12n
0590
(K12n
0590
)
A knot diagram
1
Linearized knot diagam
3 7 11 7 10 2 4 12 3 5 8 9
Solving Sequence
8,11
12 9
1,4
3 7 5 2 6 10
c
11
c
8
c
12
c
3
c
7
c
4
c
2
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h24558310627155u
21
+ 76226351773367u
20
+ ··· + 826989943797623b − 264882167921191,
− 1.11442 × 10
15
u
21
− 1.96283 × 10
15
u
20
+ ··· + 5.78893 × 10
15
a + 1.82405 × 10
14
, u
22
+ u
21
+ ··· − 12u + 7i
I
u
2
= h−u
7
+ 5u
5
+ u
4
− 7u
3
− 2u
2
+ b + 2u, u
10
− 8u
8
+ 23u
6
− u
5
− 29u
4
+ 4u
3
+ 15u
2
+ a − 4u − 1,
u
11
− 8u
9
− u
8
+ 23u
7
+ 5u
6
− 28u
5
− 7u
4
+ 13u
3
+ 2u
2
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 33 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
= h2.46×10
13
u
21
+7.62×10
13
u
20
+· · ·+8.27×10
14
b−2.65×10
14
, −1.11×
10
15
u
21
−1.96×10
15
u
20
+· · ·+5.79×10
15
a+1.82×10
14
, u
22
+u
21
+· · ·−12u+7i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
0.192509u
21
+ 0.339065u
20
+ ··· + 1.66279u − 0.0315092
−0.0296960u
21
− 0.0921733u
20
+ ··· − 0.726566u + 0.320297
a
3
=
0.162813u
21
+ 0.246892u
20
+ ··· + 0.936221u + 0.288788
−0.0296960u
21
− 0.0921733u
20
+ ··· − 0.726566u + 0.320297
a
7
=
−0.308808u
21
− 0.423908u
20
+ ··· − 4.90570u + 3.29659
−0.0596766u
21
− 0.0810654u
20
+ ··· + 1.56726u + 0.227107
a
5
=
−0.193496u
21
− 0.277468u
20
+ ··· + 1.30428u + 0.534875
0.117215u
21
+ 0.139319u
20
+ ··· − 0.863885u − 0.561596
a
2
=
−0.340275u
21
− 0.520418u
20
+ ··· − 1.11925u + 4.17997
−0.0351453u
21
− 0.0475187u
20
+ ··· − 0.171350u − 0.289676
a
6
=
0.285354u
21
+ 0.292486u
20
+ ··· − 3.40992u − 0.148449
−0.0727430u
21
− 0.200381u
20
+ ··· + 1.21818u + 0.857957
a
10
=
−0.115641u
21
− 0.306721u
20
+ ··· − 4.96241u + 2.03078
0.148193u
21
+ 0.158198u
20
+ ··· + 1.75028u − 0.764596
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
5642994687059
826989943797623
u
21
+
135118301910616
826989943797623
u
20
+ ··· +
9106726180669764
826989943797623
u −
9982429531246890
826989943797623
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 44u
21
+ ··· + 111391u + 14641
c
2
, c
6
u
22
− 22u
20
+ ··· − 95u − 121
c
3
u
22
+ 3u
21
+ ··· − 13u − 5
c
4
, c
7
u
22
+ 5u
21
+ ··· + 2u + 1
c
5
, c
10
u
22
− u
21
+ ··· − 123u − 173
c
8
, c
11
, c
12
u
22
+ u
21
+ ··· − 12u + 7
c
9
u
22
− 6u
21
+ ··· − 66255u − 6691
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
− 140y
21
+ ··· − 15971954947y + 214358881
c
2
, c
6
y
22
− 44y
21
+ ··· − 111391y + 14641
c
3
y
22
− 3y
21
+ ··· − 319y + 25
c
4
, c
7
y
22
+ 21y
21
+ ··· − 40y + 1
c
5
, c
10
y
22
− 9y
21
+ ··· − 332065y + 29929
c
8
, c
11
, c
12
y
22
