12n
0768
(K12n
0768
)
A knot diagram
1
Linearized knot diagam
4 6 9 10 11 3 12 11 2 5 8 9
Solving Sequence
5,10
11 6
2,4
1 9 3 7 8 12
c
10
c
5
c
4
c
1
c
9
c
3
c
6
c
8
c
12
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.92778 × 10
21
u
25
+ 2.56531 × 10
22
u
24
+ ··· + 3.87990 × 10
23
b − 1.26738 × 10
24
,
2.71108 × 10
23
u
25
+ 1.09424 × 10
24
u
24
+ ··· + 1.20277 × 10
25
a − 1.27374 × 10
26
, u
26
− u
25
+ ··· + 12u + 31i
I
u
2
= h−u
13
+ 6u
11
+ u
10
− 14u
9
− 6u
8
+ 14u
7
+ 15u
6
− 3u
5
− 18u
4
− 5u
3
+ 9u
2
+ b + 3u − 1,
− u
13
+ u
12
+ 6u
11
− 4u
10
− 15u
9
+ 2u
8
+ 19u
7
+ 13u
6
− 12u
5
− 22u
4
+ 10u
2
+ a + 5u − 1,
u
14
− 7u
12
− u
11
+ 19u
10
+ 7u
9
− 23u
8
− 20u
7
+ 8u
6
+ 28u
5
+ 7u
4
− 17u
3
− 6u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 40 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
= h−2.93 × 10
21
u
25
+ 2.57 × 10
22
u
24
+ · · · + 3.88 × 10
23
b − 1.27 ×
10
24
, 2.71 × 10
23
u
25
+ 1.09 × 10
24
u
24
+ · · · + 1.20 × 10
25
a − 1.27 ×
10
26
, u
26
− u
25
+ · · · + 12u + 31i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
2
=
−0.0225403u
25
− 0.0909766u
24
+ ··· + 5.26309u + 10.5900
0.00754601u
25
− 0.0661178u
24
+ ··· − 2.55257u + 3.26654
a
4
=
u
u
a
1
=
−0.0460308u
25
− 0.0975533u
24
+ ··· + 3.67107u + 8.88673
−0.0159445u
25
− 0.0726945u
24
+ ··· − 4.14458u + 1.56324
a
9
=
−0.0228681u
25
− 0.107777u
24
+ ··· − 0.424141u + 6.30289
0.105094u
25
− 0.0463805u
24
+ ··· − 7.82285u − 2.16744
a
3
=
−0.0714029u
25
− 0.131292u
24
+ ··· + 7.63526u + 15.5015
0.00756581u
25
− 0.0451761u
24
+ ··· − 2.33987u + 1.11953
a
7
=
−0.466848u
25
+ 0.229048u
24
+ ··· + 25.8666u + 8.47181
0.0444585u
25
+ 0.0166367u
24
+ ··· − 2.65167u − 3.76153
a
8
=
−0.192283u
25
− 0.0910018u
24
+ ··· + 9.67536u + 12.5203
−0.00596389u
25
− 0.0322573u
24
+ ··· − 0.739325u + 2.56438
a
12
=
0.235482u
25
− 0.154199u
24
+ ··· − 20.6106u − 7.26827
0.0223603u
25
− 0.0205281u
24
+ ··· − 3.27390u − 0.0676271
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
46652502203510047014961
129330126954099989822173
u
25
+
99744336497876668876271
387990380862299969466519
u
24
+ ··· +
7467042075154247618912122
387990380862299969466519
u −
1919651047154391762384890
387990380862299969466519
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
− 24u
24
+ ··· − 352u + 41
c
2
, c
6
u
26
− 4u
25
+ ··· − 322u − 529
c
3
u
26
− u
25
+ ··· − 12769u − 1781
c
4
, c
5
, c
10
u
26
+ u
25
+ ··· − 12u + 31
c
7
, c
8
, c
11
u
26
+ u
25
+ ··· − 34u − 4
c
9
u
26
+ 2u
25
+ ··· + 36u − 19
c
12
u
26
− u
25
+ ··· − 304218u − 40564
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 48y
25
+ ··· + 19924y + 1681
c
2
, c
6
y
26
+ 12y
25
+ ··· − 348082y + 279841
c
3
y
26
+ 69y
25
+ ··· − 266466469y + 3171961
c
4
, c
5
, c
10
y
26
− 33y
25
+ ··· − 12482y + 961
c
7
, c
8
, c
11
y
26
+ 43y
25
+ ··· − 716y + 16
c
9
y
26
+ 6y
25
+ ··· + 1782y + 361
c
12
y
26
+ 147y
25
+ ··· − 80260701260y + 1645438096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.764982 + 0.335496I
