12n
0080
(K12n
0080
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 9 3 11 5 7 12 8 10
Solving Sequence
7,11
8
3,12
6 4 10 1 9 5 2
c
7
c
11
c
6
c
3
c
10
c
12
c
9
c
5
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h11188446734u
40
+ 14624888210u
39
+ ··· + 38482965369b − 18882609362,
− 471858393299u
40
− 603105686012u
39
+ ··· + 38482965369a + 737953573358,
u
41
+ 2u
40
+ ··· − u − 1i
I
u
2
= hb, 3u
7
+ u
6
− 4u
5
− 4u
4
+ 5u
3
+ 3u
2
+ a − u − 5, u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 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
= h1.12×10
10
u
40
+1.46×10
10
u
39
+· · ·+3.85×10
10
b−1.89×10
10
, −4.72×
10
11
u
40
−6.03×10
11
u
39
+· · ·+3.85×10
10
a+7.38×10
11
, u
41
+2u
40
+· · ·−u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
3
=
12.2615u
40
+ 15.6720u
39
+ ··· + 6.84293u − 19.1761
−0.290738u
40
− 0.380035u
39
+ ··· + 1.44537u + 0.490674
a
12
=
u
−u
3
+ u
a
6
=
−3.45682u
40
− 4.64440u
39
+ ··· − 4.50872u + 4.12223
0.869179u
40
+ 1.73038u
39
+ ··· − 0.134593u − 0.868998
a
4
=
9.93735u
40
+ 13.3319u
39
+ ··· + 8.96828u − 18.9057
2.61664u
40
+ 5.42032u
39
+ ··· − 0.00831753u − 2.41607
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
5
=
−2.06817u
40
− 2.86404u
39
+ ··· − 3.90693u + 2.53206
−0.127787u
40
+ 0.140106u
39
+ ··· − 0.336106u + 0.527977
a
2
=
10.3978u
40
+ 13.4001u
39
+ ··· + 5.65679u − 18.2402
0.964836u
40
+ 2.33975u
39
+ ··· + 1.11758u − 0.565279
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1507194298715
12827655123
u
40
+
1944574441589
12827655123
u
39
+ ··· +
274922951097
4275885041
u −
2104450199471
12827655123
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 53u
40
+ ··· + 381u + 1
c
2
, c
4
u
41
− 9u
40
+ ··· − 29u + 1
c
3
, c
6
u
41
+ 7u
40
+ ··· + 2176u − 256
c
5
, c
8
u
41
− 2u
40
+ ··· + u − 1
c
7
, c
11
u
41
− 2u
40
+ ··· − u + 1
c
9
u
41
+ 2u
40
+ ··· + 12241u + 8353
c
10
, c
12
u
41
− 12u
40
+ ··· + 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
− 121y
40
+ ··· + 73433y −1
c
2
, c
4
y
41
− 53y
40
+ ··· + 381y −1
c
3
, c
6
y
41
+ 51y
40
+ ··· + 2801664y −65536
c
5
, c
8
y
41
+ 42y
39
+ ··· + 5y −1
c
7
, c
11
y
41
− 12y
40
+ ··· + 5y −1
c
9
y
41
+ 36y
40
+ ··· + 949291005y −69772609
c
10
, c
12
y
41
+ 36y
40
+ ··· + 37y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.645157 + 0.706341I
a = 0.129346 + 0.199509I
b = −0.707026 − 0.083217I
0.314025 + 0.700329I 9.78155 + 0.42575I
u = −0.645157 − 0.706341I
a = 0.129346 − 0.199509I
b = −0.707026 + 0.083217I
0.314025 − 0.700329I 9.78155 − 0.42575I
u = 1.06514
a = 1.21465
b = −0.619887
5.55858 18.6510
u = 0.878483 + 0.257640I
a = 1.36190 + 1.41715I
b = −0.395484 − 1.274730I
0.35281 + 3.72015I 8.01877 − 8.67695I
u = 0.878483 − 0.257640I
a = 1.36190 − 1.41715I
b = −0.395484 + 1.274730I
0.35281 − 3.72015I 8.01877 + 8.67695I
u = 0.863822 + 0.692291I
a = −0.710177 + 0.248903I
b = 0.360833 − 0.064088I
−2.33978 + 2.66185I 4.87613 − 3.55699I
u = 0.863822 − 0.692291I
a = −0.710177 − 0.248903I
b = 0.360833 + 0.064088I
−2.33978 − 2.66185I 4.87613 + 3.55699I
u = 0.850650 + 0.773602I
a = 0.65587 − 1.74300I
b = −0.305447 − 0.669138I
−4.76432 + 2.07222I 3.47881 − 9.56031I
u = 0.850650 − 0.773602I
a = 0.65587 + 1.74300I
b = −0.305447 + 0.669138I
−4.76432 − 2.07222I 3.47881 + 9.56031I
u = 1.113340 + 0.298427I
