12n
0169
(K12n
0169
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 11 10 3 5 12 6 7 9
Solving Sequence
6,10 3,7
8 11 12 5 2 1 4 9
c
6
c
7
c
10
c
11
c
5
c
2
c
1
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
31
+ u
30
+ ··· + b + 2u, u
31
− u
30
+ ··· + a − 5u, u
32
− 2u
31
+ ··· + 5u − 1i
I
u
2
= hb + u, u
2
+ a + 2, u
3
+ 2u − 1i
I
u
3
= h−u
2
+ b − u, u
3
+ u
2
+ a + 2u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
31
+u
30
+· · ·+b+2u, u
31
−u
30
+· · ·+a−5u, u
32
−2u
31
+· · ·+5u−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
−u
31
+ u
30
+ ··· − 7u
2
+ 5u
u
31
− u
30
+ ··· + 3u
2
− 2u
a
7
=
1
u
2
a
8
=
u
13
+ 6u
11
+ 13u
9
+ 12u
7
+ 6u
5
+ 4u
3
+ u
−u
13
− 5u
11
− 7u
9
+ 2u
5
− 3u
3
+ u
a
11
=
−u
u
a
12
=
−u
3
− 2u
−u
5
− u
3
+ u
a
5
=
u
2
+ 1
−u
2
a
2
=
u
25
+ 12u
23
+ ··· − 4u
2
+ 3u
u
27
− u
26
+ ··· + 3u
2
− 2u
a
1
=
u
11
+ 6u
9
+ 12u
7
+ 8u
5
+ u
3
+ 2u
u
13
+ 5u
11
+ 7u
9
− 2u
5
+ 3u
3
− u
a
4
=
u
31
− u
30
+ ··· + u + 1
−u
31
+ u
30
+ ··· + u
2
− u
a
9
=
u
7
+ 4u
5
+ 4u
3
u
9
+ 3u
7
+ u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
31
+ 8u
30
− 69u
29
+ 118u
28
− 516u
27
+ 760u
26
− 2191u
25
+
2774u
24
− 5768u
23
+ 6188u
22
− 9542u
21
+ 8335u
20
− 9334u
19
+ 5857u
18
− 4125u
17
+
549u
16
+ 634u
15
− 1814u
14
+ 878u
13
− 598u
12
− 232u
11
− 15u
10
+ 140u
9
− 465u
8
+
354u
7
− 156u
6
− 44u
5
+ 113u
4
− 54u
3
− 27u
2
+ 31u − 23
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 44u
31
+ ··· + 48u + 1
c
2
, c
4
u
32
− 8u
31
+ ··· + 24u
2
− 1
c
3
, c
7
u
32
− u
31
+ ··· + 192u + 128
c
5
, c
6
, c
10
u
32
+ 2u
31
+ ··· − 5u − 1
c
8
u
32
+ 2u
31
+ ··· − 3u − 1
c
9
, c
12
u
32
− 6u
31
+ ··· − 39u + 19
c
11
u
32
− 2u
31
+ ··· − 40u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 104y
31
+ ··· − 1812y + 1
c
2
, c
4
y
32
− 44y
31
+ ··· − 48y + 1
c
3
, c
7
y
32
− 45y
31
+ ··· − 12288y + 16384
c
5
, c
6
, c
10
y
32
+ 30y
31
+ ··· − 9y + 1
c
8
y
32
− 66y
31
+ ··· − 9y + 1
c
9
, c
12
y
32
+ 18y
31
+ ··· − 1673y + 361
c
11
y
32
+ 6y
31
+ ··· + 368y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.066411 + 1.151990I
a = 1.410670 + 0.024945I
b = −2.76292 − 0.86053I
−0.138725 + 1.342600I −14.3827 − 1.1828I
u = −0.066411 − 1.151990I
a = 1.410670 − 0.024945I
b = −2.76292 + 0.86053I
−0.138725 − 1.342600I −14.3827 + 1.1828I
u = 0.513081 + 0.643079I
a = −0.646531 + 0.515194I
b = 1.64282 − 0.89836I
−9.52014 + 3.30104I −13.33638 + 0.19936I
u = 0.513081 − 0.643079I
a = −0.646531 − 0.515194I
b = 1.64282 + 0.89836I
−9.52014 − 3.30104I −13.33638 − 0.19936I
u = 0.737634 + 0.348273I
a = 2.23105 − 1.72413I
b = −0.0831051 − 0.0958486I
−10.57590 − 7.60354I −15.1269 + 5.2212I
u = 0.737634 − 0.348273I
a = 2.23105 + 1.72413I
b = −0.0831051 + 0.0958486I
−10.57590 + 7.60354I −15.1269 − 5.2212I
u = −0.298547 + 1.193750I
a = −1.40936 − 1.46969I
b = 2.94438 + 2.80689I
−11.30460 + 3.80890I −14.0876 − 3.1596I
u = −0.298547 − 1.193750I
a = −1.40936 + 1.46969I
b = 2.94438 − 2.80689I
−11.30460 − 3.80890I −14.0876 + 3.1596I
u = −0.748139
a = 3.59916