a = 0.043942 − 0.279919I
b = 1.019140 − 0.401366I
−1.38320 − 3.64635I 1.35824 + 7.32767I
u = 0.764982 − 0.335496I
a = 0.043942 + 0.279919I
b = 1.019140 + 0.401366I
−1.38320 + 3.64635I 1.35824 − 7.32767I
u = 0.777607 + 0.118039I
a = 2.55907 − 0.95852I
b = −0.478055 − 0.742412I
−13.15060 − 0.37776I −1.22408 − 1.34498I
u = 0.777607 − 0.118039I
a = 2.55907 + 0.95852I
b = −0.478055 + 0.742412I
−13.15060 + 0.37776I −1.22408 + 1.34498I
u = 0.842907 + 0.881469I
a = −0.571747 + 0.134450I
b = −0.469429 − 0.674553I
−1.203930 − 0.496037I 0.79686 − 2.54300I
u = 0.842907 − 0.881469I
a = −0.571747 − 0.134450I
b = −0.469429 + 0.674553I
−1.203930 + 0.496037I 0.79686 + 2.54300I
u = −0.719777
a = −0.187725
b = −1.16272
2.17524 2.79410
u = 0.159981 + 0.568954I
a = 0.445478 + 0.461168I
b = −0.321692 + 0.362997I
0.333623 − 1.008960I 5.28376 + 6.66934I
u = 0.159981 − 0.568954I
a = 0.445478 − 0.461168I
b = −0.321692 − 0.362997I
0.333623 + 1.008960I 5.28376 − 6.66934I
u = −1.40068 + 0.23284I
a = −0.23778 + 1.47903I
b = 0.413279 + 0.877413I
−4.79645 + 4.01069I 0.34699 − 9.12487I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40068 − 0.23284I
a = −0.23778 − 1.47903I
b = 0.413279 − 0.877413I
−4.79645 − 4.01069I 0.34699 + 9.12487I
u = −0.571043
a = 2.07972
b = 0.907867
2.93787 −5.61080
u = −0.484610 + 0.243441I
a = −1.71290 + 0.53126I
b = 0.461065 + 0.861987I
−3.51192 + 0.62346I −2.84586 − 0.67416I
u = −0.484610 − 0.243441I
a = −1.71290 − 0.53126I
b = 0.461065 − 0.861987I
−3.51192 − 0.62346I −2.84586 + 0.67416I
u = 1.52698 + 0.20871I
a = 0.133555 + 1.387270I
b = −1.051380 + 0.928731I
−10.29450 − 2.62985I −1.70383 + 1.78381I
u = 1.52698 − 0.20871I
a = 0.133555 − 1.387270I
b = −1.051380 − 0.928731I
−10.29450 + 2.62985I −1.70383 − 1.78381I
u = −0.82721 + 1.33251I
a = 0.352400 − 0.194950I
b = −0.445225 − 1.242010I
−15.2105 + 4.3640I −1.41577 − 3.17462I
u = −0.82721 − 1.33251I
a = 0.352400 + 0.194950I
b = −0.445225 + 1.242010I
−15.2105 − 4.3640I −1.41577 + 3.17462I
u = 1.65746 + 0.14713I
a = 0.159581 + 0.992092I
b = 0.250667 + 1.107520I
−5.67875 − 1.02437I −0.521896 − 0.910763I
u = 1.65746 − 0.14713I
a = 0.159581 − 0.992092I
b = 0.250667 − 1.107520I
−5.67875 + 1.02437I −0.521896 + 0.910763I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.77337 + 0.06842I
a = 0.294799 − 0.960179I
b = 1.44079 − 1.23613I
16.7710 + 1.3129I −1.252333 − 0.225172I
u = −1.77337 − 0.06842I
a = 0.294799 + 0.960179I
b = 1.44079 + 1.23613I
16.7710 − 1.3129I −1.252333 + 0.225172I
u = 1.74642 + 0.44620I
a = −0.080704 − 1.254640I
b = 1.18154 − 1.35502I
16.0643 − 11.0020I −1.09431 + 4.10898I
u = 1.74642 − 0.44620I
a = −0.080704 + 1.254640I
b = 1.18154 + 1.35502I
16.0643 + 11.0020I −1.09431 − 4.10898I
u = −1.84505 + 0.14739I
a = −0.154266 − 1.018760I
b = −0.87327 − 1.36223I
−11.74950 + 5.10402I −1.81943 − 2.88501I
u = −1.84505 − 0.14739I
a = −0.154266 + 1.018760I
b = −0.87327 + 1.36223I
−11.74950 − 5.10402I −1.81943 + 2.88501I
7
II.