− 37y
21
+ ··· − 186y + 49
c
9
y
22
− 122y
21
+ ··· − 1381946559y + 44769481
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.813375 + 0.695520I
a = −0.621272 + 0.888438I
b = 0.930849 − 0.474906I
−3.57261 + 1.24203I −13.51654 − 2.69011I
u = −0.813375 − 0.695520I
a = −0.621272 − 0.888438I
b = 0.930849 + 0.474906I
−3.57261 − 1.24203I −13.51654 + 2.69011I
u = 1.14577
a = −1.21118
b = −0.786013
−5.50697 −18.3200
u = 0.689625 + 0.112216I
a = 0.60149 + 2.27758I
b = 0.783398 − 0.626729I
−2.94663 + 3.18256I −14.1931 − 4.5682I
u = 0.689625 − 0.112216I
a = 0.60149 − 2.27758I
b = 0.783398 + 0.626729I
−2.94663 − 3.18256I −14.1931 + 4.5682I
u = −1.42854 + 0.12438I
a = −0.118525 + 0.124764I
b = −0.515254 − 0.715646I
−3.21068 − 1.94100I −8.44193 + 4.49457I
u = −1.42854 − 0.12438I
a = −0.118525 − 0.124764I
b = −0.515254 + 0.715646I
−3.21068 + 1.94100I −8.44193 − 4.49457I
u = 0.439809 + 0.342493I
a = 0.658925 − 0.629517I
b = 0.104940 + 1.110730I
2.79807 − 1.24078I −2.39806 + 5.78935I
u = 0.439809 − 0.342493I
a = 0.658925 + 0.629517I
b = 0.104940 − 1.110730I
2.79807 + 1.24078I −2.39806 − 5.78935I
u = −1.47818 + 0.29462I
a = −0.196203 + 1.126150I
b = −0.95775 − 1.05246I
−10.23510 − 1.31699I −13.69750 + 1.12150I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.47818 − 0.29462I
a = −0.196203 − 1.126150I
b = −0.95775 + 1.05246I
−10.23510 + 1.31699I −13.69750 − 1.12150I
u = −0.389548
a = 0.489094
b = 0.423427
−0.651086 −15.1770
u = 0.055531 + 0.375870I
a = 3.27936 − 0.40059I
b = −0.735830 + 0.136693I
−1.05201 + 2.37866I −7.37484 + 0.30199I
u = 0.055531 − 0.375870I
a = 3.27936 + 0.40059I
b = −0.735830 − 0.136693I
−1.05201 − 2.37866I −7.37484 − 0.30199I
u = 1.54618 + 0.54222I
a = 0.045294 + 0.967197I
b = −1.17072 − 0.88115I
−11.07800 − 6.11809I −13.7689 + 3.7652I
u = 1.54618 − 0.54222I
a = 0.045294 − 0.967197I
b = −1.17072 + 0.88115I
−11.07800 + 6.11809I −13.7689 − 3.7652I
u = −1.78595
a = 1.55694
b = 0.706857
−16.3042 −23.6750
u = 1.92376 + 0.13131I
a = −0.017132 + 0.779055I
b = 1.13005 − 1.28185I
16.3680 − 1.4414I −13.10276 + 0.15786I
u = 1.92376 − 0.13131I
a = −0.017132 − 0.779055I
b = 1.13005 + 1.28185I
16.3680 + 1.4414I −13.10276 − 0.15786I
u = −1.92464 + 0.18067I
a = 0.224483 + 0.973602I
b = 1.24119 − 1.08462I
15.7619 + 10.2221I −12.91781 − 4.03002I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.92464 − 0.18067I
a = 0.224483 − 0.973602I
b = 1.24119 + 1.08462I
15.7619 − 10.2221I −12.91781 + 4.03002I
u = 2.00938
a = −0.119138
b = 1.03398
−17.7473 −16.0060
7
II. I
u
2
=
h−u
7
+5u
5
+u
4
−7u
3
−2u
2
+b+2u, u
10
−8u
8
+· · ·+a−1, u
11
−8u
9