b = −0.104790
−14.9662 −18.4420
u = −0.606144 + 0.421198I
a = 0.546855 + 0.039840I
b = 0.102884 + 0.314041I
2.64609 + 1.95373I −5.09513 − 3.50992I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.606144 − 0.421198I
a = 0.546855 − 0.039840I
b = 0.102884 − 0.314041I
2.64609 − 1.95373I −5.09513 + 3.50992I
u = 0.087660 + 1.285000I
a = −0.478992 + 0.327569I
b = 1.049280 − 0.185331I
3.25069 − 1.60094I −6.41790 + 3.90851I
u = 0.087660 − 1.285000I
a = −0.478992 − 0.327569I
b = 1.049280 + 0.185331I
3.25069 + 1.60094I −6.41790 − 3.90851I
u = 0.636337 + 0.293336I
a = −1.79875 + 1.45255I
b = 0.098008 − 0.391373I
−1.49855 − 3.68796I −14.9081 + 6.2088I
u = 0.636337 − 0.293336I
a = −1.79875 − 1.45255I
b = 0.098008 + 0.391373I
−1.49855 + 3.68796I −14.9081 − 6.2088I
u = −0.570243 + 0.210892I
a = −2.18842 − 0.55206I
b = −0.306041 − 0.510133I
−2.65931 + 1.04311I −15.7710 − 5.3018I
u = −0.570243 − 0.210892I
a = −2.18842 + 0.55206I
b = −0.306041 + 0.510133I
−2.65931 − 1.04311I −15.7710 + 5.3018I
u = −0.223261 + 1.390520I
a = 1.39534 + 1.47490I
b = −2.07768 − 2.46657I
2.48306 + 3.95929I −10.08448 − 4.02414I
u = −0.223261 − 1.390520I
a = 1.39534 − 1.47490I
b = −2.07768 + 2.46657I
2.48306 − 3.95929I −10.08448 + 4.02414I
u = 0.17860 + 1.41277I
a = −0.040682 + 0.405502I
b = 0.92757 − 1.07299I
4.99294 − 1.76578I −7.97967 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.17860 − 1.41277I
a = −0.040682 − 0.405502I
b = 0.92757 + 1.07299I
4.99294 + 1.76578I −7.97967 + 0.I
u = 0.24778 + 1.41576I
a = 0.414031 − 1.342460I
b = −1.22414 + 3.01107I
3.97101 − 6.92815I −10.07608 + 5.79570I
u = 0.24778 − 1.41576I
a = 0.414031 + 1.342460I
b = −1.22414 − 3.01107I
3.97101 + 6.92815I −10.07608 − 5.79570I
u = 0.362402 + 0.388238I
a = 1.219120 − 0.628084I
b = −0.742946 + 0.324636I
−0.624317 + 0.441347I −12.18762 + 0.45370I
u = 0.362402 − 0.388238I
a = 1.219120 + 0.628084I
b = −0.742946 − 0.324636I
−0.624317 − 0.441347I −12.18762 − 0.45370I
u = 0.28464 + 1.44733I
a = −0.48786 + 2.28909I
b = 0.97778 − 4.42043I
−4.81611 − 11.32490I −11.18818 + 5.52166I
u = 0.28464 − 1.44733I
a = −0.48786 − 2.28909I
b = 0.97778 + 4.42043I
−4.81611 + 11.32490I −11.18818 − 5.52166I
u = −0.22536 + 1.45912I
a = −0.502797 − 0.538242I
b = 0.712769 + 0.835870I
8.69537 + 5.01097I 0. − 2.91597I
u = −0.22536 − 1.45912I
a = −0.502797 + 0.538242I
b = 0.712769 − 0.835870I
8.69537 − 5.01097I 0. + 2.91597I
u = 0.13561 + 1.49035I
a = −0.978732 + 0.012239I
b = 0.422885 − 0.072441I
−2.61419 + 1.12722I −9.88189 + 0.I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.13561 − 1.49035I
a = −0.978732 − 0.012239I
b = 0.422885 + 0.072441I
−2.61419 − 1.12722I −9.88189 + 0.I
u = 0.360560
a = 1.03093
b = −0.258292
−0.601323 −16.4090
8
II. I
u
2
= hb + u, u
2
+ a + 2, u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
−u
2
− 2
−u
a
7
=
1
u
2
a
8
=
1
u
2
a
11
=
−u
u
a
12
=
−1
−u
2
− u + 1
a
5
=
u
2
+ 1
−u
2
a
2
=
−2u
2
− 3
u
2
− u
a
1
=
−u
2
− 1
u
2
a
4
=
−u
2
− 2
−u
a
9
=
u
u
2
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
