12a
0507
(K12a
0507
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 2 6 4 12 1 5 10
Solving Sequence
2,7
3 1 6 8 4
9,11
5 10 12
c
2
c
1
c
6
c
7
c
3
c
8
c
5
c
10
c
12
c
4
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
63
+ u
62
+ ··· + b + 1, u
64
+ 2u
63
+ ··· + a + 2, u
65
+ 2u
64
+ ··· + u + 1i
I
u
2
= hu
7
− u
5
+ u
4
+ u
3
+ b + 1, u
7
− u
5
+ u
4
+ u
3
+ a + 1, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 73 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
63
+ u
62
+ · · · + b + 1, u
64
+ 2u
63
+ · · · + a + 2, u
65
+ 2u
64
+ · · · + u + 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
6
=
u
u
a
8
=
−u
3
−u
3
+ u
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
9
=
u
13
− 2u
11
+ 3u
9
− 2u
7
− u
u
13
− 3u
11
+ 5u
9
− 6u
7
+ 4u
5
− 3u
3
+ u
a
11
=
−u
64
− 2u
63
+ ··· + 6u − 2
−u
63
− u
62
+ ··· + 2u − 1
a
5
=
u
18
− 3u
16
+ 6u
14
− 7u
12
+ 5u
10
− 3u
8
− u
2
+ 1
u
18
− 4u
16
+ 9u
14
− 14u
12
+ 15u
10
− 14u
8
+ 10u
6
− 6u
4
+ 3u
2
a
10
=
−u
64
− u
63
+ ··· + 7u − 2
−u
64
− u
63
+ ··· + u − 1
a
12
=
−u
64
+ 11u
62
+ ··· + 7u − 1
−2u
64
− u
63
+ ··· + u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
64
+ 6u
63
+ ··· − 13u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
65
+ 24u
64
+ ··· + 17u + 1
c
2
, c
6
u
65
− 2u
64
+ ··· + u − 1
c
3
, c
4
, c
8
u
65
+ 2u
64
+ ··· + 72u − 36
c
5
, c
11
u
65
+ u
64
+ ··· + 896u + 256
c
9
, c
10
, c
12
u
65
− 9u
64
+ ··· − 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
65
+ 36y
64
+ ··· + 41y −1
c
2
, c
6
y
65
− 24y
64
+ ··· + 17y −1
c
3
, c
4
, c
8
y
65
− 72y
64
+ ··· + 29736y −1296
c
5
, c
11
y
65
+ 51y
64
+ ··· + 507904y −65536
c
9
, c
10
, c
12
y
65
− 69y
64
+ ··· + 49y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.713224 + 0.687444I
a = −1.24609 − 1.61970I
b = −0.365613 − 1.272500I
1.29911 + 1.50054I −6.43332 − 4.09954I
u = 0.713224 − 0.687444I
a = −1.24609 + 1.61970I
b = −0.365613 + 1.272500I
1.29911 − 1.50054I −6.43332 + 4.09954I
u = −0.548659 + 0.822542I
a = 1.67765 − 1.87113I
b = 0.03464 − 1.54590I
−12.9972 − 8.8431I −12.85103 + 3.48991I
u = −0.548659 − 0.822542I
a = 1.67765 + 1.87113I
b = 0.03464 + 1.54590I
−12.9972 + 8.8431I −12.85103 − 3.48991I
u = 0.791470 + 0.591517I
a = 1.45052 + 0.22949I
b = 0.791445 − 0.191921I
−0.45144 − 1.99183I −13.03479 + 1.78533I
u = 0.791470 − 0.591517I
a = 1.45052 − 0.22949I
b = 0.791445 + 0.191921I
−0.45144 + 1.99183I −13.03479 − 1.78533I
u = −1.015680 + 0.168873I
a = 1.072100 + 0.272547I
b = 1.00409 − 1.15227I
−10.47320 + 4.58721I −18.5571 − 4.5138I
u = −1.015680 − 0.168873I
a = 1.072100 − 0.272547I
b = 1.00409 + 1.15227I
−10.47320 − 4.58721I −18.5571 + 4.5138I
u = 0.700275 + 0.763983I
a = 0.48960 + 2.63017I
b = −0.64142 + 2.04420I
−4.38452 + 4.13952I −11.02353 − 2.97070I
u = 0.700275 − 0.763983I
a = 0.48960 − 2.63017I
b = −0.64142 − 2.04420I
−4.38452 − 4.13952I −11.02353 + 2.97070I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.533819 + 0.798870I
