12n
0322
(K12n
0322
)
A knot diagram
1
Linearized knot diagam
3 6 11 7 2 10 11 3 12 5 10 9
Solving Sequence
5,10
11
7,12
4 3 6 2 1 9 8
c
10
c
11
c
4
c
3
c
6
c
2
c
1
c
9
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−25033252u
22
+ 121557101u
21
+ ··· + 292671322b + 456806862,
157446473u
22
− 71514300u
21
+ ··· + 146335661a − 334256756, u
23
− u
22
+ ··· − 4u + 1i
I
u
2
= h−u
5
b + 2u
4
b − u
5
− u
3
b + b
2
− 2bu + 2b − 2u, −u
4
+ a − 1, u
6
+ u
4
+ 2u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 35 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−2.50 × 10
7
u
22
+ 1.22 × 10
8
u
21
+ · · · + 2.93 × 10
8
b + 4.57 × 10
8
, 1.57 ×
10
8
u
22
− 7.15 × 10
7
u
21
+ · · · + 1.46 × 10
8
a − 3.34 × 10
8
, u
23
− u
22
+ · · · − 4u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−1.07593u
22
+ 0.488700u
21
+ ··· − 6.60771u + 2.28418
0.0855337u
22
− 0.415337u
21
+ ··· + 1.72095u − 1.56082
a
12
=
u
2
+ 1
u
2
a
4
=
−1.74252u
22
+ 1.41007u
21
+ ··· − 7.48338u + 5.61498
−0.341312u
22
− 0.0869202u
21
+ ··· − 0.317822u + 0.422860
a
3
=
−2.02915u
22
+ 1.43160u
21
+ ··· − 8.21390u + 5.70538
−0.505038u
22
+ 0.0146154u
21
+ ··· − 1.09158u + 0.687959
a
6
=
−0.990393u
22
+ 0.0733639u
21
+ ··· − 4.88676u + 0.723360
0.0855337u
22
− 0.415337u
21
+ ··· + 1.72095u − 1.56082
a
2
=
−0.0458208u
22
+ 0.256242u
21
+ ··· + 1.37542u + 1.65212
−0.626056u
22
+ 0.0915497u
21
+ ··· − 1.35086u + 1.57783
a
1
=
−u
6
− u
4
− 2u
2
− 1
−u
6
− u
2
a
9
=
u
4
+ u
2
+ 1
u
4
a
8
=
−0.580827u
22
− 0.0431012u
21
+ ··· − 3.61378u + 0.136133
0.107132u
22
− 0.226022u
21
+ ··· + 1.07904u − 1.52412
(ii) Obstruction class = −1
(iii) Cusp Shapes =
670390078
146335661
u
22
−
445304479
146335661
u
21
+ ··· +
1755269129
146335661
u −
1610811133
146335661
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 19u
22
+ ··· − 12u + 1
c
2
, c
5
u
23
+ u
22
+ ··· + 6u + 1
c
3
u
23
+ 5u
22
+ ··· + 138708u + 29957
c
4
, c
8
u
23
+ u
22
+ ··· − 140u + 25
c
6
u
23
+ 7u
22
+ ··· − 11462u − 5383
c
7
u
23
+ u
22
+ ··· + 376u − 7
c
9
, c
11
, c
12
u
23
− 3u
22
+ ··· + 6u + 1
c
10
u
23
− u
22
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 23y
22
+ ··· − 984y −1
c
2
, c
5
y
23
− 19y
22
+ ··· − 12y −1
c
3
y
23
− 73y
22
+ ··· − 10832245444y −897421849
c
4
, c
8
y
23
+ 43y
22
+ ··· + 8150y −625
c
6
y
23
− 41y
22
+ ··· + 193389604y −28976689
c
7
y
23
+ 43y
22
+ ··· + 163692y −49
c
9
, c
11
, c
12
y
23
+ 39y
22
+ ··· + 30y −1
c
10
y
23
+ 3y
22
+ ··· + 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.638291 + 0.756340I
a = −0.053920 + 0.672113I
b = −1.27006 − 0.66289I
