12n
0850
(K12n
0850
)
A knot diagram
1
Linearized knot diagam
4 8 10 8 11 12 3 1 4 5 6 10
Solving Sequence
5,10
11 6 12
1,8
4 2 3 7 9
c
10
c
5
c
11
c
12
c
4
c
1
c
3
c
7
c
9
c
2
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
− 5u
9
− 4u
8
+ 8u
7
− u
6
− 16u
5
+ 19u
4
+ 20u
3
− 15u
2
+ 2b + 5u + 4,
− 3u
10
− 10u
9
− u
8
+ 14u
7
− 17u
6
− 17u
5
+ 47u
4
+ 13u
3
− 25u
2
+ 2a + 18u + 7,
u
11
+ 5u
10
+ 6u
9
− 4u
8
− 3u
7
+ 14u
6
− 5u
5
− 30u
4
− u
3
+ 9u
2
− 10u − 4i
I
u
2
= h−u
5
+ 4u
3
+ b − 3u, u
5
− 5u
3
+ u
2
+ a + 6u − 3, u
7
− u
6
− 5u
5
+ 5u
4
+ 6u
3
− 7u
2
+ u + 1i
I
u
3
= hb, a + 1, u + 1i
I
u
4
= h−a
3
+ b + a + 1, a
4
+ a
3
− 2a − 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 23 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
10
−5u
9
+· · ·+2b+4, −3u
10
−10u
9
+· · ·+2a+7, u
11
+5u
10
+· · ·−10u−4i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
4
− 3u
2
+ 1
−u
4
+ 2u
2
a
8
=
3
2
u
10
+ 5u
9
+ ··· − 9u −
7
2
1
2
u
10
+
5
2
u
9
+ ··· −
5
2
u − 2
a
4
=
1
4
u
10
+
3
4
u
9
+ ··· −
5
4
u − 1
−
1
2
u
10
−
3
2
u
9
+ ··· +
5
2
u + 1
a
2
=
−
1
2
u
9
−
1
2
u
8
+ ··· +
1
2
u +
3
2
5
2
u
10
+
17
2
u
9
+ ··· −
23
2
u − 6
a
3
=
−
1
4
u
10
−
3
4
u
9
+ ··· −
3
4
u
2
+
5
4
u
−
1
2
u
10
−
3
2
u
9
+ ··· +
5
2
u + 1
a
7
=
u
3
− 2u
u
5
− 3u
3
+ u
a
9
=
−u
10
−
5
2
u
9
+ ··· +
7
2
u +
5
2
−
1
2
u
10
−
5
2
u
9
+ ··· +
7
2
u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
10
− 25u
9
− 6u
8
+ 36u
7
− 35u
6
− 50u
5
+ 113u
4
+ 43u
3
− 65u
2
+ 46u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
− 7u
10
+ ··· − 2u + 2
c
2
, c
3
, c
7
c
9
u
11
+ 14u
9
+ 4u
8
+ 46u
7
+ 67u
6
− 66u
5
− 7u
4
− 30u
3
− 9u
2
− 3u − 1
c
4
, c
8
u
11
+ u
10
+ ··· − 4u − 1
c
5
, c
6
, c
10
c
11
u
11
+ 5u
10
+ ··· − 10u − 4
c
12
u
11
− 3u
10
+ ··· − 3192u − 576
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 31y
10
+ ··· + 200y −4
c
2
, c
3
, c
7
c
9
y
11
+ 28y
10
+ ··· − 9y −1
c
4
, c
8
y
11
+ 17y
10
+ ··· + 24y −1
c
5
, c
6
, c
10
c
11
y
11
− 13y
10
+ ··· + 172y −16
c
12
y
11
+ 67y
10
+ ··· + 7508160y −331776
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.18814
a = 0.348347
b = −0.566087
−5.45154 −15.3520
u = 0.651462 + 1.063750I
a = 1.81578 + 0.65642I
b = −1.77237 + 0.06609I
12.65630 − 3.45618I −10.65599 + 2.10885I
u = 0.651462 − 1.063750I
a = 1.81578 − 0.65642I
b = −1.77237 − 0.06609I
12.65630 + 3.45618I −10.65599 − 2.10885I
u = 0.496165 + 0.539201I
a = −1.042860 − 0.661841I
b = 1.138470 − 0.357731I
2.08534 − 1.86536I −10.69284 + 5.33447I
u = 0.496165 − 0.539201I
a = −1.042860 + 0.661841I
b = 1.138470 + 0.357731I
2.08534 + 1.86536I −10.69284 − 5.33447I
u = −1.53157 + 0.15203I
a = −0.354779 + 0.587363I
b = 1.12977 + 1.06011I
−4.66784 + 4.30939I −14.0390 − 6.9085I
u = −1.53157 − 0.15203I
