12a
0840
(K12a
0840
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 1 3 8 7 6
Solving Sequence
6,11
7 12 8
1,4
2 9 3 10 5
c
6
c
11
c
7
c
12
c
1
c
8
c
3
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
70
+ u
69
+ ··· + b + 2u, u
70
+ u
69
+ ··· + a + 8u, u
71
+ 2u
70
+ ··· + 10u
2
− 1i
I
u
2
= hb + 1, u
6
− 3u
4
+ u
3
+ 2u
2
+ a − 2u + 1, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 79 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
70
+u
69
+· · ·+b+2u, u
70
+u
69
+· · ·+a+8u, u
71
+2u
70
+· · ·+10u
2
−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
3
− 2u
−u
3
+ u
a
4
=
−u
70
− u
69
+ ··· − 20u
2
− 8u
−u
70
− u
69
+ ··· − 9u
2
− 2u
a
2
=
−2u
70
− 2u
69
+ ··· − 22u
2
− 9u
−u
70
− u
69
+ ··· − 9u
2
− 2u
a
9
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
3
=
−3u
70
− 3u
69
+ ··· − 8u + 1
−u
70
− u
69
+ ··· − 10u
2
− 3u
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
12
+ 5u
10
− 9u
8
+ 6u
6
− u
2
+ 1
−u
14
+ 6u
12
− 13u
10
+ 10u
8
+ 2u
6
− 4u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
70
+ 4u
69
+ ··· + 10u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
71
− 9u
70
+ ··· − 8u + 1
c
3
, c
9
u
71
− u
70
+ ··· + 896u + 256
c
5
, c
8
u
71
− 2u
70
+ ··· + 1648u − 505
c
6
, c
7
, c
11
u
71
+ 2u
70
+ ··· + 10u
2
− 1
c
10
, c
12
u
71
− 6u
70
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
71
− 73y
70
+ ··· + 40y −1
c
3
, c
9
y
71
+ 51y
70
+ ··· + 376832y −65536
c
5
, c
8
y
71
− 60y
70
+ ··· + 2629044y −255025
c
6
, c
7
, c
11
y
71
− 60y
70
+ ··· + 20y −1
c
10
, c
12
y
71
+ 36y
70
+ ··· + 20y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.986659 + 0.246337I
a = 0.840333 + 0.355711I
b = −0.851935 + 0.777616I
−4.39759 + 2.62166I −12.54325 + 0.I
u = 0.986659 − 0.246337I
a = 0.840333 − 0.355711I
b = −0.851935 − 0.777616I
−4.39759 − 2.62166I −12.54325 + 0.I
u = 0.996077 + 0.329643I
a = −2.15290 − 1.03284I
b = 2.27588 − 0.46222I
−11.10650 + 6.59599I 0
u = 0.996077 − 0.329643I
a = −2.15290 + 1.03284I
b = 2.27588 + 0.46222I
−11.10650 − 6.59599I 0
u = −0.908611 + 0.214665I
a = −2.60723 + 0.20772I
b = 1.91874 + 0.49065I
−6.64883 − 0.06384I −13.96488 − 0.64877I
u = −0.908611 − 0.214665I
a = −2.60723 − 0.20772I
b = 1.91874 − 0.49065I
−6.64883 + 0.06384I −13.96488 + 0.64877I
u = 0.809385 + 0.207067I
a = 0.362023 − 0.291325I
b = −0.557314 − 0.483316I
−4.57830 − 2.54230I −13.6596 + 4.1821I
u = 0.809385 − 0.207067I
a = 0.362023 + 0.291325I
b = −0.557314 + 0.483316I
−4.57830 + 2.54230I −13.6596 − 4.1821I
u = 0.766372 + 0.323559I
a = −1.75720 + 0.29981I
b = 1.83541 − 0.37016I
−11.57750 − 6.28991I −15.1916 + 4.8659I
u = 0.766372 − 0.323559I
a = −1.75720 − 0.29981I
b = 1.83541 + 0.37016I
−11.57750 + 6.28991I −15.1916 − 4.8659I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.154890 + 0.240784I
a = 0.924878 + 0.182653I
b = −0.298204 − 0.725011I
−1.31391 + 0.68641I 0
u = −1.154890 − 0.240784I
a = 0.924878 − 0.182653I
b = −0.298204 + 0.725011I
−1.31391 − 0.68641I 0
u = 0.181853 + 0.789212I
a = −3.78833 − 2.50123I
b = 2.82607 + 0.74359I