a = −1.46665 + 1.17530I
b = −0.159563 + 0.626147I
−5.93028 − 4.47135I −11.14289 + 3.00215I
u = −0.533819 − 0.798870I
a = −1.46665 − 1.17530I
b = −0.159563 − 0.626147I
−5.93028 + 4.47135I −11.14289 − 3.00215I
u = 0.518412 + 0.795387I
a = −1.70961 − 0.08232I
b = −0.94816 − 1.34212I
−8.23637 + 1.60432I −12.14095 − 0.18134I
u = 0.518412 − 0.795387I
a = −1.70961 + 0.08232I
b = −0.94816 + 1.34212I
−8.23637 − 1.60432I −12.14095 + 0.18134I
u = −0.812099 + 0.670235I
a = 1.52706 + 0.04285I
b = 1.43769 − 0.56681I
2.51447 + 2.05945I 0. − 3.62781I
u = −0.812099 − 0.670235I
a = 1.52706 − 0.04285I
b = 1.43769 + 0.56681I
2.51447 − 2.05945I 0. + 3.62781I
u = −0.705301 + 0.621394I
a = −1.92385 + 0.84589I
b = −1.55110 + 1.65367I
−1.222500 + 0.017209I −9.24300 − 1.12644I
u = −0.705301 − 0.621394I
a = −1.92385 − 0.84589I
b = −1.55110 − 1.65367I
−1.222500 − 0.017209I −9.24300 + 1.12644I
u = 0.937657
a = −1.59986
b = −0.575570
−5.51198 −17.1080
u = −0.505383 + 0.783995I
a = 0.778262 − 0.689429I
b = −0.327984 + 0.239802I
−6.10999 + 1.30884I −11.57224 − 2.61156I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.505383 − 0.783995I
a = 0.778262 + 0.689429I
b = −0.327984 − 0.239802I
−6.10999 − 1.30884I −11.57224 + 2.61156I
u = 0.552421 + 0.745317I
a = 0.802012 − 0.057028I
b = 0.441507 + 0.474951I
−1.66866 + 1.26457I −4.48163 − 0.64966I
u = 0.552421 − 0.745317I
a = 0.802012 + 0.057028I
b = 0.441507 − 0.474951I
−1.66866 − 1.26457I −4.48163 + 0.64966I
u = −0.921663 + 0.079540I
a = −0.505185 + 0.420325I
b = −0.43418 + 1.48058I
−3.61711 + 2.05218I −17.0578 − 5.0395I
u = −0.921663 − 0.079540I
a = −0.505185 − 0.420325I
b = −0.43418 − 1.48058I
−3.61711 − 2.05218I −17.0578 + 5.0395I
u = −0.467653 + 0.791706I
a = 0.021320 + 0.804749I
b = 1.170190 − 0.600782I
−13.4836 + 5.4777I −13.28643 − 3.27596I
u = −0.467653 − 0.791706I
a = 0.021320 − 0.804749I
b = 1.170190 + 0.600782I
−13.4836 − 5.4777I −13.28643 + 3.27596I
u = 0.984595 + 0.489919I
a = 0.486559 − 0.702071I
b = 1.188930 + 0.454218I
−8.70414 − 1.49648I 0
u = 0.984595 − 0.489919I
a = 0.486559 + 0.702071I
b = 1.188930 − 0.454218I
−8.70414 + 1.49648I 0
u = 0.923412 + 0.605274I
a = −0.457168 − 0.927717I
b = −0.76064 − 1.67724I
−0.89013 − 2.73765I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.923412 − 0.605274I
a = −0.457168 + 0.927717I
b = −0.76064 + 1.67724I
−0.89013 + 2.73765I 0
u = −0.888492 + 0.663349I
a = −0.41584 + 1.37057I
b = 0.137087 + 1.392720I
2.27977 + 3.10358I 0
u = −0.888492 − 0.663349I
a = −0.41584 − 1.37057I
b = 0.137087 − 1.392720I
2.27977 − 3.10358I 0
u = −1.11106
a = 0.266855
b = −0.278254
−7.21985 −10.5080
u = −0.858807 + 0.738316I
a = −1.78012 − 2.04437I
b = −2.43706 − 1.24263I
−1.89391 + 2.79805I 0
u = −0.858807 − 0.738316I
a = −1.78012 + 2.04437I
b = −2.43706 + 1.24263I
−1.89391 − 2.79805I 0
u = 1.146070 + 0.010400I
a = −0.399146 − 0.884665I
b = −0.22151 − 2.08621I
