12a
0735
(K12a
0735
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 1 12 11 2 5 4 7 6
Solving Sequence
1,5 6,10
4 11 9 3 2 8 12 7
c
5
c
4
c
10
c
9
c
3
c
1
c
8
c
12
c
6
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
37
+ 2u
36
+ ··· + 4b + 2, −u
36
− 23u
34
+ ··· + 4a − 2, u
38
+ 2u
37
+ ··· + 5u + 2i
I
u
2
= h2u
4
a − 2u
3
a + a
2
u + 5u
2
a − 4au + b + a + 2u − 2,
2u
4
a
2
− u
4
a + 6a
2
u
2
+ 2u
3
a − 3u
4
+ a
3
− 5u
2
a + u
3
+ 2a
2
+ 7au − 10u
2
− 2a + 3u − 5,
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= hu
3
+ b + 2u, −u
3
− u
2
+ a − 2u − 2, u
4
+ 3u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 57 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
=
h2u
37
+2u
36
+· · ·+4b+2, −u
36
−23u
34
+· · ·+4a−2, u
38
+2u
37
+· · ·+5u+2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
10
=
1
4
u
36
+
23
4
u
34
+ ··· +
5
2
u +
1
2
−
1
2
u
37
−
1
2
u
36
+ ··· −
1
4
u −
1
2
a
4
=
1
4
u
33
+ 5u
31
+ ··· −
1
4
u + 1
−
1
4
u
33
−
21
4
u
31
+ ··· −
1
2
u
2
−
1
2
u
a
11
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
9
=
−
1
2
u
37
−
1
4
u
36
+ ··· +
1
4
u
2
+
9
4
u
−
1
2
u
37
−
1
2
u
36
+ ··· −
1
4
u −
1
2
a
3
=
−
1
2
u
37
− u
36
+ ··· −
15
4
u − 2
−
3
4
u
33
−
3
4
u
32
+ ··· −
1
2
u − 1
a
2
=
1
4
u
32
+
21
4
u
30
+ ··· −
1
2
u −
1
2
1
4
u
32
+ 5u
30
+ ··· +
5
4
u
2
+ u
a
8
=
u
4
+ 3u
2
+ 1
u
6
+ 4u
4
+ 3u
2
a
12
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
37
− 4u
36
+ ··· − 4u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
38
+ 17u
37
+ ··· + 194u + 25
c
2
, c
8
u
38
− u
37
+ ··· − 6u + 5
c
3
u
38
− 2u
37
+ ··· + 400u + 800
c
4
, c
9
, c
10
u
38
− u
37
+ ··· − 4u + 5
c
5
, c
6
, c
7
c
11
, c
12
u
38
− 2u
37
+ ··· − 5u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
38
+ 13y
37
+ ··· + 8514y + 625
c
2
, c
8
y
38
+ 17y
37
+ ··· + 194y + 25
c
3
y
38
− 6y
37
+ ··· + 7276800y + 640000
c
4
, c
9
, c
10
y
38
+ 37y
37
+ ··· + 34y + 25
c
5
, c
6
, c
7
c
11
, c
12
y
38
+ 48y
37
+ ··· + 19y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339832 + 0.921512I
a = 0.416428 + 0.809179I
b = 0.816368 − 0.218889I
−0.53116 + 6.72152I −6.76982 − 7.69269I
u = −0.339832 − 0.921512I
a = 0.416428 − 0.809179I
b = 0.816368 + 0.218889I
−0.53116 − 6.72152I −6.76982 + 7.69269I
u = −0.325570 + 0.993591I
a = −1.55595 − 1.85235I
b = −0.257004 + 1.389260I
7.06845 + 5.49844I 0.54907 − 4.52833I
u = −0.325570 − 0.993591I
a = −1.55595 + 1.85235I
b = −0.257004 − 1.389260I
7.06845 − 5.49844I 0.54907 + 4.52833I
u = 0.394709 + 0.970581I
a = 1.66954 − 1.57123I
b = 0.33472 + 1.39633I
4.59569 − 10.87090I −2.77827 + 8.43141I
u = 0.394709 − 0.970581I
a = 1.66954 + 1.57123I
b = 0.33472 − 1.39633I
4.59569 + 10.87090I −2.77827 − 8.43141I
u = −0.009913 + 0.916024I
a = −0.106232 + 0.617839I
b = −0.435699 − 0.647770I
