12n
0318
(K12n
0318
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 2 4 10 12 6 4 8 9
Solving Sequence
5,10
4
8,11
12 3 7 6 2 1 9
c
4
c
10
c
11
c
3
c
7
c
6
c
2
c
1
c
9
c
5
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−46u
9
+ 96u
8
− 471u
7
+ 631u
6
− 1426u
5
+ 1075u
4
− 1285u
3
+ 249u
2
+ 61b − 157u − 203,
− 3u
9
+ 5u
8
− 28u
7
+ 29u
6
− 74u
5
+ 37u
4
− 51u
3
+ u
2
+ a − 5u − 8,
u
10
− 2u
9
+ 10u
8
− 13u
7
+ 29u
6
− 22u
5
+ 24u
4
− 8u
3
+ 3u
2
+ 2u − 1i
I
u
2
= h4u
9
− 38u
8
+ 23u
7
− 323u
6
+ 100u
5
− 821u
4
+ 395u
3
− 595u
2
+ 185b + 617u − 243,
− 439u
9
− 177u
8
− 3588u
7
− 117u
6
− 9310u
5
+ 2461u
4
− 8895u
3
+ 5315u
2
+ 185a − 3937u + 1278,
u
10
+ 8u
8
− 3u
7
+ 21u
6
− 14u
5
+ 22u
4
− 20u
3
+ 13u
2
− 6u + 1i
* 2 irreducible components of dim
C
= 0, with total 20 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−46u
9
+96u
8
+· · ·+61b−203, −3u
9
+5u
8
+· · ·+a−8, u
10
−2u
9
+· · ·+2u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
8
=
3u
9
− 5u
8
+ 28u
7
− 29u
6
+ 74u
5
− 37u
4
+ 51u
3
− u
2
+ 5u + 8
0.754098u
9
− 1.57377u
8
+ ··· + 2.57377u + 3.32787
a
11
=
u
u
3
+ u
a
12
=
7.27869u
9
− 12.0164u
8
+ ··· + 22.0164u + 22.2951
3.27869u
9
− 5.01639u
8
+ ··· + 9.01639u + 8.29508
a
3
=
−7.70492u
9
+ 12.6885u
8
+ ··· − 21.6885u − 22.3934
−3.40984u
9
+ 5.37705u
8
+ ··· − 8.37705u − 8.78689
a
7
=
3u
9
− 5u
8
+ 28u
7
− 29u
6
+ 74u
5
− 37u
4
+ 51u
3
− u
2
+ 5u + 8
0.754098u
9
− 1.57377u
8
+ ··· + 3.57377u + 4.32787
a
6
=
3.75410u
9
− 6.57377u
8
+ ··· + 7.57377u + 11.3279
0.803279u
9
− 1.45902u
8
+ ··· + 4.45902u + 4.26230
a
2
=
1.26230u
9
− 1.72131u
8
+ ··· + 2.72131u + 3.98361
0.983607u
9
− 1.70492u
8
+ ··· + 0.704918u + 1.68852
a
1
=
51.8033u
9
− 86.4590u
8
+ ··· + 129.459u + 150.262
19.7377u
9
− 31.2787u
8
+ ··· + 54.2787u + 58.0164
a
9
=
−20.5410u
9
+ 33.7377u
8
+ ··· − 52.7377u − 58.2787
−8.31148u
9
+ 14.6066u
8
+ ··· − 18.6066u − 23.9180
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
62
61
u
9
+
201
61
u
8
−
725
61
u
7
+
1487
61
u
6
−
2349
61
u
5
+
3080
61
u
4
−
2090
61
u
3
+
1667
61
u
2
−
445
61
u +
103
61
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 22u
9
+ ··· + 5917u + 49
c
2
, c
5
u
10
+ 2u
9
+ ··· − 59u + 7
c
3
u
10
+ 3u
9
+ ··· + 12u − 13
c
4
, c
10
u
10
+ 2u
9
+ 10u
8
+ 13u
7
+ 29u
6
+ 22u
5
+ 24u
4
+ 8u
3
+ 3u
2
− 2u − 1
c
6
u
10
− 3u
9
+ ··· + 110u − 25
c
7
u
10