−11.82220 − 2.96438I 0
u = 1.146070 − 0.010400I
a = −0.399146 + 0.884665I
b = −0.22151 + 2.08621I
−11.82220 + 2.96438I 0
u = −0.955422 + 0.633656I
a = 1.50663 − 1.77636I
b = 0.69599 − 1.98085I
−1.98303 + 4.94470I 0
u = −0.955422 − 0.633656I
a = 1.50663 + 1.77636I
b = 0.69599 + 1.98085I
−1.98303 − 4.94470I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.14899
a = −0.795555
b = 0.455262
−14.0593 −18.0700
u = 1.156610 + 0.029881I
a = 0.959308 + 0.424441I
b = 0.45584 + 1.79793I
−19.1194 − 7.3744I 0
u = 1.156610 − 0.029881I
a = 0.959308 − 0.424441I
b = 0.45584 − 1.79793I
−19.1194 + 7.3744I 0
u = 0.956552 + 0.664807I
a = 1.79937 + 0.87406I
b = 1.97712 + 1.72485I
0.56939 − 6.73062I 0
u = 0.956552 − 0.664807I
a = 1.79937 − 0.87406I
b = 1.97712 − 1.72485I
0.56939 + 6.73062I 0
u = 0.981422 + 0.702007I
a = −2.70793 − 0.03543I
b = −3.06189 − 1.09293I
−5.22876 − 9.69725I 0
u = 0.981422 − 0.702007I
a = −2.70793 + 0.03543I
b = −3.06189 + 1.09293I
−5.22876 + 9.69725I 0
u = 1.040310 + 0.653623I
a = −0.290402 − 0.811750I
b = 0.140732 − 1.060590I
−3.08664 − 6.59798I 0
u = 1.040310 − 0.653623I
a = −0.290402 + 0.811750I
b = 0.140732 + 1.060590I
−3.08664 + 6.59798I 0
u = −1.064460 + 0.646027I
a = −0.450969 + 0.418508I
b = −1.12284 + 1.29370I
−7.74953 + 4.07259I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.064460 − 0.646027I
a = −0.450969 − 0.418508I
b = −1.12284 − 1.29370I
−7.74953 − 4.07259I 0
u = −1.074890 + 0.630986I
a = 0.307606 + 0.701773I
b = 1.351880 − 0.171332I
−15.2686 − 0.1519I 0
u = −1.074890 − 0.630986I
a = 0.307606 − 0.701773I
b = 1.351880 + 0.171332I
−15.2686 + 0.1519I 0
u = 1.066140 + 0.653704I
a = 0.82798 + 1.79881I
b = −0.17076 + 2.34794I
−9.85329 − 7.04987I 0
u = 1.066140 − 0.653704I
a = 0.82798 − 1.79881I
b = −0.17076 − 2.34794I
−9.85329 + 7.04987I 0
u = −1.063690 + 0.660571I
a = 1.27520 − 1.15879I
b = 1.66073 − 2.19821I
−7.49945 + 9.95718I 0
u = −1.063690 − 0.660571I
a = 1.27520 + 1.15879I
b = 1.66073 + 2.19821I
−7.49945 − 9.95718I 0
u = −1.068690 + 0.673015I
a = −2.21159 + 1.26271I
b = −2.42067 + 2.56720I
−14.5530 + 14.4409I 0
u = −1.068690 − 0.673015I
a = −2.21159 − 1.26271I
b = −2.42067 − 2.56720I
−14.5530 − 14.4409I 0
u = 0.683701
a = 0.602547
b = 0.0385752
−1.02307 −9.31980
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.211468 + 0.588502I
a = −0.736413 − 0.786410I
b = 1.212590 + 0.233332I
−6.70249 − 2.38114I −11.80434 + 2.89982I
u = 0.211468 − 0.588502I
a = −0.736413 + 0.786410I
b = 1.212590 − 0.233332I
−6.70249 + 2.38114I −11.80434 − 2.89982I
u = 0.212478 + 0.317081I
a = 0.920282 + 1.026800I
b = −0.309776 − 0.311699I
−0.431525 − 0.957888I −7.32117 + 6.97948I
u = 0.212478 − 0.317081I
a = 0.920282 − 1.026800I
b = −0.309776 + 0.311699I
−0.431525 + 0.957888I −7.32117 − 6.97948I
u = −0.301622
a = −2.67493
b = −1.17462
−2.05866 −0.865930
11
II. I
u
2
= hu
7
− u
5
+ u
4
+ u
3
+ b + 1, u
7
− u
5
+ u
4
+ u
3
+ a + 1, u
8
− u
7
−
