12n
0388
(K12n
0388
)
A knot diagram
1
Linearized knot diagam
3 6 11 9 2 12 1 3 12 4 10 7
Solving Sequence
4,11 3,6
2 1 5 10 12 7 8 9
c
3
c
2
c
1
c
5
c
10
c
11
c
6
c
7
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
25
+ u
24
+ ··· + 4b − 4, −2u
26
+ 3u
25
+ ··· + 4a − 2, u
27
− 2u
26
+ ··· − 2u
2
+ 2i
I
u
2
= h−u
2
+ b − u − 1, −u
3
− 2u
2
+ 2a − u, u
4
+ u
2
+ 2i
I
u
3
= h−a
2
u − a
2
− au + b + a − 2, a
3
+ 2a
2
u − 3au + u, u
2
+ u + 1i
I
u
4
= hb + u, a + u − 1, u
2
+ 1i
I
u
5
= hu
3
+ u
2
+ b − 1, a − u − 1, u
4
+ 1i
I
v
1
= ha, b − 1, v −1i
* 6 irreducible components of dim
C
= 0, with total 44 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−u
25
+u
24
+· · ·+4b−4, −2u
26
+3u
25
+· · ·+4a−2, u
27
−2u
26
+· · ·−2u
2
+2i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
3
=
1
−u
2
a
6
=
1
2
u
26
−
3
4
u
25
+ ··· −
1
2
u +
1
2
1
4
u
25
−
1
4
u
24
+ ··· +
1
2
u + 1
a
2
=
−
1
4
u
22
− u
20
+ ··· −
1
2
u +
1
2
1
2
u
17
+
3
2
u
15
+ ··· − u
2
+ u
a
1
=
−
1
4
u
24
−
5
4
u
22
+ ··· +
1
2
u +
1
2
1
4
u
26
+ u
24
+ ··· − u
2
+ u
a
5
=
u
10
+ u
8
+ 2u
6
+ u
4
+ u
2
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
10
=
u
u
a
12
=
u
3
u
3
+ u
a
7
=
−
1
4
u
22
− u
20
+ ··· −
3
2
u +
1
2
1
4
u
24
+ u
22
+ ··· +
1
2
u
2
− u
a
8
=
−u
7
− 2u
5
− 2u
3
− 2u
u
9
+ u
7
+ u
5
− u
a
9
=
u
5
+ u
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
26
− 4u
25
+ 10u
24
− 16u
23
+ 28u
22
− 46u
21
+ 58u
20
− 88u
19
+
86u
18
− 130u
17
+ 114u
16
− 168u
15
+ 124u
14
− 168u
13
+ 116u
12
− 166u
11
+ 96u
10
−
118u
9
+ 40u
8
− 68u
7
+ 20u
6
− 44u
5
− 6u
4
+ 8u
3
− 20u
2
+ 2u − 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
27
− 2u
26
+ ··· − 1367u + 256
c
2
, c
5
u
27
+ 2u
26
+ ··· − 13u + 16
c
3
, c
10
u
27
− 2u
26
+ ··· − 2u
2
+ 2
c
4
u
27
− 5u
26
+ ··· + 4404u + 1706
c
6
, c
7
, c
12
u
27
− 2u
26
+ ··· + 19u + 16
c
8
u
27
+ 4u
26
+ ··· − 358556u + 54322
c
9
, c
11
u
27
− 10u
26
+ ··· + 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
27
+ 74y
26
+ ··· + 1160081y −65536
c
2
, c
5
y
27
+ 2y
26
+ ··· − 1367y −256
c
3
, c
10
y
27
+ 10y
26
+ ··· + 8y −4
c
4
y
27
+ 49y
26
+ ··· − 17215544y −2910436
c
6
, c
7
, c
12
y
27
− 46y
26
+ ··· − 2839y −256
c
8
y
27
+ 82y
26
+ ··· + 70277723880y −2950879684
c
9
, c
11
y
27
+ 14y
26
+ ··· + 1024y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.697654 + 0.748808I
a = 1.026780 + 0.002420I
b = 1.06126 − 1.23166I
−3.52813 + 0.07338I −9.79527 − 0.84081I
u = 0.697654 − 0.748808I
a = 1.026780 − 0.002420I
b = 1.06126 + 1.23166I
−3.52813 − 0.07338I −9.79527 + 0.84081I
u = 0.864921 + 0.447342I
a = −0.587169 + 0.347747I
b = −0.479795 − 0.989527I
12.22120 − 2.26521I −0.78342 + 1.87468I
u = 0.864921 − 0.447342I
a = −0.587169 − 0.347747I
b = −0.479795 + 0.989527I
12.22120 + 2.26521I −0.78342 − 1.87468I
u = −0.770627 + 0.681378I
a = 0.728531 − 0.074268I
b = 0.086141 + 1.355080I
0.458304 − 0.114158I 0.422197 − 0.454382I
u = −0.770627 − 0.681378I
a = 0.728531 + 0.074268I
b = 0.086141 − 1.355080I