−8.59298 − 10.74520I −11.21931 + 6.66626I
u = 0.181853 − 0.789212I
a = −3.78833 + 2.50123I
b = 2.82607 − 0.74359I
−8.59298 + 10.74520I −11.21931 − 6.66626I
u = −0.066855 + 0.801312I
a = 1.80010 − 0.25938I
b = −1.259410 − 0.508282I
−1.15216 + 4.20230I −9.31977 − 4.21160I
u = −0.066855 − 0.801312I
a = 1.80010 + 0.25938I
b = −1.259410 + 0.508282I
−1.15216 − 4.20230I −9.31977 + 4.21160I
u = 0.176589 + 0.764676I
a = 1.26518 + 1.55971I
b = −1.31967 − 0.80362I
−1.90510 − 6.52099I −9.14939 + 6.60717I
u = 0.176589 − 0.764676I
a = 1.26518 − 1.55971I
b = −1.31967 + 0.80362I
−1.90510 + 6.52099I −9.14939 − 6.60717I
u = −0.185138 + 0.751263I
a = −3.20382 + 3.96451I
b = 2.40680 − 1.58953I
−4.30951 + 3.86319I −10.53421 − 3.73969I
u = −0.185138 − 0.751263I
a = −3.20382 − 3.96451I
b = 2.40680 + 1.58953I
−4.30951 − 3.86319I −10.53421 + 3.73969I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.23414
a = −2.32831
b = −0.376389
−6.28970 0
u = −1.190850 + 0.346873I
a = 0.457682 − 0.424063I
b = −1.37788 + 0.35953I
−4.59392 − 0.04944I 0
u = −1.190850 − 0.346873I
a = 0.457682 + 0.424063I
b = −1.37788 − 0.35953I
−4.59392 + 0.04944I 0
u = 1.239300 + 0.109593I
a = 0.445590 − 0.103058I
b = −0.159999 − 0.708388I
−4.58430 − 2.00294I 0
u = 1.239300 − 0.109593I
a = 0.445590 + 0.103058I
b = −0.159999 + 0.708388I
−4.58430 + 2.00294I 0
u = 0.186710 + 0.731693I
a = 0.677426 + 0.479857I
b = −0.176307 + 0.203264I
−2.39470 − 1.09775I −10.39559 + 0.93933I
u = 0.186710 − 0.731693I
a = 0.677426 − 0.479857I
b = −0.176307 − 0.203264I
−2.39470 + 1.09775I −10.39559 − 0.93933I
u = −0.129455 + 0.743382I
a = 0.91484 − 1.53124I
b = −0.739107 + 0.752454I
1.69012 + 2.95686I −2.27360 − 4.11680I
u = −0.129455 − 0.743382I
a = 0.91484 + 1.53124I
b = −0.739107 − 0.752454I
1.69012 − 2.95686I −2.27360 + 4.11680I
u = 0.232613 + 0.714723I
a = −1.49386 − 3.33234I
b = 1.33349 + 1.10636I
−9.77913 + 2.47980I −12.49777 + 0.44362I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.232613 − 0.714723I
a = −1.49386 + 3.33234I
b = 1.33349 − 1.10636I
−9.77913 − 2.47980I −12.49777 − 0.44362I
u = −0.038065 + 0.746925I
a = −1.26831 − 0.69126I
b = 1.057950 + 0.468795I
3.58845 + 2.01767I −2.00784 − 4.42424I
u = −0.038065 − 0.746925I
a = −1.26831 + 0.69126I
b = 1.057950 − 0.468795I
3.58845 − 2.01767I −2.00784 + 4.42424I
u = −1.234340 + 0.297863I
a = 0.077706 + 0.925364I
b = 1.063820 − 0.074072I
−0.08058 + 1.76105I 0
u = −1.234340 − 0.297863I
a = 0.077706 − 0.925364I
b = 1.063820 + 0.074072I
−0.08058 − 1.76105I 0
u = 1.263550 + 0.265912I
a = 1.184650 + 0.148415I
b = −1.40923 + 0.68906I
−3.14372 − 2.42246I 0
u = 1.263550 − 0.265912I
a = 1.184650 − 0.148415I
b = −1.40923 − 0.68906I
−3.14372 + 2.42246I 0
u = 0.029470 + 0.697447I
a = 2.38285 + 1.53386I
b = −1.62005 − 0.27761I
0.662530 − 1.041100I −7.27548 − 0.38248I
u = 0.029470 − 0.697447I
a = 2.38285 − 1.53386I
b = −1.62005 + 0.27761I
0.662530 + 1.041100I −7.27548 + 0.38248I
u = −1.291470 + 0.286170I
a = 0.10379 − 2.25863I
b = −1.84247 − 0.07524I
−3.46345 + 4.60561I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.291470 − 0.286170I
