12n
0303
(K12n
0303
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 2 5 11 12 5 7 8 10
Solving Sequence
2,5
6 3
7,10
11 1 9 4 12 8
c
5
c
2
c
6
c
10
c
1
c
9
c
4
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
23
+ u
22
+ ··· + b − 2u, −3u
23
− 6u
22
+ ··· + 2a + 7, u
26
+ 3u
25
+ ··· − 9u
2
+ 1i
I
u
2
= hb, a
2
+ au − u
2
− a + 2u − 1, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
= hu
23
+u
22
+· · ·+ b−2u, −3u
23
−6u
22
+· · ·+ 2a + 7 , u
26
+3u
25
+· · ·−9u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
10
=
3
2
u
23
+ 3u
22
+ ··· − 4u −
7
2
−u
23
− u
22
+ ··· + 2u
2
+ 2u
a
11
=
u
25
+
7
2
u
24
+ ··· −
11
2
u − 3
−
1
2
u
25
−
3
2
u
24
+ ··· +
3
2
u −
1
2
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
5
2
u
23
+ 4u
22
+ ··· − 6u −
7
2
−u
23
− u
22
+ ··· + 2u
2
+ 2u
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
12
=
1
2
u
23
+ u
22
+ ··· + 2u +
1
2
−u
2
a
8
=
1
2
u
24
− 2u
22
+ ··· −
7
2
u + 1
−
1
2
u
25
−
3
2
u
24
+ ··· +
3
2
u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
11
2
u
25
− 12u
24
+
1
2
u
23
+ 30u
22
− 13u
21
− 105u
20
− 42u
19
+
337
2
u
18
+ 57u
17
− 325u
16
− 196u
15
+ 356u
14
+ 198u
13
−
979
2
u
12
− 283u
11
+
789
2
u
10
+
353
2
u
9
−
809
2
u
8
−
297
2
u
7
+ 250u
6
+ 41u
5
− 154u
4
− 16u
3
+
149
2
u
2
+ 13u −
33
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
26
+ 5u
25
+ ··· + 18u + 1
c
2
, c
5
u
26
+ 3u
25
+ ··· − 9u
2
+ 1
c
3
u
26
− 3u
25
+ ··· + 6516u + 1009
c
4
, c
9
u
26
− u
25
+ ··· + 96u + 64
c
7
, c
8
, c
10
c
11
u
26
+ 4u
25
+ ··· − 3u + 1
c
12
u
26
+ 28u
24
+ ··· − 31u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
26
+ 35y
25
+ ··· − 74y + 1
c
2
, c
5
y
26
− 5y
25
+ ··· − 18y + 1
c
3
y
26
+ 95y
25
+ ··· − 36103574y + 1018081
c
4
, c
9
y
26
− 35y
25
+ ··· − 58368y + 4096
c
7
, c
8
, c
10
c
11
y
26
− 28y
25
+ ··· − 27y + 1
c
12
y
26
+ 56y
25
+ ··· − 391y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.697878 + 0.750786I
a = 0.950274 + 0.282069I
b = −0.969430 + 0.478608I
3.30312 − 1.30372I −2.90180 + 1.28901I
u = 0.697878 − 0.750786I
a = 0.950274 − 0.282069I
b = −0.969430 − 0.478608I
3.30312 + 1.30372I −2.90180 − 1.28901I
u = −1.05556
a = −1.13582
b = −1.20261
−6.55918 −13.9020
u = −0.714104 + 0.530028I
a = −0.308272 + 1.024340I
b = −0.222426 + 0.888370I
−7.39553 + 1.99902I −12.68261 − 2.64464I
u = −0.714104 − 0.530028I
a = −0.308272 − 1.024340I
b = −0.222426 − 0.888370I
−7.39553 − 1.99902I −12.68261 + 2.64464I
u = 0.426539 + 0.776149I
a = −1.244140 − 0.230526I
b = 1.405990 + 0.009237I
−1.37386 + 1.46827I −7.40504 − 0.61110I
u = 0.426539 − 0.776149I
a = −1.244140 + 0.230526I
b = 1.405990 − 0.009237I
−1.37386 − 1.46827I −7.40504 + 0.61110I
u = 0.924653 + 0.644299I
a = −0.192140 − 0.877647I
b = 0.922849 + 0.070450I
2.53585 − 3.91698I −4.21369 + 6.80514I
u = 0.924653 − 0.644299I
a = −0.192140 + 0.877647I
b = 0.922849 − 0.070450I
2.53585 + 3.91698I −4.21369 − 6.80514I
u = 1.032140 + 0.523292I
a = 0.16264 + 1.48931I
b = −1.372650 + 0.239227I
−3.36202 − 6.28703I −10.76890 + 5.79025I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.032140 − 0.523292I
a = 0.16264 − 1.48931I
b = −1.372650 − 0.239227I
−3.36202 + 6.28703I −10.76890 − 5.79025I
u = −0.826543
a = 0.499620
b = 0.424946
−1.35750 −5.74160
u = 0.904709 + 0.855635I
a = −0.749040 + 0.396223I
b = 0.015583 − 1.255880I
0.45711 − 3.16518I −10.09379 + 2.71963I
