12a
0794
(K12a
0794
)
A knot diagram
1
Linearized knot diagam
3 8 11 12 1 9 10 2 7 6 4 5
Solving Sequence
4,12
5 1 6 11
3,8
2 9 10 7
c
4
c
12
c
5
c
11
c
3
c
2
c
8
c
10
c
7
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
41
+ 25u
39
+ ··· + b − u, u
45
+ u
44
+ ··· + a + 1, u
47
+ 2u
46
+ ··· − 2u − 1i
I
u
2
= hb + u, a + 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 49 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
41
+25u
39
+· · ·+b−u, u
45
+u
44
+· · ·+a+1, u
47
+2u
46
+· · ·−2u−1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
1
=
−u
−u
3
+ u
a
6
=
−u
2
+ 1
−u
4
+ 2u
2
a
11
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
8
=
−u
45
− u
44
+ ··· − 9u − 1
u
41
− 25u
39
+ ··· + 8u
2
+ u
a
2
=
u
7
− 4u
5
+ 4u
3
− 2u
u
7
− 3u
5
+ u
a
9
=
−2u
46
− u
45
+ ··· − 20u
2
− 7u
−2u
46
+ 58u
44
+ ··· + 4u + 2
a
10
=
−u
7
+ 4u
5
− 4u
3
+ 2u
−u
9
+ 5u
7
− 7u
5
+ 2u
3
+ u
a
7
=
−u
46
− u
45
+ ··· − 19u
2
− 7u
−u
46
+ 29u
44
+ ··· + 3u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
46
+ 11u
45
+ ··· − u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
47
+ 15u
46
+ ··· − 8u − 16
c
2
, c
8
u
47
− u
46
+ ··· + 4u + 4
c
3
, c
4
, c
5
c
11
, c
12
u
47
+ 2u
46
+ ··· − 2u − 1
c
6
, c
7
, c
9
u
47
− 3u
46
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
47
+ 31y
46
+ ··· + 16672y −256
c
2
, c
8
y
47
+ 15y
46
+ ··· − 8y −16
c
3
, c
4
, c
5
c
11
, c
12
y
47
− 60y
46
+ ··· + 18y −1
c
6
, c
7
, c
9
y
47
− 39y
46
+ ··· + 37y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.922487 + 0.402677I
a = −0.207419 + 0.251166I
b = 1.74840 + 0.71063I
−3.29083 − 10.41820I −13.6071 + 8.5049I
u = 0.922487 − 0.402677I
a = −0.207419 − 0.251166I
b = 1.74840 − 0.71063I
−3.29083 + 10.41820I −13.6071 − 8.5049I
u = 0.879259 + 0.378637I
a = 0.457803 − 0.144632I
b = −1.55193 − 0.79861I
1.34378 − 6.10986I −9.33039 + 6.97853I
u = 0.879259 − 0.378637I
a = 0.457803 + 0.144632I
b = −1.55193 + 0.79861I
1.34378 + 6.10986I −9.33039 − 6.97853I
u = −0.938794
a = −0.801843
b = −1.64853
−5.57924 −16.5310
u = −0.868977 + 0.342519I
a = 0.139578 − 0.719067I
b = 0.79264 − 1.61212I
−2.03144 + 4.33676I −12.48751 − 4.75672I
u = −0.868977 − 0.342519I
a = 0.139578 + 0.719067I
b = 0.79264 + 1.61212I
−2.03144 − 4.33676I −12.48751 + 4.75672I
u = 1.058930 + 0.178373I
a = −0.112920 + 0.790693I
b = −0.911890 + 0.265216I
−9.70306 − 3.81664I −19.0886 + 0.I
u = 1.058930 − 0.178373I
a = −0.112920 − 0.790693I
b = −0.911890 − 0.265216I
−9.70306 + 3.81664I −19.0886 + 0.I
u = 0.921531 + 0.084288I
a = −0.162135 − 1.014360I
b = 0.416091 + 0.386772I
