12n
0198
(K12n
0198
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 11 9 3 12 5 6 10 8
Solving Sequence
5,11 3,6
2 1 4 10 12 9 7 8
c
5
c
2
c
1
c
4
c
10
c
11
c
9
c
6
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
21
− u
20
+ ··· + b − u, −u
21
− u
20
+ ··· − 3u
3
+ a, u
23
+ 2u
22
+ ··· + 2u + 1i
I
u
2
= hb + 1, −u
3
+ u
2
+ a, u
4
+ u
2
+ u + 1i
I
u
3
= hb + 1, −u
3
+ a − u + 1, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 33 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
21
−u
20
+· · ·+b−u, −u
21
−u
20
+· · ·−3u
3
+a, u
23
+2u
22
+· · ·+2u+1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
u
21
+ u
20
+ ··· + u
4
+ 3u
3
u
21
+ u
20
+ ··· + u
2
+ u
a
6
=
1
−u
2
a
2
=
2u
21
+ 2u
20
+ ··· + u
2
+ u
u
21
+ u
20
+ ··· + u
2
+ u
a
1
=
u
19
+ 4u
17
+ 8u
15
+ 8u
13
+ 3u
11
− 2u
9
− 2u
7
+ u
3
−u
21
− 5u
19
+ ··· − u
3
− u
a
4
=
3u
21
+ 3u
20
+ ··· + 3u + 1
u
21
+ u
20
+ ··· + u
2
+ 2u
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
5
+ u
3
+ u
a
9
=
u
3
u
3
+ u
a
7
=
u
8
+ u
6
+ u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
8
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
22
+ 12u
21
+ 34u
20
+ 68u
19
+ 118u
18
+ 192u
17
+ 251u
16
+ 335u
15
+ 353u
14
+ 392u
13
+
355u
12
+316u
11
+243u
10
+163u
9
+106u
8
+46u
7
+14u
6
+11u
5
+15u
4
+30u
3
+15u
2
+13u+6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 47u
22
+ ··· − 11u + 1
c
2
, c
4
u
23
− 11u
22
+ ··· − 9u + 1
c
3
, c
7
u
23
− u
22
+ ··· + 2048u + 1024
c
5
, c
10
u
23
− 2u
22
+ ··· + 2u − 1
c
6
u
23
− 10u
22
+ ··· + 120u − 31
c
8
, c
12
u
23
+ 24u
21
+ ··· + 2u + 1
c
9
u
23
+ 2u
22
+ ··· − 15u
2
− 8
c
11
u
23
+ 12u
22
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 211y
22
+ ··· − 215y −1
c
2
, c
4
y
23
− 47y
22
+ ··· − 11y −1
c
3
, c
7
y
23
+ 63y
22
+ ··· + 8912896y −1048576
c
5
, c
10
y
23
+ 12y
22
+ ··· − 2y −1
c
6
y
23
− 12y
22
+ ··· − 6122y −961
c
8
, c
12
y
23
+ 48y
22
+ ··· − 2y −1
c
9
y
23
− 12y
22
+ ··· − 240y −64
c
11
y
23
+ 24y
21
+ ··· + 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.695674 + 0.794301I
a = −0.38303 + 1.50526I
b = 2.14604 − 0.04602I
−15.4705 + 2.6332I −1.74115 − 2.82837I
u = 0.695674 − 0.794301I
a = −0.38303 − 1.50526I
b = 2.14604 + 0.04602I
−15.4705 − 2.6332I −1.74115 + 2.82837I
u = −0.851428 + 0.257921I
a = 0.102425 + 0.901741I
b = 2.23097 − 0.23648I
−18.5278 + 5.0308I −2.42173 − 1.77619I
u = −0.851428 − 0.257921I
a = 0.102425 − 0.901741I
b = 2.23097 + 0.23648I
−18.5278 − 5.0308I −2.42173 + 1.77619I
u = −0.483954 + 1.020520I
a = 0.825387 − 0.274069I
b = 0.223453 + 0.115878I
−0.56646 − 3.01940I 3.36749 + 3.11832I
u = −0.483954 − 1.020520I
a = 0.825387 + 0.274069I
b = 0.223453 − 0.115878I
−0.56646 + 3.01940I 3.36749 − 3.11832I
u = 0.364715 + 1.105530I
a = 0.392472 − 0.466255I
b = −0.334457 − 0.705017I
−3.74014 + 1.10612I −5.73854 − 0.32981I
u = 0.364715 − 1.105530I
a = 0.392472 + 0.466255I
b = −0.334457 + 0.705017I
−3.74014 − 1.10612I −5.73854 + 0.32981I