+ 3u
9
− 20u
7
− 41u
6
+ 294u
5
− 405u
4
+ 219u
3
+ 58u
2
− 33u − 9
c
8
, c
11
, c
12
u
10
− 3u
9
+ ··· + 37u − 29
c
9
u
10
+ u
9
+ ··· − 25u − 167
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 34y
9
+ ··· − 33340185y + 2401
c
2
, c
5
y
10
− 22y
9
+ ··· − 5917y + 49
c
3
y
10
− 19y
9
+ ··· − 2666y + 169
c
4
, c
10
y
10
+ 16y
9
+ ··· − 10y + 1
c
6
y
10
− 27y
9
+ ··· − 23900y + 625
c
7
y
10
− 9y
9
+ ··· − 2133y + 81
c
8
, c
11
, c
12
y
10
− 21y
9
+ ··· + 3097y + 841
c
9
y
10
− 17y
9
+ ··· − 55401y + 27889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.058067 + 0.959343I
a = −1.02272 + 1.68359I
b = −0.56593 − 1.71994I
2.74627 + 1.52551I 1.61031 − 1.19705I
u = 0.058067 − 0.959343I
a = −1.02272 − 1.68359I
b = −0.56593 + 1.71994I
2.74627 − 1.52551I 1.61031 + 1.19705I
u = 0.369407 + 0.683035I
a = 0.045927 − 0.332128I
b = 0.679565 + 0.238520I
−1.42142 − 0.67316I −3.66351 + 1.21479I
u = 0.369407 − 0.683035I
a = 0.045927 + 0.332128I
b = 0.679565 − 0.238520I
−1.42142 + 0.67316I −3.66351 − 1.21479I
u = −0.387852
a = 1.30915
b = −0.163807
0.892115 12.2090
u = 0.346316
a = 11.5322
b = 4.45406
5.57991 1.58660
u = 0.45233 + 1.77782I
a = −0.557671 + 0.304853I
b = −0.225028 − 1.028840I
−9.90958 + 3.92064I 1.45511 − 3.03765I
u = 0.45233 − 1.77782I
a = −0.557671 − 0.304853I
b = −0.225028 + 1.028840I
−9.90958 − 3.92064I 1.45511 + 3.03765I
u = 0.14096 + 1.98796I
a = 1.61376 − 0.88253I
b = −3.53373 + 2.38309I
15.2183 + 8.0662I 1.70029 − 2.38226I
u = 0.14096 − 1.98796I
a = 1.61376 + 0.88253I
b = −3.53373 − 2.38309I
15.2183 − 8.0662I 1.70029 + 2.38226I
5
II. I
u
2
= h4u
9
− 38u
8
+ · · · + 185b − 243, −439u
9
− 177u
8
+ · · · + 185a +
1278, u
10
+ 8u
8
+ · · · − 6u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
8
=
2.37297u
9
+ 0.956757u
8
+ ··· + 21.2811u − 6.90811
−0.0216216u
9
+ 0.205405u
8
+ ··· − 3.33514u + 1.31351
a
11
=
u
u
3
+ u
a
12
=
1.37297u
9
+ 0.956757u
8
+ ··· + 9.28108u − 0.908108
0.432432u
9
− 0.108108u
8
+ ··· + 8.70270u − 2.27027
a
3
=
−0.745946u
9
+ 0.0864865u
8
+ ··· − 7.56216u + 3.81622
0.745946u
9
− 0.0864865u
8
+ ··· + 7.56216u − 1.81622
a
7
=
2.37297u
9
+ 0.956757u
8
+ ··· + 21.2811u − 6.90811
0.389189u
9
+ 0.302703u
8
+ ··· + 0.0324324u + 0.356757
a
6
=
2.35135u
9
+ 1.16216u
8
+ ··· + 17.9459u − 5.59459
0.437838u
9
+ 0.340541u
8
+ ··· + 1.28649u + 0.151351
a
2
=
−3.11892u
9
− 0.870270u
8
