12n
0680
(K12n
0680
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 10 3 12 11 5 7 8 9
Solving Sequence
7,12 4,8
3 6 11 9 1 10 5 2
c
7
c
3
c
6
c
11
c
8
c
12
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h3449u
20
+ 3669u
19
+ ··· + 71498b − 42457, −23077u
20
− 82531u
19
+ ··· + 71498a + 43556,
u
21
+ 4u
20
+ ··· − 2u + 1i
I
u
2
= hb, u
5
+ 2u
4
+ 4u
3
+ 4u
2
+ a + 3u + 2, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
I
u
3
= h−au + b − u, −u
2
a + a
2
+ au − 3u
2
+ 2u − 4, u
3
− u
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
= h3449u
20
+ 3669u
19
+ · · · + 71498b − 42457, −23077u
20
− 82531u
19
+
· · · + 71498a + 43556, u
21
+ 4u
20
+ · · · − 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
0.322764u
20
+ 1.15431u
19
+ ··· + 4.24348u − 0.609192
−0.0482391u
20
− 0.0513161u
19
+ ··· − 0.0347422u + 0.593821
a
8
=
1
u
2
a
3
=
0.274525u
20
+ 1.10300u
19
+ ··· + 4.20873u − 0.0153711
−0.0482391u
20
− 0.0513161u
19
+ ··· − 0.0347422u + 0.593821
a
6
=
−0.224552u
20
− 1.13479u
19
+ ··· − 2.45920u + 0.935718
0.166117u
20
+ 0.596254u
19
+ ··· + 0.165739u − 0.324387
a
11
=
u
u
3
+ u
a
9
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
10
=
u
3
+ 2u
u
3
+ u
a
5
=
−0.406179u
20
− 1.57648u
19
+ ··· − 2.97737u + 0.847101
−0.0482391u
20
− 0.0513161u
19
+ ··· − 0.0347422u − 0.406179
a
2
=
0.0868835u
20
+ 0.255951u
19
+ ··· + 1.16347u + 0.290428
0.0482391u
20
+ 0.0513161u
19
+ ··· + 0.0347422u + 0.406179
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
58626
35749
u
20
−
459843
71498
u
19
+ ··· +
105825
71498
u −
777975
71498
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
21
− 10u
20
+ ··· − 4u − 1
c
3
, c
6
u
21
+ 4u
20
+ ··· − 128u + 64
c
5
, c
9
u
21
− 2u
20
+ ··· + 224u + 64
c
7
, c
8
, c
11
u
21
− 4u
20
+ ··· − 2u − 1
c
10
, c
12
u
21
+ 4u
20
+ ··· − 304u − 97
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
21
− 4y
20
+ ··· + 152y −1
c
3
, c
6
y
21
+ 30y
20
+ ··· + 90112y −4096
c
5
, c
9
y
21
+ 28y
20
+ ··· + 82944y −4096
c
7
, c
8
, c
11
y
21
+ 22y
20
+ ··· − 10y −1
c
10
, c
12
y
21
+ 14y
20
+ ··· − 131266y −9409
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.904622 + 0.417722I
a = 1.45274 − 0.01148I
b = 0.92182 − 2.09929I
6.05963 + 7.34750I −14.4555 − 4.2838I
u = −0.904622 − 0.417722I
a = 1.45274 + 0.01148I
b = 0.92182 + 2.09929I
6.05963 − 7.34750I −14.4555 + 4.2838I
u = −0.766571 + 0.752408I
a = −1.037380 + 0.144450I
b = 0.29405 + 2.35369I
7.09375 − 1.76941I −12.87851 − 0.26190I
u = −0.766571 − 0.752408I
a = −1.037380 − 0.144450I
b = 0.29405 − 2.35369I
7.09375 + 1.76941I −12.87851 + 0.26190I
u = 0.182461 + 1.208850I
a = −0.276557 − 0.201585I
b = −0.119527 + 0.423528I
2.74625 − 2.07596I −5.86030 + 3.17371I
u = 0.182461 − 1.208850I
a = −0.276557 + 0.201585I