u = 0.518947 + 1.123690I
a = −0.425430 + 0.139303I
b = 0.028115 + 0.688429I
−2.62716 + 6.50806I −2.44317 − 6.43144I
u = 0.518947 − 1.123690I
a = −0.425430 − 0.139303I
b = 0.028115 − 0.688429I
−2.62716 − 6.50806I −2.44317 + 6.43144I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.213623 + 0.731187I
a = −0.66221 − 1.37892I
b = −0.994946 + 0.319082I
−2.09274 + 1.02920I −5.56905 − 0.54720I
u = 0.213623 − 0.731187I
a = −0.66221 + 1.37892I
b = −0.994946 − 0.319082I
−2.09274 − 1.02920I −5.56905 + 0.54720I
u = −0.449726 + 1.155180I
a = −2.55174 + 1.46474I
b = −1.68340 − 0.16500I
−6.16024 − 4.07736I −6.51340 + 3.55333I
u = −0.449726 − 1.155180I
a = −2.55174 − 1.46474I
b = −1.68340 + 0.16500I
−6.16024 + 4.07736I −6.51340 − 3.55333I
u = −0.490965 + 0.550455I
a = 0.503661 + 0.280792I
b = 0.106838 − 0.230098I
0.861404 − 1.022110I 5.67905 + 4.33251I
u = −0.490965 − 0.550455I
a = 0.503661 − 0.280792I
b = 0.106838 + 0.230098I
0.861404 + 1.022110I 5.67905 − 4.33251I
u = −0.276568 + 1.232250I
a = 2.87251 − 0.51904I
b = 2.32781 − 0.18898I
16.1704 + 1.4493I −7.34390 + 0.38241I
u = −0.276568 − 1.232250I
a = 2.87251 + 0.51904I
b = 2.32781 + 0.18898I
16.1704 − 1.4493I −7.34390 − 0.38241I
u = 0.653892 + 0.258897I
a = 0.330772 + 0.501247I
b = −0.038798 − 0.528699I
−0.16241 − 1.94681I 1.33418 + 3.65595I
u = 0.653892 − 0.258897I
a = 0.330772 − 0.501247I
b = −0.038798 + 0.528699I
−0.16241 + 1.94681I 1.33418 − 3.65595I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.568317 + 1.180040I
a = 2.01303 − 2.60561I
b = 2.23769 + 0.30194I
18.1942 − 10.2590I −5.29878 + 5.24355I
u = −0.568317 − 1.180040I
a = 2.01303 + 2.60561I
b = 2.23769 − 0.30194I
18.1942 + 10.2590I −5.29878 − 5.24355I
u = −0.651787
a = −1.03569
b = −1.49862
−3.01079 −2.62200
7
II. I
u
2
= hb + 1, −u
3
+ u
2
+ a, u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
u
3
− u
2
−1
a
6
=
1
−u
2
a
2
=
u
3
− u
2
− 1
−1
a
1
=
−1
0
a
4
=
u
3
− u
2
−1
a
10
=
−u
u
3
+ u
a
12
=
−u
3
−u
2
a
9
=
u
3
u
3
+ u
a
7
=
u
3
+ u
2
+ u + 1
u
a
8
=
u
3
+ u
2
+ u + 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
− 4u
2
− u − 2
8
(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
8
u
4
+ u
2
+ u + 1
c
6
u
4
− 2u
3
+ 3u
2
− u + 1
c
9
u
4
− 3u
3
+ 4u
2
− 3u + 2
c
10
, c
12
u
4
+ u
2
− u + 1
c
11
u
4
+ 2u
3
+ 3u
2
+ u + 1
9
(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
8
, c
10
c
12
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
6
, c
11
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
9
y
4
− y
3
+ 2y
2
+ 7y + 4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = 0.442547 + 0.966840I
b = −1.00000
−0.66484 − 1.39709I −0.08162 + 2.95607I
u = −0.547424 − 0.585652I
a = 0.442547 − 0.966840I
b = −1.00000
−0.66484 + 1.39709I −0.08162 − 2.95607I
u = 0.547424 + 1.120870I
a = −0.94255 − 1.62772I
b = −1.00000
−4.26996 + 7.64338I −4.41838 − 7.23121I
u = 0.547424 − 1.120870I
a = −0.94255 + 1.62772I
b = −1.00000
−4.26996 − 7.64338I −4.41838 + 7.23121I
11
III. I
u
3
