12n
0152
(K12n
0152
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 12 11 4 5 6 7 10 9
Solving Sequence
6,11 3,7
4 10 12 5 2 9 1 8
c
6
c
3
c
10
c
11
c
5
c
2
c
9
c
12
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
42
− u
41
+ ··· − u
3
+ b, −u
42
+ u
41
+ ··· + a − 1, u
44
− 2u
43
+ ··· − 2u + 1i
I
u
2
= hb + u, −u
3
+ a − u + 1, u
4
+ u
2
− u + 1i
I
u
3
= h−u
5
− u
4
− 2u
3
− 2u
2
+ b − 2u − 1, u
4
+ u
2
+ a, u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 54 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
42
−u
41
+· · ·−u
3
+b, −u
42
+u
41
+· · ·+a−1, u
44
−2u
43
+· · ·−2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
42
− u
41
+ ··· + 4u
3
+ 1
−u
42
+ u
41
+ ··· − 5u
4
+ u
3
a
7
=
1
u
2
a
4
=
−u
43
+ 3u
42
+ ··· − 2u + 2
u
43
− 3u
42
+ ··· − u
2
+ u
a
10
=
u
u
3
+ u
a
12
=
u
3
u
5
+ u
3
+ u
a
5
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
2
=
u
39
− u
38
+ ··· + u + 1
u
41
− u
40
+ ··· + u
3
+ u
2
a
9
=
−u
3
u
3
+ u
a
1
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
8
=
u
21
+ 4u
19
+ 9u
17
+ 12u
15
+ 12u
13
+ 10u
11
+ 9u
9
+ 6u
7
+ 3u
5
+ u
u
23
+ 5u
21
+ ··· + 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
43
− 8u
42
+ ··· + 11u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
44
+ 5u
43
+ ··· + 5u + 1
c
2
, c
4
u
44
− 11u
43
+ ··· − 9u + 1
c
3
, c
7
u
44
− u
43
+ ··· − 2048u + 1024
c
5
u
44
+ 10u
43
+ ··· + 510u + 61
c
6
, c
10
u
44
+ 2u
43
+ ··· + 2u + 1
c
8
u
44
+ 2u
43
+ ··· − 12568908u + 4045417
c
9
u
44
− 2u
43
+ ··· − 48u + 72
c
11
u
44
− 22u
43
+ ··· − 2u + 1
c
12
u
44
− 2u
43
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
44
+ 79y
43
+ ··· + 63y + 1
c
2
, c
4
y
44
− 5y
43
+ ··· − 5y + 1
c
3
, c
7
y
44
− 63y
43
+ ··· − 17301504y + 1048576
c
5
y
44
− 10y
43
+ ··· + 42582y + 3721
c
6
, c
10
y
44
+ 22y
43
+ ··· + 2y + 1
c
8
y
44
+ 118y
43
+ ··· + 59158715268238y + 16365398703889
c
9
y
44
− 18y
43
+ ··· − 39312y + 5184
c
11
y
44
+ 2y
43
+ ··· + 22y + 1
c
12
y
44
+ 58y
43
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.650835 + 0.753787I
a = 0.08846 − 1.78587I
b = 0.081004 + 0.538151I
7.53342 + 6.30806I −1.89413 − 5.29683I
u = −0.650835 − 0.753787I
a = 0.08846 + 1.78587I
b = 0.081004 − 0.538151I
7.53342 − 6.30806I −1.89413 + 5.29683I
u = −0.632673 + 0.802636I
a = 0.267451 + 1.071980I
b = −0.836036 − 0.256227I
7.68178 − 1.36168I −1.47660 − 0.85903I
u = −0.632673 − 0.802636I
a = 0.267451 − 1.071980I
b = −0.836036 + 0.256227I
7.68178 + 1.36168I −1.47660 + 0.85903I
u = 0.524889 + 0.986478I
a = 0.205755 − 0.498882I
b = −0.239518 + 0.643052I
−0.15574 − 2.57093I 0.70156 + 2.32156I
u = 0.524889 − 0.986478I
a = 0.205755 + 0.498882I
b = −0.239518 − 0.643052I
−0.15574 + 2.57093I 0.70156 − 2.32156I
u = −0.238245 + 1.098530I
a = −1.052800 − 0.393118I
b = 0.056688 + 0.785185I
4.15035 − 1.19914I 5.43856 + 1.99279I
u = −0.238245 − 1.098530I
a = −1.052800 + 0.393118I
b = 0.056688 − 0.785185I
4.15035 + 1.19914I 5.43856 − 1.99279I
u = 0.401111 + 1.053070I