0.458304 + 0.114158I 0.422197 + 0.454382I
u = −0.878083 + 0.542301I
a = −0.606653 + 0.267104I
b = −1.19080 − 1.81606I
11.64340 − 6.38697I −1.19981 + 2.11434I
u = −0.878083 − 0.542301I
a = −0.606653 − 0.267104I
b = −1.19080 + 1.81606I
11.64340 + 6.38697I −1.19981 − 2.11434I
u = 0.100635 + 1.097690I
a = −0.85192 + 1.71040I
b = −0.550049 + 1.253330I
6.60434 + 0.15878I 5.87983 + 0.02677I
u = 0.100635 − 1.097690I
a = −0.85192 − 1.71040I
b = −0.550049 − 1.253330I
6.60434 − 0.15878I 5.87983 − 0.02677I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.669390 + 0.942059I
a = −0.88412 + 1.56066I
b = 0.57563 + 1.56765I
−2.93763 − 5.34360I −7.41989 + 6.92820I
u = 0.669390 − 0.942059I
a = −0.88412 − 1.56066I
b = 0.57563 − 1.56765I
−2.93763 + 5.34360I −7.41989 − 6.92820I
u = 0.686168 + 0.447957I
a = 0.405997 − 0.432049I
b = −0.812403 + 0.706687I
1.78018 + 2.00372I −0.59189 − 2.09997I
u = 0.686168 − 0.447957I
a = 0.405997 + 0.432049I
b = −0.812403 − 0.706687I
1.78018 − 2.00372I −0.59189 + 2.09997I
u = 0.040008 + 1.207920I
a = −0.99416 − 2.11968I
b = −0.43164 − 1.73930I
18.1739 − 4.5997I 4.36505 + 2.18290I
u = 0.040008 − 1.207920I
a = −0.99416 + 2.11968I
b = −0.43164 + 1.73930I
18.1739 + 4.5997I 4.36505 − 2.18290I
u = 0.601091 + 1.054470I
a = 0.61079 − 1.66086I
b = −1.15984 − 0.98023I
3.48901 − 6.97695I 1.70016 + 6.49908I
u = 0.601091 − 1.054470I
a = 0.61079 + 1.66086I
b = −1.15984 + 0.98023I
3.48901 + 6.97695I 1.70016 − 6.49908I
u = −0.710977 + 1.000460I
a = −1.42612 − 0.55763I
b = −0.14675 − 1.63948I
1.40487 + 5.73270I 1.73930 − 4.91111I
u = −0.710977 − 1.000460I
a = −1.42612 + 0.55763I
b = −0.14675 + 1.63948I
1.40487 − 5.73270I 1.73930 + 4.91111I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.074391 + 0.718228I
a = 0.98792 − 1.24993I
b = −0.006103 − 0.217922I
0.94642 + 1.42613I 1.85713 − 5.82586I
u = −0.074391 − 0.718228I
a = 0.98792 + 1.24993I
b = −0.006103 + 0.217922I
0.94642 − 1.42613I 1.85713 + 5.82586I
u = 0.638164 + 1.116710I
a = −1.48999 + 0.10418I
b = −0.054880 + 0.864766I
14.2506 − 3.2919I 1.72901 + 2.48936I
u = 0.638164 − 1.116710I
a = −1.48999 − 0.10418I
b = −0.054880 − 0.864766I
14.2506 + 3.2919I 1.72901 − 2.48936I
u = −0.688671 + 1.094800I
a = 1.11378 + 2.20355I
b = −1.22699 + 2.17626I
13.3230 + 12.1918I 0.72580 − 6.39342I
u = −0.688671 − 1.094800I
a = 1.11378 − 2.20355I
b = −1.22699 − 2.17626I
13.3230 − 12.1918I 0.72580 + 6.39342I
u = −0.350564
a = 0.932661
b = 0.672452
−1.03514 −11.2560
7
II. I
u
2
= h−u
2
+ b − u − 1, −u
3
− 2u
2
+ 2a − u, u
4
+ u
2
+ 2i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
3
=
1
−u
2
a
6
=
1
2
u
3
+ u
2
+
1
2
u
u
2
+ u + 1
a
2
=
1
2
u
3
+ u
2
+
1
2
u + 1
u + 1
a
1
=
1
2
u
3
+ u
2
+
1
2
u
u
2
+ u + 1
a
5
=
−1
u
2
a
10
=
u
u
a
12
=
u
3
u
3
+ u
a
7
=
−
1
2
u
3
+ u
2
+
1
2
u
−u
3
+ u
2
+ 1
a
8
=
−u
3
−u
3
− u
a
9
=
−u
3
− u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
(u − 1)
4
c
2
, c
12
(u + 1)
4
c
3
, c
4
, c
8
c
10
u
4
+ u
2
+ 2
c
9
(u
2
+ u + 2)
2
c
11
(u
2