a = −0.354779 − 0.587363I
b = 1.12977 − 1.06011I
−4.66784 − 4.30939I −14.0390 + 6.9085I
u = −0.330126
a = 0.768686
b = −0.139205
−0.487897 −20.3390
u = −1.64929 + 0.39522I
a = 0.914271 − 0.873133I
b = −1.77328 − 0.21395I
5.23140 + 8.93346I −13.21655 − 3.59394I
u = −1.64929 − 0.39522I
a = 0.914271 + 0.873133I
b = −1.77328 + 0.21395I
5.23140 − 8.93346I −13.21655 + 3.59394I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.79155
a = 0.218139
b = −0.739878
−16.4464 −7.09980
6
II. I
u
2
= h−u
5
+ 4u
3
+ b − 3u, u
5
− 5u
3
+ u
2
+ a + 6u − 3, u
7
− u
6
− 5u
5
+
5u
4
+ 6u
3
− 7u
2
+ u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
4
− 3u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
5
+ 5u
3
− u
2
− 6u + 3
u
5
− 4u
3
+ 3u
a
4
=
−u
6
+ u
5
+ 4u
4
− 5u
3
− 2u
2
+ 7u − 4
u
4
− 3u
2
+ 1
a
2
=
−2u
6
+ 2u
5
+ 10u
4
− 10u
3
− 11u
2
+ 14u − 4
u
6
− 5u
4
+ u
3
+ 6u
2
− 3u
a
3
=
−u
6
+ u
5
+ 5u
4
− 5u
3
− 5u
2
+ 7u − 3
u
4
− 3u
2
+ 1
a
7
=
u
3
− 2u
u
5
− 3u
3
+ u
a
9
=
−u
5
+ 5u
3
− u
2
− 6u + 4
−u
3
+ 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
6
− 2u
5
− 12u
4
+ 5u
3
+ 16u
2
− 3u − 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
− 6u
6
+ 12u
5
− 11u
4
+ 7u
3
− 3u
2
− 1
c
2
, c
9
u
7
− u
6
+ u
5
− 2u
4
− u
2
+ 1
c
3
, c
7
u
7
+ u
6
+ u
5
+ 2u
4
+ u
2
− 1
c
4
, c
8
u
7
− u
5
− 2u
3
+ u
2
− u + 1
c
5
, c
6
u
7
+ u
6
− 5u
5
− 5u
4
+ 6u
3
+ 7u
2
+ u − 1
c
10
, c
11
u
7
− u
6
− 5u
5
+ 5u
4
+ 6u
3
− 7u
2
+ u + 1
c
12
u
7
− 5u
6
+ 7u
5
− 5u
4
− 4u
3
− 7u
2
+ 5u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
− 12y
6
+ 26y
5
+ 11y
4
− 29y
3
− 31y
2
− 6y −1
c
2
, c
3
, c
7
c
9
y
7
+ y
6
− 3y
5
− 6y
4
− 2y
3
+ 3y
2
+ 2y −1
c
4
, c
8
y
7
− 2y
6
− 3y
5
+ 2y
4
+ 6y
3
+ 3y
2
− y −1
c
5
, c
6
, c
10
c
11
y
7
− 11y
6
+ 47y
5
− 97y
4
+ 98y
3
− 47y
2
+ 15y −1
c
12
y
7
− 11y
6
− 9y
5
− 141y
4
+ 26y
3
− 79y
2
+ 39y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.602602 + 0.366097I
a = −0.802378 − 0.958802I
b = 1.74200 − 0.23095I
3.42389 − 1.23175I −6.47743 + 5.11160I
u = 0.602602 − 0.366097I
a = −0.802378 + 0.958802I
b = 1.74200 + 0.23095I
3.42389 + 1.23175I −6.47743 − 5.11160I
u = 1.50894
a = 1.02523
b = −1.39324
−10.4234 −15.1670
u = −1.59539 + 0.14916I
a = −0.285687 + 0.511963I
b = 1.59460 + 0.65214I
−4.17528 + 3.26775I −10.72162 − 1.02180I
u = −1.59539 − 0.14916I
a = −0.285687 − 0.511963I
b = 1.59460 − 0.65214I
−4.17528 − 3.26775I −10.72162 + 1.02180I
u = −0.293328
a = 4.54991
b = −0.781203
−4.14361 −7.95280
u = 1.76997
a = −0.399006
b = 0.501243
−16.8289 −28.4820
10
III. I
u
3
= hb, a + 1, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
10
=
1
0
a
11
=
1
1
a
6
=
1
0
a
12
=
0
1
a
1
=
−1
1
a
8
=
−1
0
a
4
=
−1
−1
a
2
=
−3
−1
a
3
=
−2
−1
a
7
=
1
1
a
9
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u − 2