2.92919 − 1.46585I −0.29190 + 4.46440I
u = −0.009913 − 0.916024I
a = −0.106232 − 0.617839I
b = −0.435699 + 0.647770I
2.92919 + 1.46585I −0.29190 − 4.46440I
u = −0.091063 + 1.155310I
a = −0.37132 − 2.12708I
b = −0.03930 + 1.42339I
9.62190 + 2.62526I 1.76254 − 3.39036I
u = −0.091063 − 1.155310I
a = −0.37132 + 2.12708I
b = −0.03930 − 1.42339I
9.62190 − 2.62526I 1.76254 + 3.39036I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.475449 + 0.649846I
a = −0.634708 + 0.373029I
b = 0.237413 − 1.333010I
2.66568 + 3.67725I −3.88495 − 1.85217I
u = 0.475449 − 0.649846I
a = −0.634708 − 0.373029I
b = 0.237413 + 1.333010I
2.66568 − 3.67725I −3.88495 + 1.85217I
u = −0.326550 + 0.707311I
a = 0.365685 + 0.985287I
b = 0.598316 + 0.079539I
−1.81820 − 0.62723I −9.78117 − 0.63046I
u = −0.326550 − 0.707311I
a = 0.365685 − 0.985287I
b = 0.598316 − 0.079539I
−1.81820 + 0.62723I −9.78117 + 0.63046I
u = −0.453897 + 0.489215I
a = 0.911640 + 0.461033I
b = −0.076008 − 1.314890I
4.18206 + 0.81384I −1.66334 − 4.28381I
u = −0.453897 − 0.489215I
a = 0.911640 − 0.461033I
b = −0.076008 + 1.314890I
4.18206 − 0.81384I −1.66334 + 4.28381I
u = 0.634985 + 0.149850I
a = −1.313290 − 0.234988I
b = −0.296893 − 1.359070I
1.15622 − 7.39365I −7.23665 + 6.75615I
u = 0.634985 − 0.149850I
a = −1.313290 + 0.234988I
b = −0.296893 + 1.359070I
1.15622 + 7.39365I −7.23665 − 6.75615I
u = −0.551325 + 0.214770I
a = 1.406100 − 0.007220I
b = 0.191390 − 1.321750I
3.34928 + 2.52485I −4.06452 − 3.23642I
u = −0.551325 − 0.214770I
a = 1.406100 + 0.007220I
b = 0.191390 + 1.321750I
3.34928 − 2.52485I −4.06452 + 3.23642I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.556094 + 0.096366I
a = −1.338990 + 0.086931I
b = −0.729346 + 0.157605I
−3.64092 + 3.67836I −13.1775 − 5.2178I
u = −0.556094 − 0.096366I
a = −1.338990 − 0.086931I
b = −0.729346 − 0.157605I
−3.64092 − 3.67836I −13.1775 + 5.2178I
u = 0.06495 + 1.55948I
a = 0.124474 − 1.120870I
b = −0.146346 + 1.269640I
9.94200 + 1.81077I 0
u = 0.06495 − 1.55948I
a = 0.124474 + 1.120870I
b = −0.146346 − 1.269640I
9.94200 − 1.81077I 0
u = 0.232971 + 0.274282I
a = 0.783127 + 0.921776I
b = 0.170162 + 0.354512I
−0.605257 − 0.910546I −10.00383 + 7.24439I
u = 0.232971 − 0.274282I
a = 0.783127 − 0.921776I
b = 0.170162 − 0.354512I
−0.605257 + 0.910546I −10.00383 − 7.24439I
u = −0.05271 + 1.63939I
a = −0.104717 − 0.802437I
b = −0.537806 + 0.097751I
6.35089 + 0.56961I 0
u = −0.05271 − 1.63939I
a = −0.104717 + 0.802437I
b = −0.537806 − 0.097751I
6.35089 − 0.56961I 0
u = −0.01436 + 1.69488I
a = −0.095169 − 0.847455I
b = 0.572206 + 0.761360I
12.20970 − 1.29201I 0
u = −0.01436 − 1.69488I
a = −0.095169 + 0.847455I
b = 0.572206 − 0.761360I
12.20970 + 1.29201I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.08704 + 1.69484I
a = −0.001733 − 0.692432I