a = −0.924275 − 0.254329I
b = 0.940261 + 0.336917I
1.12314 − 1.53132I 0.534904 + 0.918296I
u = 0.401111 − 1.053070I
a = −0.924275 + 0.254329I
b = 0.940261 − 0.336917I
1.12314 + 1.53132I 0.534904 − 0.918296I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.806426 + 0.270607I
a = 0.528240 − 1.204320I
b = 2.74166 + 0.03423I
10.03900 + 8.32938I −1.08270 − 4.13842I
u = 0.806426 − 0.270607I
a = 0.528240 + 1.204320I
b = 2.74166 − 0.03423I
10.03900 − 8.32938I −1.08270 + 4.13842I
u = 0.605593 + 0.584352I
a = −0.793404 + 0.445925I
b = 0.360026 − 0.418150I
−1.33551 − 1.91461I −1.93635 + 4.43568I
u = 0.605593 − 0.584352I
a = −0.793404 − 0.445925I
b = 0.360026 + 0.418150I
−1.33551 + 1.91461I −1.93635 − 4.43568I
u = −0.470021 + 1.059020I
a = −2.48227 + 1.10683I
b = 2.56335 + 1.29163I
−0.51801 + 3.33956I 1.74785 − 5.30450I
u = −0.470021 − 1.059020I
a = −2.48227 − 1.10683I
b = 2.56335 − 1.29163I
−0.51801 − 3.33956I 1.74785 + 5.30450I
u = 0.801520 + 0.234624I
a = −0.402454 + 0.655071I
b = −2.50289 + 0.41075I
10.56280 + 0.28731I −0.340463 + 0.065100I
u = 0.801520 − 0.234624I
a = −0.402454 − 0.655071I
b = −2.50289 − 0.41075I
10.56280 − 0.28731I −0.340463 − 0.065100I
u = −0.344708 + 1.132750I
a = 1.04475 − 1.24375I
b = −1.275840 − 0.094053I
5.34462 + 1.30032I 5.86228 − 1.34664I
u = −0.344708 − 1.132750I
a = 1.04475 + 1.24375I
b = −1.275840 + 0.094053I
5.34462 − 1.30032I 5.86228 + 1.34664I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.727737 + 0.345386I
a = −0.556447 + 0.549889I
b = 0.899472 + 0.608886I
−0.21118 − 3.75271I −0.89242 + 4.60335I
u = −0.727737 − 0.345386I
a = −0.556447 − 0.549889I
b = 0.899472 − 0.608886I
−0.21118 + 3.75271I −0.89242 − 4.60335I
u = 0.499458 + 1.093510I
a = 0.163389 − 0.118925I
b = −0.476367 − 0.549957I
0.35556 − 5.50410I −0.95894 + 5.50541I
u = 0.499458 − 1.093510I
a = 0.163389 + 0.118925I
b = −0.476367 + 0.549957I
0.35556 + 5.50410I −0.95894 − 5.50541I
u = 0.394783 + 0.687947I
a = −0.636590 − 0.714238I
b = 0.329853 + 0.582166I
−0.07264 − 1.54976I −0.86288 + 5.32741I
u = 0.394783 − 0.687947I
a = −0.636590 + 0.714238I
b = 0.329853 − 0.582166I
−0.07264 + 1.54976I −0.86288 − 5.32741I
u = 0.271461 + 1.191670I
a = −2.91510 − 0.51878I
b = 2.28357 − 1.65853I
14.6335 + 5.0175I 4.32906 − 1.77222I
u = 0.271461 − 1.191670I
a = −2.91510 + 0.51878I
b = 2.28357 + 1.65853I
14.6335 − 5.0175I 4.32906 + 1.77222I
u = 0.299244 + 1.194400I
a = 3.17931 + 0.29031I
b = −2.39631 + 1.63518I
14.9926 − 3.1935I 4.68937 + 2.57372I
u = 0.299244 − 1.194400I
a = 3.17931 − 0.29031I
b = −2.39631 − 1.63518I
14.9926 + 3.1935I 4.68937 − 2.57372I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.561031 + 1.111280I
a = −0.12864 + 1.49829I
b = 1.09429 − 1.16042I
2.02214 + 8.66027I 0. − 8.67025I
u = −0.561031 − 1.111280I
a = −0.12864 − 1.49829I
b = 1.09429 + 1.16042I
2.02214 − 8.66027I 0. + 8.67025I
u = −0.514177 + 1.135410I
a = 2.03093 − 0.39666I
b = −1.75747 − 1.02929I
4.18522 + 6.56824I 3.58280 − 6.10314I
u = −0.514177 − 1.135410I
a = 2.03093 + 0.39666I