−3.72700 − 1.88166I −16.5807 + 5.1639I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.921531 − 0.084288I
a = −0.162135 + 1.014360I
b = 0.416091 − 0.386772I
−3.72700 + 1.88166I −16.5807 − 5.1639I
u = 0.823939 + 0.323444I
a = −0.758346 − 0.139257I
b = 1.28476 + 0.85339I
−1.70171 − 1.67890I −12.57865 + 4.15714I
u = 0.823939 − 0.323444I
a = −0.758346 + 0.139257I
b = 1.28476 − 0.85339I
−1.70171 + 1.67890I −12.57865 − 4.15714I
u = −0.788716 + 0.376916I
a = 0.038734 + 0.507346I
b = −0.37806 + 1.48663I
1.90040 + 0.48777I −7.77695 − 1.43919I
u = −0.788716 − 0.376916I
a = 0.038734 − 0.507346I
b = −0.37806 − 1.48663I
1.90040 − 0.48777I −7.77695 + 1.43919I
u = −0.722890 + 0.438021I
a = −0.212238 − 0.290704I
b = −0.01128 − 1.43133I
−2.10005 − 3.33842I −12.59413 + 1.70307I
u = −0.722890 − 0.438021I
a = −0.212238 + 0.290704I
b = −0.01128 + 1.43133I
−2.10005 + 3.33842I −12.59413 − 1.70307I
u = −0.707872
a = 0.283787
b = 0.561800
−1.23668 −7.22640
u = −0.091614 + 0.630031I
a = 1.98364 + 0.95299I
b = −0.058376 − 0.279803I
−0.19552 + 6.93392I −8.85712 − 6.12963I
u = −0.091614 − 0.630031I
a = 1.98364 − 0.95299I
b = −0.058376 + 0.279803I
−0.19552 − 6.93392I −8.85712 + 6.12963I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.359859 + 0.483165I
a = −0.656484 − 0.454728I
b = 0.334069 + 0.541724I
−5.17789 + 1.63722I −14.9295 − 4.4051I
u = −0.359859 − 0.483165I
a = −0.656484 + 0.454728I
b = 0.334069 − 0.541724I
−5.17789 − 1.63722I −14.9295 + 4.4051I
u = −0.042621 + 0.595199I
a = −2.13353 − 0.70781I
b = 0.033126 + 0.376700I
4.14501 + 2.81372I −3.74235 − 3.47320I
u = −0.042621 − 0.595199I
a = −2.13353 + 0.70781I
b = 0.033126 − 0.376700I
4.14501 − 2.81372I −3.74235 + 3.47320I
u = 0.026672 + 0.550563I
a = 2.30737 + 0.41380I
b = −0.041983 − 0.497420I
0.68616 − 1.29934I −6.70637 + 0.78568I
u = 0.026672 − 0.550563I
a = 2.30737 − 0.41380I
b = −0.041983 + 0.497420I
0.68616 + 1.29934I −6.70637 − 0.78568I
u = 1.60695 + 0.08305I
a = −0.81808 − 1.54422I
b = −1.04952 − 1.97872I
−9.99222 + 1.51631I 0
u = 1.60695 − 0.08305I
a = −0.81808 + 1.54422I
b = −1.04952 + 1.97872I
−9.99222 − 1.51631I 0
u = 1.64487
a = −1.09171
b = −1.39259
−9.58532 0
u = 1.65147 + 0.08372I
a = 1.39395 + 1.50228I
b = 1.79104 + 1.91409I
−6.56652 − 2.13795I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.65147 − 0.08372I
a = 1.39395 − 1.50228I
b = 1.79104 − 1.91409I
−6.56652 + 2.13795I 0
u = −1.67028 + 0.07675I
a = −2.75333 + 1.26536I
b = −4.10758 + 2.30271I
−10.45730 + 3.15004I 0
u = −1.67028 − 0.07675I
a = −2.75333 − 1.26536I
b = −4.10758 − 2.30271I
−10.45730 − 3.15004I 0
