12n
0077
(K12n
0077
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 3 5 12 11 6 9 8
Solving Sequence
5,10 3,6
7 8 11 2 1 4 9 12
c
5
c
6
c
7
c
10
c
2
c
1
c
4
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
25
− u
24
+ ··· + b + 1, u
22
− 3u
20
+ ··· + a + 4u, u
26
− 2u
25
+ ··· − 2u − 1i
I
u
2
= hb + 1, u
4
− u
2
+ a + u + 2, u
5
− u
4
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 31 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
25
−u
24
+· · ·+b +1, u
22
−3u
20
+· · ·+a +4u, u
26
−2u
25
+· · ·−2u −1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−u
22
+ 3u
20
+ ··· − 4u
2
− 4u
−u
25
+ u
24
+ ··· − u − 1
a
6
=
1
u
2
a
7
=
−u
9
− 3u
5
− u
−u
9
+ u
7
− 3u
5
+ 2u
3
− u
a
8
=
−u
7
− 2u
3
−u
9
+ u
7
− 3u
5
+ 2u
3
− u
a
11
=
−u
−u
3
+ u
a
2
=
−u
25
+ u
24
+ ··· − 5u − 1
−u
25
+ u
24
+ ··· − u − 1
a
1
=
−u
9
− 3u
5
− u
−u
11
+ u
9
− 4u
7
+ 3u
5
− 3u
3
+ u
a
4
=
−2u
25
+ 2u
24
+ ··· − 7u − 1
−u
25
+ u
24
+ ··· − 2u − 1
a
9
=
u
3
u
5
− u
3
+ u
a
12
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
25
+ u
24
+ 8u
23
− 10u
22
− 26u
21
+ 30u
20
+ 57u
19
− 86u
18
−
103u
17
+ 156u
16
+ 148u
15
− 242u
14
− 181u
13
+ 278u
12
+ 205u
11
− 248u
10
− 197u
9
+
159u
8
+ 194u
7
− 60u
6
− 148u
5
+ 8u
4
+ 77u
3
+ 10u
2
− 19u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 34u
25
+ ··· + 68u + 1
c
2
, c
4
u
26
− 6u
25
+ ··· + 34u
2
− 1
c
3
, c
6
u
26
+ u
25
+ ··· + 96u + 32
c
5
, c
10
u
26
− 2u
25
+ ··· − 2u − 1
c
7
u
26
− 2u
25
+ ··· − 2u − 1
c
8
, c
9
, c
11
c
12
u
26
+ 6u
25
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 78y
25
+ ··· − 1224y + 1
c
2
, c
4
y
26
− 34y
25
+ ··· − 68y + 1
c
3
, c
6
y
26
− 33y
25
+ ··· − 1536y + 1024
c
5
, c
10
y
26
− 6y
25
+ ··· − 10y + 1
c
7
y
26
− 54y
25
+ ··· − 10y + 1
c
8
, c
9
, c
11
c
12
y
26
+ 30y
25
+ ··· + 18y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.03768
a = 3.52363
b = 1.79564
−14.3257 −18.8510
u = −0.848363 + 0.365549I
a = −1.78442 − 1.39985I
b = −0.876527 + 0.552462I
−2.09559 + 3.16364I −16.0021 − 6.5670I
u = −0.848363 − 0.365549I
a = −1.78442 + 1.39985I
b = −0.876527 − 0.552462I
−2.09559 − 3.16364I −16.0021 + 6.5670I
u = 0.743105 + 0.536823I
a = 0.594036 − 0.232636I
b = 0.265907 − 0.097994I
1.42344 − 2.05884I −4.65256 + 4.58362I
u = 0.743105 − 0.536823I
a = 0.594036 + 0.232636I
b = 0.265907 + 0.097994I
1.42344 + 2.05884I −4.65256 − 4.58362I
u = −1.016900 + 0.465737I
a = 2.21366 + 1.97366I
b = 1.76028 − 0.15334I
−11.56550 + 6.15142I −15.5995 − 5.2395I
u = −1.016900 − 0.465737I
a = 2.21366 − 1.97366I
b = 1.76028 + 0.15334I
−11.56550 − 6.15142I −15.5995 + 5.2395I
u = −0.340992 + 0.772246I
a = 0.190362 − 0.001300I
b = 1.70063 + 0.08748I
−9.33177 − 1.67049I −11.65109 + 0.28027I
u = −0.340992 − 0.772246I
a = 0.190362 + 0.001300I
b = 1.70063 − 0.08748I
−9.33177 + 1.67049I −11.65109 − 0.28027I
u = 0.780793 + 0.228604I
a = −2.51035 + 0.76802I
b = −1.168300 + 0.232886I
−2.90735 − 0.78726I −17.0746 + 5.9643I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.780793 − 0.228604I
a = −2.51035 − 0.76802I
b = −1.168300 − 0.232886I
−2.90735 + 0.78726I −17.0746 − 5.9643I
u = −0.901624 + 0.824241I
a = −0.978749 − 0.966956I
b = −1.49560 − 0.03445I
3.16228 + 3.07757I −10.53156 − 2.75315I
u = −0.901624 − 0.824241I
a = −0.978749 + 0.966956I
b = −1.49560 + 0.03445I
3.16228 − 3.07757I −10.53156 + 2.75315I
u = 0.875285 + 0.858337I
a = 0.452866 − 0.465694I
b = −0.617869 + 0.854688I
5.39540 − 0.25204I −9.68980 − 0.25577I
u = 0.875285 − 0.858337I
a = 0.452866 + 0.465694I
b = −0.617869 − 0.854688I
5.39540 + 0.25204I −9.68980 + 0.25577I