b = −0.119527 − 0.423528I
2.74625 + 2.07596I −5.86030 − 3.17371I
u = −0.734798
a = −1.01042
b = −1.32151
−10.5256 −25.3380
u = 0.200947 + 1.339400I
a = −1.31937 + 2.00455I
b = −0.552581 − 0.279603I
1.78822 − 2.54403I −24.8123 + 5.5170I
u = 0.200947 − 1.339400I
a = −1.31937 − 2.00455I
b = −0.552581 + 0.279603I
1.78822 + 2.54403I −24.8123 − 5.5170I
u = −0.341638 + 1.336290I
a = 0.828590 − 0.644391I
b = −1.256020 − 0.461833I
−6.25683 + 3.88389I −16.6815 − 2.9719I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.341638 − 1.336290I
a = 0.828590 + 0.644391I
b = −1.256020 + 0.461833I
−6.25683 − 3.88389I −16.6815 + 2.9719I
u = 0.512237
a = −5.60059
b = −0.289328
−2.52473 −76.9450
u = −0.35905 + 1.50519I
a = 0.78634 + 2.07126I
b = 1.39494 − 1.98909I
12.2217 + 11.9532I −11.90704 − 5.00504I
u = −0.35905 − 1.50519I
a = 0.78634 − 2.07126I
b = 1.39494 + 1.98909I
12.2217 − 11.9532I −11.90704 + 5.00504I
u = 0.06735 + 1.55292I
a = −1.37220 + 0.85690I
b = 1.86679 − 0.98776I
5.79361 − 0.73866I −10.20116 + 0.28003I
u = 0.06735 − 1.55292I
a = −1.37220 − 0.85690I
b = 1.86679 + 0.98776I
5.79361 + 0.73866I −10.20116 − 0.28003I
u = −0.21280 + 1.64374I
a = −0.63755 − 2.25205I
b = −0.62112 + 3.12830I
15.1889 + 1.8962I −10.30157 − 0.70895I
u = −0.21280 − 1.64374I
a = −0.63755 + 2.25205I
b = −0.62112 − 3.12830I
15.1889 − 1.8962I −10.30157 + 0.70895I
u = 0.334401
a = −0.860564
b = 0.297521
−0.669543 −14.6190
u = 0.077997 + 0.278544I
a = −0.18883 + 1.76255I
b = 0.728300 − 0.059767I
−0.764279 + 0.134030I −11.95123 + 0.33972I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.077997 − 0.278544I
a = −0.18883 − 1.76255I
b = 0.728300 + 0.059767I
−0.764279 − 0.134030I −11.95123 − 0.33972I
7
II.
I
u
2
= hb, u
5
+ 2u
4
+ 4u
3
+ 4u
2
+ a + 3u + 2, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
5
− 2u
4
− 4u
3
− 4u
2
− 3u − 2
0
a
8
=
1
u
2
a
3
=
−u
5
− 2u
4
− 4u
3
− 4u
2
− 3u − 2
0
a
6
=
1
0
a
11
=
u
u
3
+ u
a
9
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
5
− 2u
3
− u
−u
5
− u
4
− 2u
3
− u
2
− u + 1
a
10
=
u
3
+ 2u
u
3
+ u
a
5
=
u
5
+ 2u
3
+ u
u
5
+ u
4
+ 2u
3
+ u
2
+ u − 1
a
2
=
−2u
5
− 2u
4
− 6u
3
− 4u
2
− 4u − 2
−u
5
− u
4
− 2u
3
− u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
− u
4
+ 5u
3
+ 4u
2
+ 7u − 7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
7
, c
8
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
9
, c
10
, c
12
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
11
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
7
, c
8
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = −0.422181
b = 0
−9.30502 −14.4810
u = 0.138835 + 1.234450I
a = 0.26610 + 1.72116I
b = 0
1.31531 − 1.97241I −15.7816 + 4.5012I
u = 0.138835 − 1.234450I
a = 0.26610 − 1.72116I
b = 0