a = −2.09473 − 1.98305I
b = 0.58426 + 1.64109I
−6.81237 + 7.94638I 6.00000 − 5.66643I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.113340 − 0.298427I
a = −2.09473 + 1.98305I
b = 0.58426 − 1.64109I
−6.81237 − 7.94638I 6.00000 + 5.66643I
u = −1.123200 + 0.281615I
a = 0.82820 − 2.54731I
b = 0.17091 + 1.62625I
−6.69343 + 0.45475I 6.00000 + 0.74518I
u = −1.123200 − 0.281615I
a = 0.82820 + 2.54731I
b = 0.17091 − 1.62625I
−6.69343 − 0.45475I 6.00000 − 0.74518I
u = −0.821741 + 0.155454I
a = −1.49310 + 0.28695I
b = 0.000254 − 0.581031I
0.614586 − 0.353193I 8.75569 + 0.62140I
u = −0.821741 − 0.155454I
a = −1.49310 − 0.28695I
b = 0.000254 + 0.581031I
0.614586 + 0.353193I 8.75569 − 0.62140I
u = −0.829859 + 0.817277I
a = −1.080080 − 0.171992I
b = 0.03664 − 2.09736I
−6.22251 + 1.37166I 0. − 2.74151I
u = −0.829859 − 0.817277I
a = −1.080080 + 0.171992I
b = 0.03664 + 2.09736I
−6.22251 − 1.37166I 0. + 2.74151I
u = 0.739042 + 0.910928I
a = −0.097736 − 0.296625I
b = −0.10739 + 1.92349I
−14.7874 + 0.6933I 0. − 1.66266I
u = 0.739042 − 0.910928I
a = −0.097736 + 0.296625I
b = −0.10739 − 1.92349I
−14.7874 − 0.6933I 0. + 1.66266I
u = −0.746747 + 0.906172I
a = −0.099105 − 0.285226I
b = 0.82858 + 1.91790I
−14.9445 + 7.5314I 0. − 2.54363I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.746747 − 0.906172I
a = −0.099105 + 0.285226I
b = 0.82858 − 1.91790I
−14.9445 − 7.5314I 0. + 2.54363I
u = 0.010347 + 0.818382I
a = −0.096572 + 0.286828I
b = 0.37826 − 1.80255I
−10.51980 − 4.16733I 0.66529 + 2.27689I
u = 0.010347 − 0.818382I
a = −0.096572 − 0.286828I
b = 0.37826 + 1.80255I
−10.51980 + 4.16733I 0.66529 − 2.27689I
u = 0.913540 + 0.760932I
a = −0.676289 + 1.093800I
b = −0.209840 + 0.757944I
−4.56999 + 3.72466I 3.33454 + 2.09517I
u = 0.913540 − 0.760932I
a = −0.676289 − 1.093800I
b = −0.209840 − 0.757944I
−4.56999 − 3.72466I 3.33454 − 2.09517I
u = −0.893232 + 0.810220I
a = −0.95737 − 1.53330I
b = 2.34407 + 0.13131I
−8.47280 − 3.03045I 0. + 2.81396I
u = −0.893232 − 0.810220I
a = −0.95737 + 1.53330I
b = 2.34407 − 0.13131I
−8.47280 + 3.03045I 0. − 2.81396I
u = −1.010010 + 0.669120I
a = 0.541263 + 0.790129I
b = −0.710082 − 0.023150I
1.39688 − 6.02924I 12.17156 + 3.37002I
u = −1.010010 − 0.669120I
a = 0.541263 − 0.790129I
b = −0.710082 + 0.023150I
1.39688 + 6.02924I 12.17156 − 3.37002I
u = −0.944017 + 0.784096I
a = 1.80124 − 0.03860I
b = −0.19373 + 2.11976I
−5.86990 − 7.37157I 6.00000 + 8.00321I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.944017 − 0.784096I
a = 1.80124 + 0.03860I
b = −0.19373 − 2.11976I
−5.86990 + 7.37157I 6.00000 − 8.00321I
u = −1.028850 + 0.790391I
a = −2.50865 − 0.19833I
b = 0.91594 − 1.85439I
−14.0605 − 13.8029I 0
u = −1.028850 − 0.790391I
a = −2.50865 + 0.19833I
b = 0.91594 + 1.85439I
−14.0605 + 13.8029I 0
u = 1.035690 + 0.788962I
a = 2.07595 + 0.63512I
b = −0.21521 − 1.85036I
−13.8581 + 5.5876I 0
u = 1.035690 − 0.788962I
a = 2.07595 − 0.63512I
b = −0.21521 + 1.85036I
−13.8581 − 5.5876I 0
u = −0.691121
a = 7.56339
b = −0.229386
−0.704951 91.2610
u = 0.593654 + 0.309854I
a = −2.47422 + 0.91083I
b = 1.073020 − 0.556578I
−2.54134 + 1.31981I 0.58683 − 4.31452I
u = 0.593654 − 0.309854I
a = −2.47422 − 0.91083I
b = 1.073020 + 0.556578I
−2.54134 − 1.31981I 0.58683 + 4.31452I
u = −0.530048
a = −0.522918
b = −0.246323
0.790977 12.7730
u = 0.122257 + 0.387519I