u = −0.206634 + 0.252843I
a = 1.085750 − 0.522131I
b = −0.141040 − 0.446675I
−0.336379 + 0.800443I −8.09599 − 8.50563I
u = −0.206634 − 0.252843I
a = 1.085750 + 0.522131I
b = −0.141040 + 0.446675I
−0.336379 − 0.800443I −8.09599 + 8.50563I
u = 1.67880 + 0.08607I
a = −1.81262 − 1.55075I
b = −2.32616 − 1.96566I
−10.96150 − 5.96484I 0
u = 1.67880 − 0.08607I
a = −1.81262 + 1.55075I
b = −2.32616 + 1.96566I
−10.96150 + 5.96484I 0
u = −1.67889 + 0.09711I
a = 2.97433 − 0.87020I
b = 4.36294 − 1.66384I
−7.59134 + 7.93077I 0
u = −1.67889 − 0.09711I
a = 2.97433 + 0.87020I
b = 4.36294 + 1.66384I
−7.59134 − 7.93077I 0
u = −1.69341 + 0.01645I
a = −0.66308 + 1.50099I
b = −0.98851 + 2.84605I
−13.01140 + 2.24134I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.69341 − 0.01645I
a = −0.66308 − 1.50099I
b = −0.98851 − 2.84605I
−13.01140 − 2.24134I 0
u = −1.69112 + 0.10759I
a = −3.04031 + 0.58781I
b = −4.39614 + 1.23773I
−12.4253 + 12.4182I 0
u = −1.69112 − 0.10759I
a = −3.04031 − 0.58781I
b = −4.39614 − 1.23773I
−12.4253 − 12.4182I 0
u = 1.69743
a = 2.16202
b = 2.74953
−14.9500 0
u = −1.72282 + 0.03849I
a = 1.207830 − 0.382415I
b = 1.73369 − 1.18160I
−19.6013 + 4.6509I 0
u = −1.72282 − 0.03849I
a = 1.207830 + 0.382415I
b = 1.73369 + 1.18160I
−19.6013 − 4.6509I 0
u = 0.239957
a = −3.06923
b = 0.661230
−2.02254 −2.40680
9
II. I
u
2
= hb + u, a + 1, u
2
− u − 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u + 1
a
1
=
−u
−u − 1
a
6
=
−u
−u
a
11
=
u
u
a
3
=
−u
−u − 1
a
8
=
−1
−u
a
2
=
−u
−u − 1
a
9
=
−1
−u
a
10
=
u
u
a
7
=
−u − 1
−2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
10
u
2
c
3
, c
4
, c
5
u
2
− u − 1
c
6
, c
7
(u − 1)
2
c
9
(u + 1)
2
c
11
, c
12
u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
c
10
y
2
c
3
, c
4
, c
5
c
11
, c
12
y
2
− 3y + 1
c
6
, c
7
, c
9
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.00000
b = 0.618034
−2.63189 −17.0000
u = 1.61803
a = −1.00000
b = −1.61803
−10.5276 −17.0000
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
u
2
(u
47
+ 15u
46
+ ··· − 8u − 16)
c
2
, c
8
u
2
(u
47
− u
46
+ ··· + 4u + 4)
c
3
, c
4
, c
5
(u
2
− u − 1)(u
47
+ 2u
46
+ ··· − 2u − 1)
c
6
, c
7
((u − 1)
2
)(u
47
− 3u
46
+ ··· − u + 1)
c
9
((u + 1)
2
)(u
47
− 3u
46
+ ··· − u + 1)
c
11
, c
12
(u
2
+ u − 1)(u
47
+ 2u
46
+ ··· − 2u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
2
(y
47
+ 31y
46
+ ··· + 16672y −256)
c
2
, c
8
y
2
(y
47
+ 15y
46
+ ··· − 8y −16)
c
3
, c
4
, c
5
c
11
, c
12
(y
2
− 3y + 1)(y
47
− 60y
46
+ ··· + 18y −1)
c
6
, c
7
, c
9
((y −1)
2
)(y
47
− 39y
46
+ ··· + 37y −1)
15