= hb + 1, −u
3
+ a − u + 1, u
6
− u
5
+ 2u
4
− 2u
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 − 2
−1
a
1
=
−1
0
a
4
=
u
3
+ u − 1
−1
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
5
+ u
3
+ u
a
9
=
u
3
u
3
+ u
a
7
=
u
4
+ u
2
− u + 1
u
5
+ 2u
3
− u
2
+ u − 1
a
8
=
u
4
+ u
2
− u + 1
u
5
+ 2u
3
− u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
+ 3u
3
+ u
2
+ 4u − 5
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
, c
8
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
6
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
c
9
(u
3
+ u
2
− 1)
2
c
10
, c
12
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
11
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
8
, c
10
c
12
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
6
, c
11
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
9
(y
3
− y
2
+ 2y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = −0.122561 + 0.744862I
b = −1.00000
−1.91067 − 2.82812I −4.05004 + 3.74291I
u = −0.498832 − 1.001300I
a = −0.122561 − 0.744862I
b = −1.00000
−1.91067 + 2.82812I −4.05004 − 3.74291I
u = 0.284920 + 1.115140I
a = −1.75488
b = −1.00000
−6.04826 −7.19479 + 0.27335I
u = 0.284920 − 1.115140I
a = −1.75488
b = −1.00000
−6.04826 −7.19479 − 0.27335I
u = 0.713912 + 0.305839I
a = −0.122561 + 0.744862I
b = −1.00000
−1.91067 − 2.82812I −1.25517 + 3.34054I
u = 0.713912 − 0.305839I
a = −0.122561 − 0.744862I
b = −1.00000
−1.91067 + 2.82812I −1.25517 − 3.34054I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
10
)(u
23
+ 47u
22
+ ··· − 11u + 1)
c
2
((u − 1)
10
)(u
23
− 11u
22
+ ··· − 9u + 1)
c
3
, c
7
u
10
(u
23
− u
22
+ ··· + 2048u + 1024)
c
4
((u + 1)
10
)(u
23
− 11u
22
+ ··· − 9u + 1)
c
5
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
23
− 2u
22
+ ··· + 2u − 1)
c
6
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
23
− 10u
22
+ ··· + 120u − 31)
c
8
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
23
+ 24u
21
+ ··· + 2u + 1)
c
9
((u
3
+ u
2
− 1)
2
)(u
4
− 3u
3
+ ··· − 3u + 2)(u
23
+ 2u
22
+ ··· − 15u
2
− 8)
c
10
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
23
− 2u
22
+ ··· + 2u − 1)
c
11
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
23
+ 12u
22
+ ··· − 2u − 1)
c
12
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
23
+ 24u
21
+ ··· + 2u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
10
)(y
23
− 211y
22
+ ··· − 215y −1)
c
2
, c
4
((y −1)
10
)(y
23
− 47y
22
+ ··· − 11y −1)
c
3
, c
7
y
10
(y
23
+ 63y
22
+ ··· + 8912896y −1048576)
c
5
, c
10
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
23
+ 12y
22
+ ··· − 2y −1)
c
6
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
23
− 12y
22
+ ··· − 6122y −961)
c
8
, c
12
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
23
+ 48y
22
+ ··· − 2y −1)
c
9
(y
3
− y
2
+ 2y −1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
23
− 12y
22
+ ··· − 240y −64)
c
11
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
23
+ 24y
21
+ ··· + 2y −1)
17