a = −1.73330 + 0.73700I
b = 0.199239 + 0.977182I
−1.72184 − 1.26203I −0.75083 + 2.53377I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.122257 − 0.387519I
a = −1.73330 − 0.73700I
b = 0.199239 − 0.977182I
−1.72184 + 1.26203I −0.75083 − 2.53377I
9
II. I
u
2
= hb, 3u
7
+ u
6
− 4u
5
− 4u
4
+ 5u
3
+ 3u
2
+ a − u − 5, u
8
+ u
7
− u
6
−
2u
5
+ u
4
+ 2u
3
− 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
3
=
−3u
7
− u
6
+ 4u
5
+ 4u
4
− 5u
3
− 3u
2
+ u + 5
0
a
12
=
u
−u
3
+ u
a
6
=
1
0
a
4
=
−3u
7
− u
6
+ 4u
5
+ 4u
4
− 5u
3
− 3u
2
+ u + 5
0
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
5
=
−u
5
− u
u
7
− u
5
+ 2u
3
− u
a
2
=
−3u
7
− u
6
+ 5u
5
+ 4u
4
− 5u
3
− 3u
2
+ 2u + 5
−u
7
+ u
5
− 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9u
7
+ 6u
6
− 8u
5
− 14u
4
+ 15u
3
+ 9u
2
− 4u − 15
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
7
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
8
, c
9
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
11
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
12
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
8
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
7
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
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
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = 0.615431 − 0.295452I
b = 0
−0.604279 + 1.131230I 1.78185 − 1.82144I
u = −0.570868 − 0.730671I
a = 0.615431 + 0.295452I
b = 0
−0.604279 − 1.131230I 1.78185 + 1.82144I
u = 0.855237 + 0.665892I
a = −1.68119 − 0.49658I
b = 0
−3.80435 + 2.57849I −2.57592 − 5.06085I
u = 0.855237 − 0.665892I
a = −1.68119 + 0.49658I
b = 0
−3.80435 − 2.57849I −2.57592 + 5.06085I
u = 1.09818
a = 0.532015
b = 0
4.85780 6.04790
u = −1.031810 + 0.655470I
a = 0.473764 + 0.240160I
b = 0
0.73474 − 6.44354I 3.16642 + 7.92550I
u = −1.031810 − 0.655470I
a = 0.473764 − 0.240160I
b = 0
0.73474 + 6.44354I 3.16642 − 7.92550I
u = −0.603304
a = 4.65198
b = 0
−0.799899 −13.7930
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
41
+ 53u
40
+ ··· + 381u + 1)
c
2
((u − 1)
8
)(u
41
− 9u
40
+ ··· − 29u + 1)
c
3
, c
6
u
8
(u
41
+ 7u
40
+ ··· + 2176u − 256)
c
4
((u + 1)
8
)(u
41
− 9u
40
+ ··· − 29u + 1)
c
5
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
41
− 2u
40
+ ··· + u − 1)
c
7
(u
8
+ u
7
+ ··· − 2u − 1)(u
41
− 2u
40
+ ··· − u + 1)
c
8
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
41
− 2u
40
+ ··· + u − 1)
c
9
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
41
+ 2u
40
+ ··· + 12241u + 8353)
c
10
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
41
− 12u
40
+ ··· + 5u − 1)
c
11
(u
8
− u
7
+ ··· + 2u − 1)(u
41
− 2u
40
+ ··· − u + 1)
c
12
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
41
− 12u
40
+ ··· + 5u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
41
− 121y
40
+ ··· + 73433y −1)
c
2
, c
4
((y −1)
8
)(y
41
− 53y
40
+ ··· + 381y −1)
c
3
, c
6
y
8
(y
41
+ 51y
40
+ ··· + 2801664y −65536)
c
5
, c
8
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
41
+ 42y
39
+ ··· + 5y −1)
c
7
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
41
− 12y
40
+ ··· + 5y −1)
c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
41
+ 36y
40
+ ··· + 949291005y −69772609)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
41
+ 36y
40
+ ··· + 37y −1)
15