c
2
, c
4
, c
8
c
9
, c
10
, c
11
c
12
u + 1
c
3
, c
5
, c
6
c
7
u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y −4
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
10
c
11
, c
12
y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−6.57974 −24.0000
14
IV. I
u
4
= h−a
3
+ b + a + 1, a
4
+ a
3
− 2a − 1, u − 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
0
a
11
=
1
1
a
6
=
−1
0
a
12
=
0
1
a
1
=
−1
1
a
8
=
a
a
3
− a − 1
a
4
=
a
2
−a
3
− a
2
+ a + 2
a
2
=
−a − 2
a
3
− 1
a
3
=
−a
3
+ a + 2
−a
3
− a
2
+ a + 2
a
7
=
−1
−1
a
9
=
−a
3
+ a + 1
2a
3
− a − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −14
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u − 1)
2
c
2
, c
3
, c
7
c
9
u
4
− u
3
+ 2u
2
− 4u + 1
c
4
, c
8
u
4
+ u
3
− 2u − 1
c
5
, c
6
, c
10
c
11
(u − 1)
4
c
12
(u + 1)
4
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 3y + 1)
2
c
2
, c
3
, c
7
c
9
y
4
+ 3y
3
− 2y
2
− 12y + 1
c
4
, c
8
y
4
− y
3
+ 2y
2
− 4y + 1
c
5
, c
6
, c
10
c
11
, c
12
(y −1)
4
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.15372
b = −0.618034
−5.59278 −14.0000
u = 1.00000
a = −0.809017 + 0.981593I
b = 1.61803
2.30291 −14.0000
u = 1.00000
a = −0.809017 − 0.981593I
b = 1.61803
2.30291 −14.0000
u = 1.00000
a = −0.535687
b = −0.618034
−5.59278 −14.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 2)(u
2
+ u − 1)
2
(u
7
− 6u
6
+ 12u
5
− 11u
4
+ 7u
3
− 3u
2
− 1)
· (u
11
− 7u
10
+ ··· − 2u + 2)
c
2
, c
9
(u + 1)(u
4
− u
3
+ 2u
2
− 4u + 1)(u
7
− u
6
+ u
5
− 2u
4
− u
2
+ 1)
· (u
11
+ 14u
9
+ 4u
8
+ 46u
7
+ 67u
6
− 66u
5
− 7u
4
− 30u
3
− 9u
2
− 3u − 1)
c
3
, c
7
(u − 1)(u
4
− u
3
+ 2u
2
− 4u + 1)(u
7
+ u
6
+ u
5
+ 2u
4
+ u
2
− 1)
· (u
11
+ 14u
9
+ 4u
8
+ 46u
7
+ 67u
6
− 66u
5
− 7u
4
− 30u
3
− 9u
2
− 3u − 1)
c
4
, c
8
(u + 1)(u
4
+ u
3
− 2u − 1)(u
7
− u
5
− 2u
3
+ u
2
− u + 1)
· (u
11
+ u
10
+ ··· − 4u − 1)
c
5
, c
6
(u − 1)
5
(u
7
+ u
6
− 5u
5
− 5u
4
+ 6u
3
+ 7u
2
+ u − 1)
· (u
11
+ 5u
10
+ ··· − 10u − 4)
c
10
, c
11
(u − 1)
4
(u + 1)(u
7
− u
6
− 5u
5
+ 5u
4
+ 6u
3
− 7u
2
+ u + 1)
· (u
11
+ 5u
10
+ ··· − 10u − 4)
c
12
(u + 1)
5
(u
7
− 5u
6
+ 7u
5
− 5u
4
− 4u
3
− 7u
2
+ 5u + 1)
· (u
11
− 3u
10
+ ··· − 3192u − 576)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −4)(y
2
− 3y + 1)
2
(y
7
− 12y
6
+ ··· − 6y −1)
· (y
11
− 31y
10
+ ··· + 200y −4)
c
2
, c
3
, c
7
c
9
(y −1)(y
4
+ 3y
3
− 2y
2
− 12y + 1)
· (y
7
+ y
6
+ ··· + 2y −1)(y
11
+ 28y
10
+ ··· − 9y −1)
c
4
, c
8
(y −1)(y
4
− y
3
+ 2y
2
− 4y + 1)(y
7
− 2y
6
+ ··· − y −1)
· (y
11
+ 17y
10
+ ··· + 24y −1)
c
5
, c
6
, c
10
c
11
(y −1)
5
(y
7
− 11y
6
+ 47y
5
− 97y
4
+ 98y
3
− 47y
2
+ 15y −1)
· (y
11
− 13y
10
+ ··· + 172y −16)
c
12
(y −1)
5
(y
7
− 11y
6
− 9y
5
− 141y
4
+ 26y
3
− 79y
2
+ 39y −1)
· (y
11
+ 67y
10
+ ··· + 7508160y −331776)
20