+ ··· − 28.8432u + 9.72432
0.313514u
9
+ 0.0216216u
8
+ ··· + 0.859459u − 0.545946
a
1
=
−3.24324u
9
− 1.18919u
8
+ ··· − 29.2703u + 9.02703
−0.702703u
9
− 0.324324u
8
+ ··· − 2.89189u + 0.189189
a
9
=
−2.24324u
9
− 1.18919u
8
+ ··· − 17.2703u + 4.02703
−1.14054u
9
− 0.664865u
8
+ ··· − 6.17838u + 2.03784
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
1244
185
u
9
+
577
185
u
8
+
10113
185
u
7
+
927
185
u
6
+
5147
37
u
5
−
5396
185
u
4
+
4552
37
u
3
−
2673
37
u
2
+
7997
185
u −
1943
185
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 10u
9
+ ··· − 13u + 1
c
2
u
10
+ 2u
9
− 3u
8
− 7u
7
+ 4u
6
+ 12u
5
− u
4
− 11u
3
− 2u
2
+ 3u + 1
c
3
u
10
+ u
9
− u
8
− u
6
− 3u
5
+ 3u
4
− u
3
+ 5u
2
− 2u − 1
c
4
u
10
+ 8u
8
− 3u
7
+ 21u
6
− 14u
5
+ 22u
4
− 20u
3
+ 13u
2
− 6u + 1
c
5
u
10
− 2u
9
− 3u
8
+ 7u
7
+ 4u
6
− 12u
5
− u
4
+ 11u
3
− 2u
2
− 3u + 1
c
6
u
10
+ 3u
9
+ u
8
− 4u
7
− 8u
6
− 9u
5
+ 11u
4
+ 31u
3
+ 24u
2
+ 8u + 1
c
7
u
10
+ u
9
+ 4u
8
+ 4u
7
− 3u
6
− 8u
5
− 13u
4
− 21u
3
− 18u
2
− 7u − 1
c
8
u
10
− 3u
9
+ 9u
7
− 8u
6
− 7u
5
+ 11u
4
− u
3
− 3u
2
+ u − 1
c
9
u
10
− u
9
+ 2u
8
+ u
7
− 11u
6
+ 10u
5
− 7u
4
+ 2u
3
+ 4u
2
+ u − 1
c
10
u
10
+ 8u
8
+ 3u
7
+ 21u
6
+ 14u
5
+ 22u
4
+ 20u
3
+ 13u
2
+ 6u + 1
c
11
, c
12
u
10
+ 3u
9
− 9u
7
− 8u
6
+ 7u
5
+ 11u
4
+ u
3
− 3u
2
− u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 10y
9
+ ··· − 33y + 1
c
2
, c
5
y
10
− 10y
9
+ ··· − 13y + 1
c
3
y
10
− 3y
9
− y
8
+ 14y
7
+ 7y
6
− 23y
5
− 5y
4
+ 19y
3
+ 15y
2
− 14y + 1
c
4
, c
10
y
10
+ 16y
9
+ ··· − 10y + 1
c
6
y
10
− 7y
9
+ ··· − 16y + 1
c
7
y
10
+ 7y
9
+ ··· − 13y + 1
c
8
, c
11
, c
12
y
10
− 9y
9
+ ··· + 5y + 1
c
9
y
10
+ 3y
9
+ ··· − 9y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.421587 + 1.150120I
a = −0.068626 + 0.779782I
b = 0.442051 + 0.121008I
−0.46461 + 1.98898I −0.06537 − 3.20823I
u = −0.421587 − 1.150120I
a = −0.068626 − 0.779782I
b = 0.442051 − 0.121008I
−0.46461 − 1.98898I −0.06537 + 3.20823I
u = 0.235261 + 0.587721I
a = −0.300901 + 0.648896I
b = 0.14149 − 1.45935I
5.08555 + 2.55932I 4.45408 − 5.37582I
u = 0.235261 − 0.587721I
a = −0.300901 − 0.648896I
b = 0.14149 + 1.45935I
5.08555 − 2.55932I 4.45408 + 5.37582I
u = 0.490498
a = 3.09930
b = 0.465102
6.94382 10.5630
u = 0.12366 + 1.64371I
a = −1.257140 − 0.128400I
b = 1.83592 + 0.29637I
−6.58180 + 1.84846I 1.62067 − 1.24709I
u = 0.12366 − 1.64371I
