69043 + 1.72437I
b = 2.74686 − 0.14673I
15.5955 + 2.6821I 1.82264 − 2.33402I
u = −0.096709 + 1.327340I
a = 0.467034 + 0.177497I
b = 0.180346 − 0.216613I
−3.51172 − 1.71507I 5.41681 + 3.29736I
u = −0.096709 − 1.327340I
a = 0.467034 − 0.177497I
b = 0.180346 + 0.216613I
−3.51172 + 1.71507I 5.41681 − 3.29736I
u = 0.303421 + 0.399714I
a = 0.70061 − 1.79618I
b = −0.761956 + 0.436521I
−1.58612 + 0.99841I 0.02750 − 3.98074I
u = 0.303421 − 0.399714I
a = 0.70061 + 1.79618I
b = −0.761956 − 0.436521I
−1.58612 − 0.99841I 0.02750 + 3.98074I
u = 0.09711 + 1.51180I
a = −0.238461 − 0.866072I
b = −1.01867 + 1.25733I
−8.01829 + 2.43510I −1.52628 − 1.69137I
u = 0.09711 − 1.51180I
a = −0.238461 + 0.866072I
b = −1.01867 − 1.25733I
−8.01829 − 2.43510I −1.52628 + 1.69137I
u = −0.29124 + 1.55535I
a = −0.51989 − 1.85777I
b = 2.80237 + 0.46328I
8.71098 − 6.75197I −1.02074 + 2.56276I
u = −0.29124 − 1.55535I
a = −0.51989 + 1.85777I
b = 2.80237 − 0.46328I
8.71098 + 6.75197I −1.02074 − 2.56276I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.404507
a = 0.562272
b = 0.102109
0.648477 15.5600
6
II. I
u
2
= hb + 1, −u
2
+ a + u − 2, u
3
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
u
2
− u + 2
−1
a
6
=
1
−u
2
a
2
=
u
2
− u + 1
−1
a
1
=
−1
0
a
4
=
u
2
− u + 2
−1
a
10
=
−u
−u − 1
a
7
=
u
2
+ 1
u
a
9
=
1
−u − 1
a
12
=
u
−u
2
a
8
=
u
2
+ 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− 3u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
8
u
3
+ 2u + 1
c
9
u
3
− 3u
2
+ 5u − 2
c
10
, c
11
, c
12
u
3
+ 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
7
y
3
c
5
, c
6
, c
8
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
9
y
3
+ y
2
+ 13y −4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.22670 + 1.46771I
a = −0.329484 − 0.802255I
b = −1.00000
−11.08570 + 5.13794I −0.78288 − 3.73768I
u = 0.22670 − 1.46771I
a = −0.329484 + 0.802255I
b = −1.00000
−11.08570 − 5.13794I −0.78288 + 3.73768I
u = −0.453398
a = 2.65897
b = −1.00000
−0.857735 3.56580
10
III. I
u
3
= hb + 1, u
3
+ a + u − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−u
3
− u + 1
−1
a
6
=
1
−u
2
a
2
=
−u
3
− u
−1
a
1
=
−1
0
a
4
=
−u
3
− u + 1
−1
a
10
=
−u
u
3
+ u
a
7
=
u
2
+ 1
−u
3
− 2u + 1
a
9
=
−u
3
− 2u
u
3
+ u
a
12
=
2u
3
− u
2
+ 3u − 3
−u
3
+ u
2
− u + 2
a
8
=
u
2
+ 1
−u
3
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
3
+ 2u
2
− 6u + 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
7
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
8
u
4
− u
3
+ 2u
2
− 2u + 1
c
9
(u
2
+ u + 1)
2
c
10
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
6
, c
8
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
9
(y
2
+ y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 0.500000 − 0.866025I
b = −1.00000
−4.93480 + 2.02988I 2.26314 − 3.67497I
u = 0.621744 − 0.440597I
a = 0.500000 + 0.866025I
b = −1.00000
−4.93480 − 2.02988I 2.26314 + 3.67497I
u = −0.121744 + 1.306620I
a = 0.500000 + 0.866025I
b = −1.00000
−4.93480 − 2.02988I −0.76314 + 2.38721I
u = −0.121744 − 1.306620I
a = 0.500000 − 0.866025I
b = −1.00000
−4.93480 + 2.02988I −0.76314 − 2.38721I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
11
+ 30u
10
+ ··· + 93u + 1)
c
2
((u − 1)
7
)(u
11
− 8u
10
+ ··· + 13u − 1)
c
3
, c
7
u
7
(u
11
− u
10
+ ··· − 64u − 128)
c
4
((u + 1)
7
)(u
11
− 8u
10
+ ··· + 13u − 1)
c
5
, c
6
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
11
− 2u
10
+ ··· + 2u − 1)
c
8
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
11
+ 12u
9
+ 38u
7
+ 2u
6
+ 14u
5
+ 12u
4
+ 13u
3
+ u
2
− 1)
c
9
((u
2
+ u + 1)
2
)(u
3
− 3u
2
+ 5u − 2)(u
11
+ 2u
10
+ ··· − 15u
2
− 8)
c
10
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
11
− 2u
10
+ ··· + 2u − 1)
c
11
, c
12
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
11
+ 12u
9
+ 38u
7
+ 2u
6
+ 14u
5
+ 12u
4
+ 13u
3
+ u
2
− 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
11
− 202y
10
+ ··· + 8901y −1)
c
2
, c
4
((y −1)
7
)(y
11
− 30y
10
+ ··· + 93y −1)
c
3
, c
7
y
7
(y
11
+ 81y
10
+ ··· + 192512y −16384)
c
5
, c
6
, c
10
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
11
+ 12y
10
+ ··· + 2y −1)
c
8
, c
11
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
11
+ 24y
10
+ ··· + 2y −1)
c
9
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
11
+ 12y
10
+ ··· − 240y −64)
16