u = 0.904709 − 0.855635I
a = −0.749040 − 0.396223I
b = 0.015583 + 1.255880I
0.45711 + 3.16518I −10.09379 − 2.71963I
u = −0.874451 + 0.958708I
a = −1.34597 + 0.77169I
b = 1.85003 − 0.54896I
6.92052 − 3.79217I −7.74212 + 0.87029I
u = −0.874451 − 0.958708I
a = −1.34597 − 0.77169I
b = 1.85003 + 0.54896I
6.92052 + 3.79217I −7.74212 − 0.87029I
u = −0.932825 + 0.952507I
a = 1.48560 − 0.97786I
b = −2.02844 + 0.13031I
13.70930 + 0.63054I −5.09956 + 0.08472I
u = −0.932825 − 0.952507I
a = 1.48560 + 0.97786I
b = −2.02844 − 0.13031I
13.70930 − 0.63054I −5.09956 − 0.08472I
u = −1.007520 + 0.885407I
a = 1.72924 − 1.22466I
b = −1.78041 − 0.65102I
6.48279 + 10.56660I −8.37835 − 5.32202I
u = −1.007520 − 0.885407I
a = 1.72924 + 1.22466I
b = −1.78041 + 0.65102I
6.48279 − 10.56660I −8.37835 + 5.32202I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.977518 + 0.926314I
a = −1.61355 + 1.11637I
b = 1.99598 + 0.29162I
13.5604 + 6.2675I −5.43427 − 4.61670I
u = −0.977518 − 0.926314I
a = −1.61355 − 1.11637I
b = 1.99598 − 0.29162I
13.5604 − 6.2675I −5.43427 + 4.61670I
u = 0.605940
a = 3.65946
b = −0.461168
−9.68489 0.985630
u = −0.507505 + 0.285828I
a = 0.417815 − 0.836271I
b = 0.046187 − 0.679181I
−0.698170 + 0.981366I −8.43360 − 6.86703I
u = −0.507505 − 0.285828I
a = 0.417815 + 0.836271I
b = 0.046187 + 0.679181I
−0.698170 − 0.981366I −8.43360 + 6.86703I
u = 0.332169
a = −2.60819
b = 0.512315
−1.32945 −6.03470
7
II. I
u
2
= hb, a
2
+ au − u
2
− a + 2u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
10
=
a
0
a
11
=
au + 2a
u
2
a − au − a
a
1
=
u
2
− 1
−u
2
a
9
=
a
0
a
4
=
1
0
a
12
=
u
2
− a + u − 2
−u
2
a
8
=
−au − 2a + u − 1
−u
2
a + au + a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a − 3u
2
− a + 8u − 13
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
+ u − 1)
3
c
10
, c
11
, c
12
(u
2
− u − 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.198308 − 1.205210I
b = 0
−5.85852 − 2.82812I −8.44207 + 3.24268I
u = 0.877439 + 0.744862I
a = −0.075747 + 0.460350I
b = 0
2.03717 − 2.82812I −5.93195 + 1.57712I
u = 0.877439 − 0.744862I
a = 0.198308 + 1.205210I
b = 0
−5.85852 + 2.82812I −8.44207 − 3.24268I
u = 0.877439 − 0.744862I
a = −0.075747 − 0.460350I
b = 0
2.03717 + 2.82812I −5.93195 − 1.57712I
u = −0.754878
a = −1.08457
b = 0
−2.10041 −19.0460
u = −0.754878
a = 2.83945
b = 0
−9.99610 −25.2060
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
2
)(u
26
+ 5u
25
+ ··· + 18u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
26
+ 3u
25
+ ··· − 9u
2
+ 1)
c
3
((u
3
− u
2
+ 2u − 1)
2
)(u
26
− 3u
25
+ ··· + 6516u + 1009)
c
4
, c
9
u
6
(u
26
− u
25
+ ··· + 96u + 64)
c
5
((u
3
− u
2
+ 1)
2
)(u
26
+ 3u
25
+ ··· − 9u
2
+ 1)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
26
+ 5u
25
+ ··· + 18u + 1)
c
7
, c
8
((u
2
+ u − 1)
3
)(u
26
+ 4u
25
+ ··· − 3u + 1)
c
10
, c
11
((u
2
− u − 1)
3
)(u
26
+ 4u
25
+ ··· − 3u + 1)
c
12
((u
2
− u − 1)
3
)(u
26
+ 28u
24
+ ··· − 31u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
26
+ 35y
25
+ ··· − 74y + 1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
26
− 5y
25
+ ··· − 18y + 1)
c
3
((y
3
+ 3y
2
+ 2y −1)
2
)(y
26
+ 95y
25
+ ··· − 3.61036 × 10
7
y + 1018081)
c
4
, c
9
y
6
(y
26
− 35y
25
+ ··· − 58368y + 4096)
c
7
, c
8
, c
10
c
11
((y
2
− 3y + 1)
3
)(y
26
− 28y
25
+ ··· − 27y + 1)
c
12
((y
2
− 3y + 1)
3
)(y
26
+ 56y
25
+ ··· − 391y + 1)
13