a = 0.10379 + 2.25863I
b = −1.84247 + 0.07524I
−3.46345 − 4.60561I 0
u = 1.289160 + 0.314090I
a = −0.647097 − 0.468432I
b = 1.042920 − 0.815117I
−0.54813 − 5.85187I 0
u = 1.289160 − 0.314090I
a = −0.647097 + 0.468432I
b = 1.042920 + 0.815117I
−0.54813 + 5.85187I 0
u = 1.306420 + 0.349875I
a = 0.68982 + 1.42591I
b = −1.128250 + 0.646449I
−5.44295 − 8.34894I 0
u = 1.306420 − 0.349875I
a = 0.68982 − 1.42591I
b = −1.128250 − 0.646449I
−5.44295 + 8.34894I 0
u = 1.358380 + 0.152222I
a = −1.17648 + 1.28390I
b = −1.337860 − 0.349570I
−11.76690 − 3.60144I 0
u = 1.358380 − 0.152222I
a = −1.17648 − 1.28390I
b = −1.337860 + 0.349570I
−11.76690 + 3.60144I 0
u = 1.344140 + 0.315904I
a = −0.347097 + 1.182250I
b = −0.972415 − 0.773651I
−2.95246 − 6.79976I 0
u = 1.344140 − 0.315904I
a = −0.347097 − 1.182250I
b = −0.972415 + 0.773651I
−2.95246 + 6.79976I 0
u = 1.38516
a = −0.406375
b = −1.07210
−7.07702 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.367690 + 0.306384I
a = 0.230808 − 0.686573I
b = −0.0933768 + 0.0315903I
−7.30405 + 4.87123I 0
u = −1.367690 − 0.306384I
a = 0.230808 + 0.686573I
b = −0.0933768 − 0.0315903I
−7.30405 − 4.87123I 0
u = −1.367790 + 0.321539I
a = −0.393117 − 1.350420I
b = −1.56646 + 0.71385I
−6.78407 + 10.45570I 0
u = −1.367790 − 0.321539I
a = −0.393117 + 1.350420I
b = −1.56646 − 0.71385I
−6.78407 − 10.45570I 0
u = 1.369780 + 0.314519I
a = 0.49273 − 3.43980I
b = 2.88614 + 1.93089I
−9.22312 − 7.72882I 0
u = 1.369780 − 0.314519I
a = 0.49273 + 3.43980I
b = 2.88614 − 1.93089I
−9.22312 + 7.72882I 0
u = −0.396518 + 0.441005I
a = 0.47669 − 1.49811I
b = −1.328800 + 0.023293I
−6.36129 + 1.57513I −13.7151 − 4.1754I
u = −0.396518 − 0.441005I
a = 0.47669 + 1.49811I
b = −1.328800 − 0.023293I
−6.36129 − 1.57513I −13.7151 + 4.1754I
u = −1.381840 + 0.290987I
a = 0.90571 + 2.40675I
b = 1.44667 − 1.68893I
−14.8854 + 1.1709I 0
u = −1.381840 − 0.290987I
a = 0.90571 − 2.40675I
b = 1.44667 + 1.68893I
−14.8854 − 1.1709I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.373550 + 0.332446I
a = −0.39571 + 3.13248I
b = 3.20219 − 0.78644I
−13.5094 + 14.8002I 0
u = −1.373550 − 0.332446I
a = −0.39571 − 3.13248I
b = 3.20219 + 0.78644I
−13.5094 − 14.8002I 0
u = −1.41996 + 0.01011I
a = −0.416396 + 0.543396I
b = −0.966187 + 0.001876I
−11.25150 + 2.82348I 0
u = −1.41996 − 0.01011I
a = −0.416396 − 0.543396I
b = −0.966187 − 0.001876I
−11.25150 − 2.82348I 0
u = 1.42247
a = 0.946420
b = 3.55445
−13.4469 0
u = −1.43211 + 0.02678I
a = 0.860128 + 0.073018I
b = 2.85973 + 0.70323I
−18.4145 + 6.9595I 0
u = −1.43211 − 0.02678I
a = 0.860128 − 0.073018I
b = 2.85973 − 0.70323I
−18.4145 − 6.9595I 0
u = −0.555980
a = 0.604259
b = −0.508930
−1.13868 −8.43750
u = −0.221093 + 0.243392I
a = 1.055700 − 0.337954I
b = 0.070341 + 0.235370I
−0.399020 + 0.809354I −8.89240 − 8.42436I
u = −0.221093 − 0.243392I
a = 1.055700 + 0.337954I
b = 0.070341 − 0.235370I
−0.399020 − 0.809354I −8.89240 + 8.42436I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.229977
a = −2.81817
b = −1.03945
−2.00887 −0.983990
12
II.