b = −0.884556 + 0.249985I
8.67383 + 8.38204I 0
u = −0.08704 − 1.69484I
a = −0.001733 + 0.692432I
b = −0.884556 − 0.249985I
8.67383 − 8.38204I 0
u = 0.10617 + 1.70777I
a = −1.09920 + 1.99516I
b = −0.36348 − 1.42519I
14.0039 − 12.8731I 0
u = 0.10617 − 1.70777I
a = −1.09920 − 1.99516I
b = −0.36348 + 1.42519I
14.0039 + 12.8731I 0
u = −0.08545 + 1.71485I
a = 1.00730 + 2.20606I
b = 0.29380 − 1.43752I
16.6465 + 7.1508I 0
u = −0.08545 − 1.71485I
a = 1.00730 − 2.20606I
b = 0.29380 + 1.43752I
16.6465 − 7.1508I 0
u = −0.01542 + 1.74340I
a = 0.18700 + 2.55613I
b = 0.05206 − 1.51390I
−19.4879 + 3.0073I 0
u = −0.01542 − 1.74340I
a = 0.18700 − 2.55613I
b = 0.05206 + 1.51390I
−19.4879 − 3.0073I 0
8
II. I
u
2
=
h2u
4
a−2u
3
a+· · ·+a−2, 2u
4
a
2
−u
4
a+· · ·−2a−5, u
5
−u
4
+4u
3
−3u
2
+3u−1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
10
=
a
−2u
4
a + 2u
3
a − a
2
u − 5u
2
a + 4au − a − 2u + 2
a
4
=
−u
4
a − a
2
u
2
+ u
3
a + 2u
4
− 4u
2
a − 2u
3
− a
2
+ au + 6u
2
− a − 4u + 4
−u
4
a
2
− 2a
2
u
2
− 2u
4
+ 2u
3
+ 3au − 4u
2
− 2a + 4u
a
11
=
u
3
+ 2u
u
4
− u
3
+ 3u
2
− 2u + 1
a
9
=
−2u
4
a + 2u
3
a − a
2
u − 5u
2
a + 4au − 2u + 2
−2u
4
a + 2u
3
a − a
2
u − 5u
2
a + 4au − a − 2u + 2
a
3
=
−u
3
a
2
− 2u
4
a + ··· − 3a + 4
−u
4
a
2
− 4u
4
+ ··· + a
2
− 3a
a
2
=
u
4
a
2
+ u
4
a + 3a
2
u
2
− u
3
a + 4u
2
a + a
2
− 4au − 2u
2
+ 3a − 4
u
4
a
2
+ 2a
2
u
2
+ 2u
4
− 2u
3
− 3au + 4u
2
+ 2a − 4u
a
8
=
u
4
+ 3u
2
+ 1
u
4
− u
3
+ 3u
2
− 2u + 1
a
12
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
3
− 16u
2
+ 12u − 14
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 10u
14
+ ··· + 3u − 1
c
2
, c
4
, c
8
c
9
, c
10
u
15
+ 5u
13
+ ··· + u + 1
c
3
(u
5
+ u
4
− u
2
+ u + 1)
3
c
5
, c
6
, c
7
c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 10y
14
+ ··· + 15y −1
c
2
, c
4
, c
8
c
9
, c
10
y
15
+ 10y
14
+ ··· + 3y −1
c
3
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
3
c
5
, c
6
, c
7
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
3
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = −0.323874 + 0.796296I
b = −0.638808 − 0.271585I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 + 0.885557I
a = −0.156756 + 0.463494I
b = 0.435133 − 0.988544I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 + 0.885557I
a = 2.13619 − 2.53516I
b = 0.203675 + 1.260130I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 − 0.885557I
a = −0.323874 − 0.796296I
b = −0.638808 + 0.271585I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.233677 − 0.885557I
a = −0.156756 − 0.463494I
b = 0.435133 + 0.988544I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.233677 − 0.885557I
a = 2.13619 + 2.53516I
b = 0.203675 − 1.260130I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.416284
a = 1.12253
b = 0.511430
−0.882183 −11.6090
u = 0.416284
a = −2.11117 + 0.66665I