I
u
2
= h−u
13
+6u
11
+· · ·+b−1, −u
13
+u
12
+· · ·+a−1, u
14
−7u
12
+· · ·+2u+1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
2
=
u
13
− u
12
+ ··· − 5u + 1
u
13
− 6u
11
+ ··· − 3u + 1
a
4
=
u
u
a
1
=
u
13
+ u
12
+ ··· − 4u
2
− 7u
u
13
+ 2u
12
+ ··· − 3u
2
− 5u
a
9
=
−u
13
+ 2u
12
+ ··· + 10u
3
+ 3
2u
12
− u
11
+ ··· − u + 1
a
3
=
u
13
− 6u
11
+ ··· − 5u
2
− 8u
u
13
− 6u
11
+ ··· − 2u + 1
a
7
=
u
13
− 5u
11
− u
10
+ 8u
9
+ 4u
8
− u
7
− 5u
6
− 6u
5
− u
3
+ 3u
2
+ 6u
u
12
+ u
11
+ ··· + u − 2
a
8
=
−3u
13
+ 4u
12
+ ··· + 20u
2
− 2u
−2u
13
+ 5u
12
+ ··· − 3u − 1
a
12
=
−4u
13
+ 5u
12
+ ··· − 5u − 7
−3u
13
+ 6u
12
+ ··· − 8u − 4
(ii) Obstruction class = 1
(iii) Cusp Shapes =
−2u
13
−3u
12
+16u
11
+20u
10
−44u
9
−56u
8
+40u
7
+84u
6
+28u
5
−71u
4
−68u
3
+24u
2
+27u+7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
− 5u
13
+ ··· − u
2
+ 1
c
2
u
14
− 3u
13
+ 8u
11
− 9u
10
+ 13u
8
− 16u
7
+ 14u
5
− 9u
4
− 3u
3
+ 4u
2
− 1
c
3
u
14
+ 4u
12
+ 5u
10
+ 2u
9
− 3u
8
+ 4u
7
− u
6
− u
5
+ u
4
+ 2u
3
+ u
2
− u − 1
c
4
, c
5
u
14
− 7u
12
+ ··· − 2u + 1
c
6
u
14
+ 3u
13
− 8u
11
− 9u
10
+ 13u
8
+ 16u
7
− 14u
5
− 9u
4
+ 3u
3
+ 4u
2
− 1
c
7
, c
8
u
14
+ 9u
12
+ ··· + 2u − 1
c
9
u
14
+ u
13
− u
12
− 2u
11
− u
10
+ u
9
+ u
8
− 4u
7
+ 3u
6
− 2u
5
− 5u
4
− 4u
2
− 1
c
10
u
14
− 7u
12
+ ··· + 2u + 1
c
11
u
14
+ 9u
12
+ ··· − 2u − 1
c
12
u
14
+ 11u
12
+ ··· − 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 17y
13
+ ··· − 2y + 1
c
2
, c
6
y
14
− 9y
13
+ ··· − 8y + 1
c
3
y
14
+ 8y
13
+ ··· − 3y + 1
c
4
, c
5
, c
10
y
14
− 14y
13
+ ··· − 16y + 1
c
7
, c
8
, c
11
y
14
+ 18y
13
+ ··· − 6y + 1
c
9
y
14
− 3y
13
+ ··· + 8y + 1
c
12
y
14
+ 22y
13
+ ··· − 4y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.388251 + 0.920456I
a = 0.055423 − 0.175899I
b = 0.169523 + 0.638462I
−1.29955 + 1.48242I 0.47756 − 5.07450I
u = −0.388251 − 0.920456I
a = 0.055423 + 0.175899I
b = 0.169523 − 0.638462I
−1.29955 − 1.48242I 0.47756 + 5.07450I
u = 1.132050 + 0.372393I
a = 1.77286 + 0.32994I
b = −0.171528 + 0.571644I
−13.37670 − 1.61202I −3.25492 + 4.00039I
u = 1.132050 − 0.372393I
a = 1.77286 − 0.32994I
b = −0.171528 − 0.571644I
−13.37670 + 1.61202I −3.25492 − 4.00039I
u = 1.28132
a = 0.283790
b = 1.39820
0.241511 −0.487210
u = −1.290010 + 0.033553I
a = −0.507609 + 0.986318I
b = −1.39611 + 0.51797I
−4.48352 − 1.89225I −0.38957 + 1.57022I
u = −1.290010 − 0.033553I
a = −0.507609 − 0.986318I