1.31531 + 1.97241I −15.7816 − 4.5012I
u = −0.408802 + 1.276380I
a = −0.417699 − 0.090629I
b = 0
−5.34051 + 4.59213I −11.43321 − 5.39767I
u = −0.408802 − 1.276380I
a = −0.417699 + 0.090629I
b = 0
−5.34051 − 4.59213I −11.43321 + 5.39767I
u = 0.413150
a = −4.27462
b = 0
−2.38379 −3.08970
11
III. I
u
3
= h−au + b − u, −u
2
a + a
2
+ au − 3u
2
+ 2u − 4, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
au + u
a
8
=
1
u
2
a
3
=
au + a + u
au + u
a
6
=
−u
2
− a + u − 3
−au − u − 1
a
11
=
u
u
2
− u + 1
a
9
=
u
2
+ 1
u
2
− u + 1
a
1
=
−1
0
a
10
=
u
2
+ 1
u
2
− u + 1
a
5
=
−u
2
− a + u − 3
−au − u − 1
a
2
=
au − u
2
− a + 2u − 4
−au − u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
2
a − 5u
2
+ 3a + 7u − 15
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)
3
c
4
, c
6
(u
2
− u − 1)
3
c
5
, c
9
u
6
c
7
, c
8
(u
3
− u
2
+ 2u − 1)
2
c
10
, c
12
(u
3
− u
2
+ 1)
2
c
11
(u
3
+ u
2
+ 2u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
(y
2
− 3y + 1)
3
c
5
, c
9
y
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(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 = 0.215080 + 1.307140I
a = −1.075750 + 0.460350I
b = −0.618034
2.03717 − 2.82812I −12.9982 + 11.8301I
u = 0.215080 + 1.307140I
a = −0.80169 − 1.20521I
b = 1.61803
−5.85852 − 2.82812I −13.61882 − 1.93520I
u = 0.215080 − 1.307140I
a = −1.075750 − 0.460350I
b = −0.618034
2.03717 + 2.82812I −12.9982 − 11.8301I
u = 0.215080 − 1.307140I
a = −0.80169 + 1.20521I
b = 1.61803
−5.85852 + 2.82812I −13.61882 + 1.93520I
u = 0.569840
a = 1.83945
b = 1.61803
−9.99610 −8.90830
u = 0.569840
a = −2.08457
b = −0.618034
−2.10041 −16.8580
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
2
+ u − 1)
3
(u
21
− 10u
20
+ ··· − 4u − 1)
c
3
u
6
(u
2
+ u − 1)
3
(u
21
+ 4u
20
+ ··· − 128u + 64)
c
4
((u + 1)
6
)(u
2
− u − 1)
3
(u
21
− 10u
20
+ ··· − 4u − 1)
c
5
u
6
(u
6
− u
5
+ ··· + u − 1)(u
21
− 2u
20
+ ··· + 224u + 64)
c
6
u
6
(u
2
− u − 1)
3
(u
21
+ 4u
20
+ ··· − 128u + 64)
c
7
, c
8
(u
3
− u
2
+ 2u − 1)
2
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
21
− 4u
20
+ ··· − 2u − 1)
c
9
u
6
(u
6
+ u
5
+ ··· − u − 1)(u
21
− 2u
20
+ ··· + 224u + 64)
c
10
, c
12
(u
3
− u
2
+ 1)
2
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
21
+ 4u
20
+ ··· − 304u − 97)
c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
21
− 4u
20
+ ··· − 2u − 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
2
− 3y + 1)
3
(y
21
− 4y
20
+ ··· + 152y −1)
c
3
, c
6
y
6
(y
2
− 3y + 1)
3
(y
21
+ 30y
20
+ ··· + 90112y −4096)
c
5
, c
9
y
6
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
21
+ 28y
20
+ ··· + 82944y −4096)
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
21
+ 22y
20
+ ··· − 10y −1)
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
21
+ 14y
20
+ ··· − 131266y −9409)
17