b = −0.255715 + 1.093700I
−0.882183 −11.6090
u = 0.416284
a = −2.11117 − 0.66665I
b = −0.255715 − 1.093700I
−0.882183 −11.6090
u = 0.05818 + 1.69128I
a = 0.154896 − 0.889970I
b = −0.549193 + 1.000850I
10.95830 − 3.33174I −2.08126 + 2.36228I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.05818 + 1.69128I
a = −0.007493 − 0.744869I
b = 0.762735 + 0.344098I
10.95830 − 3.33174I −2.08126 + 2.36228I
u = 0.05818 + 1.69128I
a = −1.25306 + 2.70311I
b = −0.213543 − 1.344950I
10.95830 − 3.33174I −2.08126 + 2.36228I
u = 0.05818 − 1.69128I
a = 0.154896 + 0.889970I
b = −0.549193 − 1.000850I
10.95830 + 3.33174I −2.08126 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.007493 + 0.744869I
b = 0.762735 − 0.344098I
10.95830 + 3.33174I −2.08126 − 2.36228I
u = 0.05818 − 1.69128I
a = −1.25306 − 2.70311I
b = −0.213543 + 1.344950I
10.95830 + 3.33174I −2.08126 − 2.36228I
13
III. I
u
3
= hu
3
+ b + 2u, −u
3
− u
2
+ a − 2u − 2, u
4
+ 3u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
10
=
u
3
+ u
2
+ 2u + 2
−u
3
− 2u
a
4
=
u
3
+ 3u
1
a
11
=
u
3
+ 2u
0
a
9
=
u
2
+ 2
−u
3
− 2u
a
3
=
u
3
+ 3u
1
a
2
=
u
3
+ 3u
u + 1
a
8
=
0
−u
2
− 1
a
12
=
u
u
3
+ u
a
7
=
u
2
+ 1
−u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
, c
8
c
9
, c
10
(u
2
+ 1)
2
c
3
u
4
c
5
, c
6
, c
7
c
11
, c
12
u
4
+ 3u
2
+ 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
4
c
2
, c
4
, c
8
c
9
, c
10
(y + 1)
4
c
3
y
4
c
5
, c
6
, c
7
c
11
, c
12
(y
2
+ 3y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034I
a = 1.61803 + 1.00000I
b = − 1.000000I
0.986960 −4.00000
u = − 0.618034I
a = 1.61803 − 1.00000I
b = 1.000000I
0.986960 −4.00000
u = 1.61803I
a = −0.618034 − 1.000000I
b = 1.000000I
8.88264 −4.00000
u = − 1.61803I
a = −0.618034 + 1.000000I
b = − 1.000000I
8.88264 −4.00000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
15
+ 10u
14
+ ··· + 3u − 1)(u
38
+ 17u
37
+ ··· + 194u + 25)
c
2
, c
8
((u
2
+ 1)
2
)(u
15
+ 5u
13
+ ··· + u + 1)(u
38
− u
37
+ ··· − 6u + 5)
c
3
u
4
(u
5
+ u
4
− u
2
+ u + 1)
3
(u
38
− 2u
37
+ ··· + 400u + 800)
c
4
, c
9
, c
10
((u
2
+ 1)
2
)(u
15
+ 5u
13
+ ··· + u + 1)(u
38
− u
37
+ ··· − 4u + 5)
c
5
, c
6
, c
7
c
11
, c
12
(u
4
+ 3u
2
+ 1)(u
5
+ u
4
+ ··· + 3u + 1)
3
(u
38
− 2u
37
+ ··· − 5u + 2)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
4
)(y
15
− 10y
14
+ ··· + 15y −1)
· (y
38
+ 13y
37
+ ··· + 8514y + 625)
c
2
, c
8
((y + 1)
4
)(y
15
+ 10y
14
+ ··· + 3y −1)(y
38
+ 17y
37
+ ··· + 194y + 25)
c
3
y
4
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
3
· (y
38
− 6y
37
+ ··· + 7276800y + 640000)
c
4
, c
9
, c
10
((y + 1)
4
)(y
15
+ 10y
14
+ ··· + 3y −1)(y
38
+ 37y
37
+ ··· + 34y + 25)
c
5
, c
6
, c
7
c
11
, c
12
(y
2
+ 3y + 1)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
3
· (y
38
+ 48y
37
+ ··· + 19y + 4)
19