u = 0.840955 + 0.925824I
a = 0.193479 + 0.004136I
b = 1.65874 − 0.26853I
−2.27912 + 3.95861I −11.45131 − 0.83940I
u = 0.840955 − 0.925824I
a = 0.193479 − 0.004136I
b = 1.65874 + 0.26853I
−2.27912 − 3.95861I −11.45131 + 0.83940I
u = 0.939409 + 0.834109I
a = −0.76938 + 1.23390I
b = −0.687301 − 0.869387I
5.19384 − 6.03805I −10.26289 + 5.25215I
u = 0.939409 − 0.834109I
a = −0.76938 − 1.23390I
b = −0.687301 + 0.869387I
5.19384 + 6.03805I −10.26289 − 5.25215I
u = −0.931297 + 0.895111I
a = 0.372833 + 0.308620I
b = 0.537292 + 0.017409I
10.10520 + 3.30342I −2.39471 − 2.26919I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.931297 − 0.895111I
a = 0.372833 − 0.308620I
b = 0.537292 − 0.017409I
10.10520 − 3.30342I −2.39471 + 2.26919I
u = 0.998011 + 0.849444I
a = 0.83951 − 1.88704I
b = 1.68410 + 0.28961I
−2.78496 − 10.50130I −12.09690 + 5.41863I
u = 0.998011 − 0.849444I
a = 0.83951 + 1.88704I
b = 1.68410 − 0.28961I
−2.78496 + 10.50130I −12.09690 − 5.41863I
u = −0.527536
a = 0.849795
b = −0.171524
−0.708379 −14.2100
u = −0.393456 + 0.342390I
a = 0.999444 + 0.387392I
b = −0.573413 − 0.271251I
−0.780751 − 0.150062I −11.06268 − 0.12594I
u = −0.393456 − 0.342390I
a = 0.999444 − 0.387392I
b = −0.573413 + 0.271251I
−0.780751 + 0.150062I −11.06268 + 0.12594I
7
II. I
u
2
= hb + 1, u
4
− u
2
+ a + u + 2, u
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−u
4
+ u
2
− u − 2
−1
a
6
=
1
u
2
a
7
=
1
u
2
a
8
=
−u
2
+ 1
u
2
a
11
=
−u
−u
3
+ u
a
2
=
−u
4
+ u
2
− u − 3
−1
a
1
=
−1
0
a
4
=
−u
4
+ u
2
− u − 2
−1
a
9
=
u
3
u
4
− u
3
− u
2
+ 1
a
12
=
−u
4
+ u
2
− 1
u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
3
+ 3u
2
− u − 14
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
6
u
5
c
4
(u + 1)
5
c
5
u
5
− u
4
+ u
2
+ u − 1
c
7
, c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
8
, c
9
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
10
u
5
+ u
4
− u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
6
y
5
c
5
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
7
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
a = −0.278580 − 1.055720I
b = −1.00000
0.17487 + 2.21397I −12.88087 − 4.04855I
u = −0.758138 − 0.584034I
a = −0.278580 + 1.055720I
b = −1.00000
0.17487 − 2.21397I −12.88087 + 4.04855I
u = 0.935538 + 0.903908I
a = −0.020316 + 0.590570I
b = −1.00000
9.31336 − 3.33174I −13.28666 + 2.53508I
u = 0.935538 − 0.903908I
a = −0.020316 − 0.590570I
b = −1.00000
9.31336 + 3.33174I −13.28666 − 2.53508I
u = 0.645200
a = −2.40221
b = −1.00000
−2.52712 −13.6650
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
26
+ 34u
25
+ ··· + 68u + 1)
c
2
((u − 1)
5
)(u
26
− 6u
25
+ ··· + 34u
2
− 1)
c
3
, c
6
u
5
(u
26
+ u
25
+ ··· + 96u + 32)
c
4
((u + 1)
5
)(u
26
− 6u
25
+ ··· + 34u
2
− 1)
c
5
(u
5
− u
4
+ u
2
+ u − 1)(u
26
− 2u
25
+ ··· − 2u − 1)
c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
26
− 2u
25
+ ··· − 2u − 1)
c
8
, c
9
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
26
+ 6u
25
+ ··· + 10u + 1)
c
10
(u
5
+ u
4
− u
2
+ u + 1)(u
26
− 2u
25
+ ··· − 2u − 1)
c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
26
+ 6u
25
+ ··· + 10u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
26
− 78y
25
+ ··· − 1224y + 1)
c
2
, c
4
((y −1)
5
)(y
26
− 34y
25
+ ··· − 68y + 1)
c
3
, c
6
y
5
(y
26
− 33y
25
+ ··· − 1536y + 1024)
c
5
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
26
− 6y
25
+ ··· − 10y + 1)
c
7
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
26
− 54y
25
+ ··· − 10y + 1)
c
8
, c
9
, c
11
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
26
+ 30y
25
+ ··· + 18y + 1)
13