− u + 2)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
12
(y −1)
4
c
3
, c
4
, c
8
c
10
(y
2
+ y + 2)
2
c
9
, c
11
(y
2
+ 3y + 4)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.676097 + 0.978318I
a = −0.97807 + 2.01465I
b = 1.17610 + 2.30119I
−0.82247 − 5.33349I −2.00000 + 5.29150I
u = 0.676097 − 0.978318I
a = −0.97807 − 2.01465I
b = 1.17610 − 2.30119I
−0.82247 + 5.33349I −2.00000 − 5.29150I
u = −0.676097 + 0.978318I
a = −0.021927 − 0.631100I
b = −0.176097 − 0.344557I
−0.82247 + 5.33349I −2.00000 − 5.29150I
u = −0.676097 − 0.978318I
a = −0.021927 + 0.631100I
b = −0.176097 + 0.344557I
−0.82247 − 5.33349I −2.00000 + 5.29150I
11
III. I
u
3
= h−a
2
u − a
2
− au + b + a − 2, a
3
+ 2a
2
u − 3au + u, u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
3
=
1
u + 1
a
6
=
a
a
2
u + a
2
+ au − a + 2
a
2
=
a
−a
2
u − a
2
+ 2a − 2
a
1
=
−a
2
u − a
2
− au + 2a − 2
−2a
2
u − a
2
+ au + 4a − 2u − 4
a
5
=
1
0
a
10
=
u
u
a
12
=
1
u + 1
a
7
=
a
2
u + a
2
− a + 2
2a
2
u + a
2
− au − 3a + 2u + 4
a
8
=
−u
−u
a
9
=
−1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u − 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1
c
2
, c
5
, c
6
c
7
, c
12
u
6
− 2u
4
+ u
3
+ u
2
− u + 1
c
3
, c
10
(u
2
+ u + 1)
3
c
4
u
6
c
8
, c
9
, c
11
(u
2
− u + 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1
c
2
, c
5
, c
6
c
7
, c
12
y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1
c
3
, c
8
, c
9
c
10
, c
11
(y
2
+ y + 1)
3
c
4
y
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.741145 + 0.632163I
b = −0.395862 + 0.291743I
2.02988I 0. − 3.46410I
u = −0.500000 + 0.866025I
a = 0.439111 − 0.046276I
b = 1.51194 + 0.59451I
2.02988I 0. − 3.46410I
u = −0.500000 + 0.866025I
a = −0.18026 − 2.31794I
b = 0.883917 − 0.886250I
2.02988I 0. − 3.46410I
u = −0.500000 − 0.866025I
a = 0.741145 − 0.632163I
b = −0.395862 − 0.291743I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 0.439111 + 0.046276I
b = 1.51194 − 0.59451I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = −0.18026 + 2.31794I
b = 0.883917 + 0.886250I
− 2.02988I 0. + 3.46410I
15
IV. I
u
4
= hb + u, a + u − 1, u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
3
=
1
1
a
6
=
−u + 1
−u
a
2
=
−u + 2
−u + 1
a
1
=
−u + 1
−u
a
5
=
−1
−1
a
10
=
u
u
a
12
=
−u
0
a
7
=
1
−u
a
8
=
u
0
a
9
=
2u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
(u − 1)
2
c
2
, c
9
, c
12
(u + 1)
2
c
3
, c
4
, c
8
c
10
u
2
+ 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
c
11
, c
12
(y −1)
2
c
3
, c
4
, c
8
c
10
(y + 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.00000 − 1.00000I
b = − 1.000000I
3.28987 4.00000
u = − 1.000000I
a = 1.00000 + 1.00000I
b = 1.000000I
3.28987 4.00000
19
V. I
u
5
= hu
3
+ u
2
+ b − 1, a − u − 1, u
4
+ 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
3
=
1
−u
2
a
6
=
u + 1
−u
3
− u
2
+ 1
a
2
=
−u
u
3
− 1
a
1
=
−u − 1
u
3
+ u
2
− 1
a
5
=
1
−u
2
a
10
=
u
u
a
12
=
u
3
u
3
+ u
a
7
=
u
3
+ u + 1
−u
2
+ u + 1
a
8
=
u
3
u
3
+ u
a
9
=
0
u