+· · ·−2u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
9
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−u
10
+ 8u
8
− 23u
6
+ u
5
+ 29u
4
− 4u
3
− 15u
2
+ 4u + 1
u
7
− 5u
5
− u
4
+ 7u
3
+ 2u
2
− 2u
a
3
=
−u
10
+ 8u
8
+ u
7
− 23u
6
− 4u
5
+ 28u
4
+ 3u
3
− 13u
2
+ 2u + 1
u
7
− 5u
5
− u
4
+ 7u
3
+ 2u
2
− 2u
a
7
=
−u
8
− u
7
+ 6u
6
+ 6u
5
− 11u
4
− 10u
3
+ 7u
2
+ 4u − 3
−u
10
+ 7u
8
+ u
7
− 17u
6
− 4u
5
+ 17u
4
+ 4u
3
− 7u
2
+ 1
a
5
=
u
10
− 8u
8
− u
7
+ 23u
6
+ 5u
5
− 28u
4
− 6u
3
+ 14u
2
− 3
−u
10
+ 7u
8
+ u
7
− 17u
6
− 4u
5
+ 16u
4
+ 4u
3
− 4u
2
a
2
=
u
8
+ u
7
− 6u
6
− 5u
5
+ 11u
4
+ 7u
3
− 7u
2
− 2u + 2
u
7
− 5u
5
− 2u
4
+ 7u
3
+ 5u
2
− 2u − 1
a
6
=
u
10
− 7u
8
− u
7
+ 18u
6
+ 5u
5
− 20u
4
− 7u
3
+ 9u
2
+ 2u − 2
−u
7
+ 5u
5
+ u
4
− 7u
3
− 3u
2
+ 2u + 1
a
10
=
u
9
+ u
8
− 6u
7
− 6u
6
+ 11u
5
+ 10u
4
− 7u
3
− 3u
2
+ 3u
−u
3
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
10
− 2u
9
− 17u
8
+ 12u
7
+ 54u
6
− 24u
5
− 73u
4
+ 21u
3
+ 35u
2
− 12u − 17
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
− 11u
10
+ ··· + 5u − 1
c
2
u
11
− 3u
10
− u
9
+ 8u
8
− 2u
7
− 10u
6
+ 5u
5
+ 6u
4
− 5u
3
− 2u
2
+ 3u − 1
c
3
u
11
− 2u
10
+ 3u
9
− 2u
8
+ u
7
+ u
6
− u
5
+ u
4
+ 5u
3
− 4u
2
− u + 1
c
4
u
11
+ 3u
9
− 6u
8
+ u
7
− 20u
6
− 4u
5
− 26u
4
− 6u
3
− 15u
2
− 4u − 3
c
5
u
11
+ 4u
9
− u
8
+ 4u
7
− 4u
6
− u
5
− 5u
4
− 2u
3
− 3u
2
− u − 1
c
6
u
11
+ 3u
10
− u
9
− 8u
8
− 2u
7
+ 10u
6
+ 5u
5
− 6u
4
− 5u
3
+ 2u
2
+ 3u + 1
c
7
u
11
+ 3u
9
+ 6u
8
+ u
7
+ 20u
6
− 4u
5
+ 26u
4
− 6u
3
+ 15u
2
− 4u + 3
c
8
u
11
− 8u
9
+ u
8
+ 23u
7
− 5u
6
− 28u
5
+ 7u
4
+ 13u
3
− 2u
2
− 2u + 1
c
9
u
11
− 5u
10
+ ··· + 5u − 1
c
10
u
11
+ 4u
9
+ u
8
+ 4u
7
+ 4u
6
− u
5
+ 5u
4
− 2u
3
+ 3u
2
− u + 1
c
11
, c
12
u
11
− 8u
9
− u
8
+ 23u
7
+ 5u
6
− 28u
5
− 7u
4
+ 13u
3
+ 2u
2
− 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 31y
10
+ ··· − 19y −1
c
2
, c
6
y
11
− 11y
10
+ ··· + 5y −1
c
3
y
11
+ 2y
10
+ ··· + 9y −1
c
4
, c
7
y
11
+ 6y
10
+ ··· − 74y −9
c
5
, c
10
y
11
+ 8y
10
+ ··· − 5y −1
c
8
, c
11
, c
12
y
11
− 16y
10
+ ··· + 8y −1
c
9
y
11
− 33y
10
+ ··· − 11y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.137090 + 0.296275I
a = −0.112692 − 0.807373I
b = −0.583202 − 0.143276I
−4.14701 − 1.11671I −15.2828 + 0.9622I
u = −1.137090 − 0.296275I
a = −0.112692 + 0.807373I
b = −0.583202 + 0.143276I
−4.14701 + 1.11671I −15.2828 − 0.9622I
u = 0.592958 + 0.265544I
a = −0.648705 − 0.517226I
b = 0.224636 + 1.262620I
2.05237 − 0.90366I −12.99258 + 0.59002I
u = 0.592958 − 0.265544I
a = −0.648705 + 0.517226I
b = 0.224636 − 1.262620I
2.05237 + 0.90366I −12.99258 − 0.59002I