b = −1.75747 + 1.02929I
4.18522 − 6.56824I 3.58280 + 6.10314I
u = −0.690430 + 0.212080I
a = −0.519384 − 0.455199I
b = −1.075590 + 0.770658I
1.55771 − 1.98048I 0.55925 + 2.67881I
u = −0.690430 − 0.212080I
a = −0.519384 + 0.455199I
b = −1.075590 − 0.770658I
1.55771 + 1.98048I 0.55925 − 2.67881I
u = 0.559852 + 1.158950I
a = −2.47252 − 2.48487I
b = 3.85040 − 0.10977I
12.6669 − 13.4080I 0. + 7.64627I
u = 0.559852 − 1.158950I
a = −2.47252 + 2.48487I
b = 3.85040 + 0.10977I
12.6669 + 13.4080I 0. − 7.64627I
u = 0.544747 + 1.166240I
a = 1.89623 + 2.56346I
b = −3.21064 − 0.17517I
13.3130 − 5.2801I 0
u = 0.544747 − 1.166240I
a = 1.89623 − 2.56346I
b = −3.21064 + 0.17517I
13.3130 + 5.2801I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.433758 + 0.454843I
a = 1.52940 + 1.29138I
b = 1.02671 − 0.97590I
−2.34698 + 0.55015I −4.01438 + 2.82078I
u = −0.433758 − 0.454843I
a = 1.52940 − 1.29138I
b = 1.02671 + 0.97590I
−2.34698 − 0.55015I −4.01438 − 2.82078I
u = 0.554531 + 0.293565I
a = 0.44996 + 1.43424I
b = 0.043378 − 0.204320I
−1.89085 + 1.22646I −5.83511 − 0.82987I
u = 0.554531 − 0.293565I
a = 0.44996 − 1.43424I
b = 0.043378 + 0.204320I
−1.89085 − 1.22646I −5.83511 + 0.82987I
9
II. I
u
2
= hb + u, −u
3
+ a − u + 1, u
4
+ u
2
− u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
3
+ u − 1
−u
a
7
=
1
u
2
a
4
=
u
3
+ u − 1
−u
a
10
=
u
u
3
+ u
a
12
=
u
3
u
2
a
5
=
−u
3
+ u
2
− u + 1
−u
2
+ u − 1
a
2
=
2u
3
− u
2
+ 2u − 2
u
2
− 2u + 1
a
9
=
−u
3
u
3
+ u
a
1
=
u
3
− u
2
+ u − 1
u
2
− u + 1
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
+ 4u
2
+ u − 6
10
(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
u
4
+ 2u
3
+ 3u
2
+ u + 1
c
6
u
4
+ u
2
− u + 1
c
8
, c
10
, c
12
u
4
+ u
2
+ u + 1
c
9
u
4
+ 3u
3
+ 4u
2
+ 3u + 2
c
11
u
4
− 2u
3
+ 3u
2
− u + 1
11
(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
11
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
6
, c
8
, c
10
c
12
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
9
y
4
− y
3
+ 2y
2
+ 7y + 4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.547424 + 0.585652I
a = −0.851808 + 0.911292I
b = −0.547424 − 0.585652I
−2.62503 − 1.39709I −7.62200 + 4.77865I
u = 0.547424 − 0.585652I
a = −0.851808 − 0.911292I
b = −0.547424 + 0.585652I
−2.62503 + 1.39709I −7.62200 − 4.77865I
u = −0.547424 + 1.120870I
a = 0.351808 + 0.720342I
b = 0.547424 − 1.120870I
0.98010 + 7.64338I −0.87800 − 5.79053I
u = −0.547424 − 1.120870I
a = 0.351808 − 0.720342I
b = 0.547424 + 1.120870I
0.98010 − 7.64338I −0.87800 + 5.79053I
13
III. I
u
3
=
h−u
5
−u
4
−2u
3
−2u
2
+b−2u−1, u
4
+u
2
+a, u
6
+u
5
+2u
4
+2u
3
+2u
2
+2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
4
− u
2
u
5
+ u
4
+ 2u
3
+ 2u
2
+ 2u + 1
a
7
=
1
u
2
a
4
=
−u
4
− u
2
u
5
+ u
4
+ 2u
3
+ 2u
2
+ 2u + 1
a
10
=
u
u
3
+ u
a
12
=
u
3
u
5
+ u
3
+ u
a
5
=
u
4
+ u
2
+ u + 1
−2u
5
− u
4
− 3u
3
− 2u
2
− 3u − 2
a
2
=
−2u
4
− 2u
2
− u − 1
3u
5
+ 2u
4
+ 5u
3
+ 4u
2
+ 5u + 3
a
9
=
−u
3
u
3
+ u
a
1
=
−u
4
− u
2
− u − 1
2u
5
+ u
4
+ 3u