− 3u − 14
9
(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
9
u
3
+ 2u − 1
c
8
, c
10
, c
12
u
3
+ 2u + 1
c
11
u
3
+ 3u
2
+ 5u + 2
10
(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
9
, c
10
, c
12
y
3
+ 4y
2
+ 4y −1
c
11
y
3
+ y
2
+ 13y −4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 0.102785 + 0.665457I
b = 0.22670 − 1.46771I
7.79580 + 5.13794I −11.21712 − 3.73768I
u = −0.22670 − 1.46771I
a = 0.102785 − 0.665457I
b = 0.22670 + 1.46771I
7.79580 − 5.13794I −11.21712 + 3.73768I
u = 0.453398
a = −2.20557
b = −0.453398
−2.43213 −15.5660
12
III. I
u
3
= h−u
2
+ b − u, u
3
+ u
2
+ a + 2u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
−u
3
− u
2
− 2u − 1
u
2
+ u
a
7
=
1
u
2
a
8
=
1
u
2
a
11
=
−u
u
a
12
=
−u
3
− 2u
−1
a
5
=
u
2
+ 1
−u
2
a
2
=
−u
3
− 2u
2
− 2u − 2
2u
2
+ u
a
1
=
−u
2
− 1
u
2
a
4
=
−u
3
− u
2
− 2u − 1
u
2
+ u
a
9
=
2u
3
+ u
2
+ 3u + 3
−u
3
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
3
− 2u
2
− 6u − 17
13
(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
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
8
, c
10
, c
12
u
4
− u
3
+ 2u
2
− 2u + 1
c
11
(u
2
− u + 1)
2
14
(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
9
, c
10
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
11
(y
2
+ y + 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −0.070696 − 0.758745I
b = −0.429304 − 0.107280I
1.64493 + 2.02988I −14.2631 − 3.6750I
u = −0.621744 − 0.440597I
a = −0.070696 + 0.758745I
b = −0.429304 + 0.107280I
1.64493 − 2.02988I −14.2631 + 3.6750I
u = 0.121744 + 1.306620I
a = 1.070700 − 0.758745I
b = −1.57070 + 1.62477I
1.64493 − 2.02988I −11.23686 + 2.38721I
u = 0.121744 − 1.306620I
a = 1.070700 + 0.758745I
b = −1.57070 − 1.62477I
1.64493 + 2.02988I −11.23686 − 2.38721I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
32
+ 44u
31
+ ··· + 48u + 1)
c
2
((u − 1)
7
)(u
32
− 8u
31
+ ··· + 24u
2
− 1)
c
3
, c
7
u
7
(u
32
− u
31
+ ··· + 192u + 128)
c
4
((u + 1)
7
)(u
32
− 8u
31
+ ··· + 24u
2
− 1)
c
5
, c
6
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
8
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
32
+ 2u
31
+ ··· − 3u − 1)
c
9
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
32
− 6u
31
+ ··· − 39u + 19)
c
10
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
11
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
32
− 2u
31
+ ··· − 40u − 8)
c
12
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
32
− 6u
31
+ ··· − 39u + 19)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
32
− 104y
31
+ ··· − 1812y + 1)
c
2
, c
4
((y −1)
7
)(y
32
− 44y
31
+ ··· − 48y + 1)
c
3
, c
7
y
7
(y
32
− 45y
31
+ ··· − 12288y + 16384)
c
5
, c
6
, c
10
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
32
+ 30y
31
+ ··· − 9y + 1)
c
8
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
32
− 66y
31
+ ··· − 9y + 1)
c
9
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
32
+ 18y
31
+ ··· − 1673y + 361)
c
11
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
32
+ 6y
31
+ ··· + 368y + 64)
18