u = 1.52480 + 0.07903I
a = −0.465943 + 0.973368I
b = −1.133880 − 0.701740I
−7.77644 − 4.37367I −13.16320 + 3.47973I
u = 1.52480 − 0.07903I
a = −0.465943 − 0.973368I
b = −1.133880 + 0.701740I
−7.77644 + 4.37367I −13.16320 − 3.47973I
u = −1.59014 + 0.09758I
a = 0.004921 − 0.710560I
b = −0.62219 + 1.27817I
−5.52612 + 2.32410I −12.39685 − 2.31746I
u = −1.59014 − 0.09758I
a = 0.004921 + 0.710560I
b = −0.62219 − 1.27817I
−5.52612 − 2.32410I −12.39685 + 2.31746I
u = −0.300210 + 0.263166I
a = −1.31732 + 2.93205I
b = 0.867895 − 0.444411I
−1.38510 + 3.17083I −10.45196 − 6.81024I
u = −0.300210 − 0.263166I
a = −1.31732 − 2.93205I
b = 0.867895 + 0.444411I
−1.38510 − 3.17083I −10.45196 + 6.81024I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.81936
a = 1.07948
b = 0.493485
−15.7834 −7.42530
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
11
− 11u
10
+ ··· + 5u − 1)(u
22
+ 44u
21
+ ··· + 111391u + 14641)
c
2
(u
11
− 3u
10
− u
9
+ 8u
8
− 2u
7
− 10u
6
+ 5u
5
+ 6u
4
− 5u
3
− 2u
2
+ 3u − 1)
· (u
22
− 22u
20
+ ··· − 95u − 121)
c
3
(u
11
− 2u
10
+ 3u
9
− 2u
8
+ u
7
+ u
6
− u
5
+ u
4
+ 5u
3
− 4u
2
− u + 1)
· (u
22
+ 3u
21
+ ··· − 13u − 5)
c
4
(u
11
+ 3u
9
− 6u
8
+ u
7
− 20u
6
− 4u
5
− 26u
4
− 6u
3
− 15u
2
− 4u − 3)
· (u
22
+ 5u
21
+ ··· + 2u + 1)
c
5
(u
11
+ 4u
9
− u
8
+ 4u
7
− 4u
6
− u
5
− 5u
4
− 2u
3
− 3u
2
− u − 1)
· (u
22
− u
21
+ ··· − 123u − 173)
c
6
(u
11
+ 3u
10
− u
9
− 8u
8
− 2u
7
+ 10u
6
+ 5u
5
− 6u
4
− 5u
3
+ 2u
2
+ 3u + 1)
· (u
22
− 22u
20
+ ··· − 95u − 121)
c
7
(u
11
+ 3u
9
+ 6u
8
+ u
7
+ 20u
6
− 4u
5
+ 26u
4
− 6u
3
+ 15u
2
− 4u + 3)
· (u
22
+ 5u
21
+ ··· + 2u + 1)
c
8
(u
11
− 8u
9
+ u
8
+ 23u
7
− 5u
6
− 28u
5
+ 7u
4
+ 13u
3
− 2u
2
− 2u + 1)
· (u
22
+ u
21
+ ··· − 12u + 7)
c
9
(u
11
− 5u
10
+ ··· + 5u − 1)(u
22
− 6u
21
+ ··· − 66255u − 6691)
c
10
(u
11
+ 4u
9
+ u
8
+ 4u
7
+ 4u
6
− u
5
+ 5u
4
− 2u
3
+ 3u
2
− u + 1)
· (u
22
− u
21
+ ··· − 123u − 173)
c
11
, c
12
(u
11
− 8u
9
− u
8
+ 23u
7
+ 5u
6
− 28u
5
− 7u
4
+ 13u
3
+ 2u
2
− 2u − 1)
· (u
22
+ u
21
+ ··· − 12u + 7)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
11
− 31y
10
+ ··· − 19y −1)
· (y
22
− 140y
21
+ ··· − 15971954947y + 214358881)
c
2
, c
6
(y
11
− 11y
10
+ ··· + 5y −1)(y
22
− 44y
21
+ ··· − 111391y + 14641)
c
3
(y
11
+ 2y
10
+ ··· + 9y −1)(y
22
− 3y
21
+ ··· − 319y + 25)
c
4
, c
7
(y
11
+ 6y
10
+ ··· − 74y −9)(y
22
+ 21y
21
+ ··· − 40y + 1)
c
5
, c
10
(y
11
+ 8y
10
+ ··· − 5y −1)(y
22
− 9y
21
+ ··· − 332065y + 29929)
c
8
, c
11
, c
12
(y
11
− 16y
10
+ ··· + 8y −1)(y
22
− 37y
21
+ ··· − 186y + 49)
c
9
(y
11
− 33y
10
+ ··· − 11y −1)
· (y
22
− 122y
21
+ ··· − 1381946559y + 44769481)
14