3
+ 2u
2
+ 3u + 2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
+ 5u
3
+ u
2
+ 4u − 1
14
(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
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
c
6
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
8
, c
10
, c
12
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
9
(u
3
− u
2
+ 1)
2
c
11
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
15
(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
11
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
6
, c
8
, c
10
c
12
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
9
(y
3
− y
2
+ 2y −1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.498832 + 1.001300I
a = 1.183530 + 0.507021I
b = −1.39861 + 0.80012I
−1.37919 − 2.82812I −7.06955 + 2.21599I
u = 0.498832 − 1.001300I
a = 1.183530 − 0.507021I
b = −1.39861 − 0.80012I
−1.37919 + 2.82812I −7.06955 − 2.21599I
u = −0.284920 + 1.115140I
a = 0.215080 − 0.841795I
b = −0.784920 + 0.841795I
2.75839 2.84423 − 0.27335I
u = −0.284920 − 1.115140I
a = 0.215080 + 0.841795I
b = −0.784920 − 0.841795I
2.75839 2.84423 + 0.27335I
u = −0.713912 + 0.305839I
a = −0.398606 + 0.800120I
b = 0.183526 + 0.507021I
−1.37919 − 2.82812I −4.27468 + 2.61835I
u = −0.713912 − 0.305839I
a = −0.398606 − 0.800120I
b = 0.183526 − 0.507021I
−1.37919 + 2.82812I −4.27468 − 2.61835I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
10
)(u
44
+ 5u
43
+ ··· + 5u + 1)
c
2
((u − 1)
10
)(u
44
− 11u
43
+ ··· − 9u + 1)
c
3
, c
7
u
10
(u
44
− u
43
+ ··· − 2048u + 1024)
c
4
((u + 1)
10
)(u
44
− 11u
43
+ ··· − 9u + 1)
c
5
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
44
+ 10u
43
+ ··· + 510u + 61)
c
6
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
44
+ 2u
43
+ ··· + 2u + 1)
c
8
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
44
+ 2u
43
+ ··· − 12568908u + 4045417)
c
9
((u
3
− u
2
+ 1)
2
)(u
4
+ 3u
3
+ ··· + 3u + 2)(u
44
− 2u
43
+ ··· − 48u + 72)
c
10
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
44
+ 2u
43
+ ··· + 2u + 1)
c
11
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
44
− 22u
43
+ ··· − 2u + 1)
c
12
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
44
− 2u
43
+ ··· − 2u + 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
10
)(y
44
+ 79y
43
+ ··· + 63y + 1)
c
2
, c
4
((y −1)
10
)(y
44
− 5y
43
+ ··· − 5y + 1)
c
3
, c
7
y
10
(y
44
− 63y
43
+ ··· − 1.73015 × 10
7
y + 1048576)
c
5
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
44
− 10y
43
+ ··· + 42582y + 3721)
c
6
, c
10
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
44
+ 22y
43
+ ··· + 2y + 1)
c
8
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
44
+ 118y
43
+ ··· + 59158715268238y + 16365398703889)
c
9
(y
3
− y
2
+ 2y −1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
44
− 18y
43
+ ··· − 39312y + 5184)
c
11
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
44
+ 2y
43
+ ··· + 22y + 1)
c
12
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
44
+ 58y
43
+ ··· + 2y + 1)
19