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
(u − 1)
4
c
3
, c
4
, c
8
c
10
u
4
+ 1
c
5
, c
6
, c
7
(u + 1)
4
c
9
, c
11
(u
2
+ 1)
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
12
(y −1)
4
c
3
, c
4
, c
8
c
10
(y
2
+ 1)
2
c
9
, c
11
(y + 1)
4
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.707107 + 0.707107I
a = 1.70711 + 0.70711I
b = 1.70711 − 1.70711I
−1.64493 −4.00000
u = 0.707107 − 0.707107I
a = 1.70711 − 0.70711I
b = 1.70711 + 1.70711I
−1.64493 −4.00000
u = −0.707107 + 0.707107I
a = 0.292893 + 0.707107I
b = 0.292893 + 0.292893I
−1.64493 −4.00000
u = −0.707107 − 0.707107I
a = 0.292893 − 0.707107I
b = 0.292893 − 0.292893I
−1.64493 −4.00000
23
VI. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
1
0
a
3
=
1
0
a
6
=
0
1
a
2
=
1
−1
a
1
=
0
−1
a
5
=
1
0
a
10
=
1
0
a
12
=
1
0
a
7
=
1
1
a
8
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
24
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
4
, c
8
c
9
, c
10
, c
11
u
c
5
, c
6
, c
7
u + 1
25
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
12
y −1
c
3
, c
4
, c
8
c
9
, c
10
, c
11
y
26
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
27
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
11
(u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1)
· (u
27
− 2u
26
+ ··· − 1367u + 256)
c
2
(u − 1)
5
(u + 1)
6
(u
6
− 2u
4
+ u
3
+ u
2
− u + 1)
· (u
27
+ 2u
26
+ ··· − 13u + 16)
c
3
, c
10
u(u
2
+ 1)(u
2
+ u + 1)
3
(u
4
+ 1)(u
4
+ u
2
+ 2)(u
27
− 2u
26
+ ··· − 2u
2
+ 2)
c
4
u
7
(u
2
+ 1)(u
4
+ 1)(u
4
+ u
2
+ 2)(u
27
− 5u
26
+ ··· + 4404u + 1706)
c
5
(u − 1)
6
(u + 1)
5
(u
6
− 2u
4
+ u
3
+ u
2
− u + 1)
· (u
27
+ 2u
26
+ ··· − 13u + 16)
c
6
, c
7
(u − 1)
6
(u + 1)
5
(u
6
− 2u
4
+ u
3
+ u
2
− u + 1)
· (u
27
− 2u
26
+ ··· + 19u + 16)
c
8
u(u
2
+ 1)(u
2
− u + 1)
3
(u
4
+ 1)(u
4
+ u
2
+ 2)
· (u
27
+ 4u
26
+ ··· − 358556u + 54322)
c
9
u(u + 1)
2
(u
2
+ 1)
2
(u
2
− u + 1)
3
(u
2
+ u + 2)
2
· (u
27
− 10u
26
+ ··· + 8u + 4)
c
11
u(u − 1)
2
(u
2
+ 1)
2
(u
2
− u + 1)
3
(u
2
− u + 2)
2
· (u
27
− 10u
26
+ ··· + 8u + 4)
c
12
(u − 1)
5
(u + 1)
6
(u
6
− 2u
4
+ u
3
+ u
2
− u + 1)
· (u
27
− 2u
26
+ ··· + 19u + 16)
28
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
11
(y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1)
· (y
27
+ 74y
26
+ ··· + 1160081y −65536)
c
2
, c
5
(y −1)
11
(y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1)
· (y
27
+ 2y
26
+ ··· − 1367y −256)
c
3
, c
10
y(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 1)
3
(y
2
+ y + 2)
2
· (y
27
+ 10y
26
+ ··· + 8y −4)
c
4
y
7
(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 2)
2
· (y
27
+ 49y
26
+ ··· − 17215544y −2910436)
c
6
, c
7
, c
12
(y −1)
11
(y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1)
· (y
27
− 46y
26
+ ··· − 2839y −256)
c
8
y(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 1)
3
(y
2
+ y + 2)
2
· (y
27
+ 82y
26
+ ··· + 70277723880y −2950879684)
c
9
, c
11
y(y −1)
2
(y + 1)
4
(y
2
+ y + 1)
3
(y
2
+ 3y + 4)
2
· (y
27
+ 14y
26
+ ··· + 1024y −16)
29