u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
6
=
u
u
a
8
=
−u
3
−u
3
+ u
a
4
=
−u
7
+ 2u
5
− 2u
3
+ 2u
−u
7
+ u
6
+ 2u
5
− u
4
− 2u
3
+ 2u
2
+ 2u − 1
a
9
=
u
2
− 1
u
4
a
11
=
−u
7
+ u
5
− u
4
− u
3
− 1
−u
7
+ u
5
− u
4
− u
3
− 1
a
5
=
u
u
a
10
=
−u
7
+ u
5
− u
4
− u
3
+ u
2
− 2
−u
7
+ u
5
− u
3
− 1
a
12
=
−u
7
+ u
5
− u
4
− u
3
− 1
−u
7
+ u
5
− u
4
− u
3
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
7
+ u
6
+ 11u
5
− 8u
4
− 11u
3
+ 7u
2
+ 4u − 23
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
2
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
3
, c
4
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
5
, c
11
u
8
c
6
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
7
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
8
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
9
, c
10
(u − 1)
8
c
12
(u + 1)
8
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
2
, c
6
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
3
, c
4
, c
8
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
5
, c
11
y
8
c
9
, c
10
, c
12
(y −1)
8
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = −0.325934 − 0.693334I
b = −0.325934 − 0.693334I
−2.68559 + 1.13123I −13.35119 − 0.17229I
u = 0.570868 − 0.730671I
a = −0.325934 + 0.693334I
b = −0.325934 + 0.693334I
−2.68559 − 1.13123I −13.35119 + 0.17229I
u = −0.855237 + 0.665892I
a = 1.03462 + 0.99451I
b = 1.03462 + 0.99451I
0.51448 + 2.57849I −6.04880 − 3.90294I
u = −0.855237 − 0.665892I
a = 1.03462 − 0.99451I
b = 1.03462 − 0.99451I
0.51448 − 2.57849I −6.04880 + 3.90294I
u = −1.09818
a = −0.801005
b = −0.801005
−8.14766 −20.2760
u = 1.031810 + 0.655470I
a = 0.842429 + 0.289836I
b = 0.842429 + 0.289836I
−4.02461 − 6.44354I −15.5815 + 4.6831I
u = 1.031810 − 0.655470I
a = 0.842429 − 0.289836I
b = 0.842429 − 0.289836I
−4.02461 + 6.44354I −15.5815 − 4.6831I
u = 0.603304
a = −1.30123
b = −1.30123
−2.48997 −20.7610
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
65
+ 24u
64
+ ··· + 17u + 1)
c
2
(u
8
− u
7
+ ··· + 2u − 1)(u
65
− 2u
64
+ ··· + u − 1)
c
3
, c
4
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
65
+ 2u
64
+ ··· + 72u − 36)
c
5
, c
11
u
8
(u
65
+ u
64
+ ··· + 896u + 256)
c
6
(u
8
+ u
7
+ ··· − 2u − 1)(u
65
− 2u
64
+ ··· + u − 1)
c
7
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
65
+ 24u
64
+ ··· + 17u + 1)
c
8
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
65
+ 2u
64
+ ··· + 72u − 36)
c
9
, c
10
((u − 1)
8
)(u
65
− 9u
64
+ ··· − 9u + 1)
c
12
((u + 1)
8
)(u
65
− 9u
64
+ ··· − 9u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
65
+ 36y
64
+ ··· + 41y −1)
c
2
, c
6
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
65
− 24y
64
+ ··· + 17y −1)
c
3
, c
4
, c
8
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
65
− 72y
64
+ ··· + 29736y −1296)
c
5
, c
11
y
8
(y
65
+ 51y
64
+ ··· + 507904y −65536)
c
9
, c
10
, c
12
((y −1)
8
)(y
65
− 69y
64
+ ··· + 49y −1)
17