I
u
2
= hb+1, u
6
−3u
4
+u
3
+2u
2
+a−2u +1, u
8
+u
7
−3u
6
−2u
5
+3u
4
+2u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
3
− 2u
−u
3
+ u
a
4
=
−u
6
+ 3u
4
− u
3
− 2u
2
+ 2u − 1
−1
a
2
=
−u
6
+ 3u
4
− 2u
2
− 1
−u
3
+ u − 1
a
9
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
3
=
−u
6
+ 3u
4
− u
3
− 2u
2
+ 2u − 1
−1
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
3
+ 2u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
7
− 6u
6
+ 2u
5
+ 16u
4
− 5u
3
− 9u
2
+ 8u − 21
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
9
u
8
c
4
(u + 1)
8
c
5
, c
8
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
6
, c
7
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
, c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
11
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
9
y
8
c
5
, c
8
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
6
, c
7
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = 0.646194 + 0.127698I
b = −1.00000
−2.68559 − 1.13123I −10.92586 + 0.21647I
u = 1.180120 − 0.268597I
a = 0.646194 − 0.127698I
b = −1.00000
−2.68559 + 1.13123I −10.92586 − 0.21647I
u = 0.108090 + 0.747508I
a = 1.43073 + 0.89199I
b = −1.00000
0.51448 − 2.57849I −8.77377 + 3.25417I
u = 0.108090 − 0.747508I
a = 1.43073 − 0.89199I
b = −1.00000
0.51448 + 2.57849I −8.77377 − 3.25417I
u = −1.37100
a = −0.966009
b = −1.00000
−8.14766 −19.8990
u = −1.334530 + 0.318930I
a = 0.142888 − 1.323540I
b = −1.00000
−4.02461 + 6.44354I −14.3478 − 4.5473I
u = −1.334530 − 0.318930I
a = 0.142888 + 1.323540I
b = −1.00000
−4.02461 − 6.44354I −14.3478 + 4.5473I
u = 0.463640
a = −0.473616
b = −1.00000
−2.48997 −19.0060
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
8
)(u
71
− 9u
70
+ ··· − 8u + 1)
c
3
, c
9
u
8
(u
71
− u
70
+ ··· + 896u + 256)
c
4
((u + 1)
8
)(u
71
− 9u
70
+ ··· − 8u + 1)
c
5
, c
8
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
71
− 2u
70
+ ··· + 1648u − 505)
c
6
, c
7
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
71
+ 2u
70
+ ··· + 10u
2
− 1)
c
10
, c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
71
− 6u
70
+ ··· − 4u + 1)
c
11
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
71
+ 2u
70
+ ··· + 10u
2
− 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
8
)(y
71
− 73y
70
+ ··· + 40y −1)
c
3
, c
9
y
8
(y
71
+ 51y
70
+ ··· + 376832y −65536)
c
5
, c
8
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
71
− 60y
70
+ ··· + 2629044y −255025)
c
6
, c
7
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
71
− 60y
70
+ ··· + 20y −1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
71
+ 36y
70
+ ··· + 20y −1)
18