b = −1.39611 − 0.51797I
−4.48352 + 1.89225I −0.38957 − 1.57022I
u = −1.38089 + 0.32443I
a = −0.496730 + 1.151660I
b = 0.353075 + 0.853393I
−5.02225 + 3.24893I −3.06027 − 0.56445I
u = −1.38089 − 0.32443I
a = −0.496730 − 1.151660I
b = 0.353075 − 0.853393I
−5.02225 − 3.24893I −3.06027 + 0.56445I
u = 1.45042 + 0.21038I
a = −0.11773 + 1.53514I
b = −0.623035 + 1.209470I
−7.62759 − 4.80166I −1.22227 + 4.11227I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.45042 − 0.21038I
a = −0.11773 − 1.53514I
b = −0.623035 − 1.209470I
−7.62759 + 4.80166I −1.22227 − 4.11227I
u = 0.434339
a = −2.22641
b = −1.03128
3.32890 16.1810
u = −0.381146 + 0.175722I
a = 1.76509 − 0.05508I
b = 0.984611 + 0.552289I
−1.22934 + 2.47209I 1.10273 − 1.06165I
u = −0.381146 − 0.175722I
a = 1.76509 + 0.05508I
b = 0.984611 − 0.552289I
−1.22934 − 2.47209I 1.10273 + 1.06165I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
14
− 5u
13
+ ··· − u
2
+ 1)(u
26
− 24u
24
+ ··· − 352u + 41)
c
2
(u
14
− 3u
13
+ 8u
11
− 9u
10
+ 13u
8
− 16u
7
+ 14u
5
− 9u
4
− 3u
3
+ 4u
2
− 1)
· (u
26
− 4u
25
+ ··· − 322u − 529)
c
3
(u
14
+ 4u
12
+ 5u
10
+ 2u
9
− 3u
8
+ 4u
7
− u
6
− u
5
+ u
4
+ 2u
3
+ u
2
− u − 1)
· (u
26
− u
25
+ ··· − 12769u − 1781)
c
4
, c
5
(u
14
− 7u
12
+ ··· − 2u + 1)(u
26
+ u
25
+ ··· − 12u + 31)
c
6
(u
14
+ 3u
13
− 8u
11
− 9u
10
+ 13u
8
+ 16u
7
− 14u
5
− 9u
4
+ 3u
3
+ 4u
2
− 1)
· (u
26
− 4u
25
+ ··· − 322u − 529)
c
7
, c
8
(u
14
+ 9u
12
+ ··· + 2u − 1)(u
26
+ u
25
+ ··· − 34u − 4)
c
9
(u
14
+ u
13
− u
12
− 2u
11
− u
10
+ u
9
+ u
8
− 4u
7
+ 3u
6
− 2u
5
− 5u
4
− 4u
2
− 1)
· (u
26
+ 2u
25
+ ··· + 36u − 19)
c
10
(u
14
− 7u
12
+ ··· + 2u + 1)(u
26
+ u
25
+ ··· − 12u + 31)
c
11
(u
14
+ 9u
12
+ ··· − 2u − 1)(u
26
+ u
25
+ ··· − 34u − 4)
c
12
(u
14
+ 11u
12
+ ··· − 2u − 1)(u
26
− u
25
+ ··· − 304218u − 40564)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
14
− 17y
13
+ ··· − 2y + 1)(y
26
− 48y
25
+ ··· + 19924y + 1681)
c
2
, c
6
(y
14
− 9y
13
+ ··· − 8y + 1)(y
26
+ 12y
25
+ ··· − 348082y + 279841)
c
3
(y
14
+ 8y
13
+ ··· − 3y + 1)
· (y
26
+ 69y
25
+ ··· − 266466469y + 3171961)
c
4
, c
5
, c
10
(y
14
− 14y
13
+ ··· − 16y + 1)(y
26
− 33y
25
+ ··· − 12482y + 961)
c
7
, c
8
, c
11
(y
14
+ 18y
13
+ ··· − 6y + 1)(y
26
+ 43y
25
+ ··· − 716y + 16)
c
9
(y
14
− 3y
13
+ ··· + 8y + 1)(y
26
+ 6y
25
+ ··· + 1782y + 361)
c
12
(y
14
+ 22y
13
+ ··· − 4y + 1)
· (y
26
+ 147y
25
+ ··· − 80260701260y + 1645438096)
14