a = −1.257140 + 0.128400I
b = 1.83592 − 0.29637I
−6.58180 − 1.84846I 1.62067 + 1.24709I
u = 0.293026
a = −1.91458
b = 0.591313
0.102739 −0.844670
u = −0.32909 + 2.03714I
a = 0.534307 − 0.226160I
b = −1.44767 + 0.48227I
−11.43200 − 3.15494I −2.86863 + 1.76027I
u = −0.32909 − 2.03714I
a = 0.534307 + 0.226160I
b = −1.44767 − 0.48227I
−11.43200 + 3.15494I −2.86863 − 1.76027I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
10
− 10u
9
+ ··· − 13u + 1)(u
10
+ 22u
9
+ ··· + 5917u + 49)
c
2
(u
10
+ 2u
9
+ ··· − 59u + 7)
· (u
10
+ 2u
9
− 3u
8
− 7u
7
+ 4u
6
+ 12u
5
− u
4
− 11u
3
− 2u
2
+ 3u + 1)
c
3
(u
10
+ u
9
− u
8
− u
6
− 3u
5
+ 3u
4
− u
3
+ 5u
2
− 2u − 1)
· (u
10
+ 3u
9
+ ··· + 12u − 13)
c
4
(u
10
+ 8u
8
− 3u
7
+ 21u
6
− 14u
5
+ 22u
4
− 20u
3
+ 13u
2
− 6u + 1)
· (u
10
+ 2u
9
+ 10u
8
+ 13u
7
+ 29u
6
+ 22u
5
+ 24u
4
+ 8u
3
+ 3u
2
− 2u − 1)
c
5
(u
10
− 2u
9
− 3u
8
+ 7u
7
+ 4u
6
− 12u
5
− u
4
+ 11u
3
− 2u
2
− 3u + 1)
· (u
10
+ 2u
9
+ ··· − 59u + 7)
c
6
(u
10
− 3u
9
+ ··· + 110u − 25)
· (u
10
+ 3u
9
+ u
8
− 4u
7
− 8u
6
− 9u
5
+ 11u
4
+ 31u
3
+ 24u
2
+ 8u + 1)
c
7
(u
10
+ u
9
+ 4u
8
+ 4u
7
− 3u
6
− 8u
5
− 13u
4
− 21u
3
− 18u
2
− 7u − 1)
· (u
10
+ 3u
9
− 20u
7
− 41u
6
+ 294u
5
− 405u
4
+ 219u
3
+ 58u
2
− 33u − 9)
c
8
(u
10
− 3u
9
+ 9u
7
− 8u
6
− 7u
5
+ 11u
4
− u
3
− 3u
2
+ u − 1)
· (u
10
− 3u
9
+ ··· + 37u − 29)
c
9
(u
10
− u
9
+ 2u
8
+ u
7
− 11u
6
+ 10u
5
− 7u
4
+ 2u
3
+ 4u
2
+ u − 1)
· (u
10
+ u
9
+ ··· − 25u − 167)
c
10
(u
10
+ 8u
8
+ 3u
7
+ 21u
6
+ 14u
5
+ 22u
4
+ 20u
3
+ 13u
2
+ 6u + 1)
· (u
10
+ 2u
9
+ 10u
8
+ 13u
7
+ 29u
6
+ 22u
5
+ 24u
4
+ 8u
3
+ 3u
2
− 2u − 1)
c
11
, c
12
(u
10
− 3u
9
+ ··· + 37u − 29)
· (u
10
+ 3u
9
− 9u
7
− 8u
6
+ 7u
5
+ 11u
4
+ u
3
− 3u
2
− u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
10
− 34y
9
+ ··· − 33340185y + 2401)(y
10
− 10y
9
+ ··· − 33y + 1)
c
2
, c
5
(y
10
− 22y
9
+ ··· − 5917y + 49)(y
10
− 10y
9
+ ··· − 13y + 1)
c
3
(y
10
− 19y
9
+ ··· − 2666y + 169)
· (y
10
− 3y
9
− y
8
+ 14y
7
+ 7y
6
− 23y
5
− 5y
4
+ 19y
3
+ 15y
2
− 14y + 1)
c
4
, c
10
(y
10
+ 16y
9
+ ··· − 10y + 1)(y
10
+ 16y
9
+ ··· − 10y + 1)
c
6
(y
10
− 27y
9
+ ··· − 23900y + 625)(y
10
− 7y
9
+ ··· − 16y + 1)
c
7
(y
10
− 9y
9
+ ··· − 2133y + 81)(y
10
+ 7y
9
+ ··· − 13y + 1)
c
8
, c
11
, c
12
(y
10
− 21y
9
+ ··· + 3097y + 841)(y
10
− 9y
9
+ ··· + 5y + 1)
c
9
(y
10
− 17y
9
+ ··· − 55401y + 27889)(y
10
+ 3y
9
+ ··· − 9y + 1)
11