−2.28180 − 5.04874I −2.29238 + 7.64932I
u = 0.638291 − 0.756340I
a = −0.053920 − 0.672113I
b = −1.27006 + 0.66289I
−2.28180 + 5.04874I −2.29238 − 7.64932I
u = 0.728457 + 0.859772I
a = −0.696631 − 0.653322I
b = 0.729953 + 0.669715I
−4.64426 − 2.76660I −5.38126 + 3.37592I
u = 0.728457 − 0.859772I
a = −0.696631 + 0.653322I
b = 0.729953 − 0.669715I
−4.64426 + 2.76660I −5.38126 − 3.37592I
u = 0.356533 + 0.788524I
a = 1.151030 + 0.050949I
b = −1.04769 + 1.09696I
−1.92628 + 1.02137I −4.36487 − 0.00463I
u = 0.356533 − 0.788524I
a = 1.151030 − 0.050949I
b = −1.04769 − 1.09696I
−1.92628 − 1.02137I −4.36487 + 0.00463I
u = −0.087548 + 0.829182I
a = −0.165345 − 0.286028I
b = 1.012130 + 0.803767I
1.16140 + 1.81818I 6.31882 − 4.29104I
u = −0.087548 − 0.829182I
a = −0.165345 + 0.286028I
b = 1.012130 − 0.803767I
1.16140 − 1.81818I 6.31882 + 4.29104I
u = −0.370099 + 0.602173I
a = 0.256458 − 0.802239I
b = −0.536912 + 0.631160I
0.076790 + 1.263270I 0.68970 − 5.37175I
u = −0.370099 − 0.602173I
a = 0.256458 + 0.802239I
b = −0.536912 − 0.631160I
0.076790 − 1.263270I 0.68970 + 5.37175I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.076080 + 0.763334I
a = −0.18671 + 1.49317I
b = 1.74353 − 0.51727I
−11.04540 + 2.65995I −6.14199 − 2.02854I
u = −1.076080 − 0.763334I
a = −0.18671 − 1.49317I
b = 1.74353 + 0.51727I
−11.04540 − 2.65995I −6.14199 + 2.02854I
u = −0.674267
a = 1.27270
b = 0.595901
−1.85226 −5.67690
u = −0.767551 + 1.147080I
a = −1.301880 + 0.405799I
b = 2.41969 + 0.70458I
−9.61269 + 4.18878I −5.20065 − 2.68941I
u = −0.767551 − 1.147080I
a = −1.301880 − 0.405799I
b = 2.41969 − 0.70458I
−9.61269 − 4.18878I −5.20065 + 2.68941I
u = −1.01450 + 1.01430I
a = 1.01669 − 1.12066I
b = −1.98063 + 0.31926I
−18.6458 + 3.7213I −3.19708 − 1.97285I
u = −1.01450 − 1.01430I
a = 1.01669 + 1.12066I
b = −1.98063 − 0.31926I
−18.6458 − 3.7213I −3.19708 + 1.97285I
u = 1.10481 + 0.91677I
a = 1.22917 + 1.03499I
b = −1.37206 + 0.92349I
15.8685 + 3.3875I −5.25744 − 0.65270I
u = 1.10481 − 0.91677I
a = 1.22917 − 1.03499I
b = −1.37206 − 0.92349I
15.8685 − 3.3875I −5.25744 + 0.65270I
u = 0.94898 + 1.10084I
a = 0.84505 + 1.21860I
b = −2.63899 − 1.28859I
16.5464 − 10.8721I −4.62089 + 4.80987I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.94898 − 1.10084I
a = 0.84505 − 1.21860I
b = −2.63899 + 1.28859I
16.5464 + 10.8721I −4.62089 − 4.80987I
u = 0.375835 + 0.114241I
a = −1.23027 − 2.62777I
b = −0.856913 + 0.343009I
−1.84255 − 2.10614I −6.71348 + 2.97651I
u = 0.375835 − 0.114241I
a = −1.23027 + 2.62777I
b = −0.856913 − 0.343009I
−1.84255 + 2.10614I −6.71348 − 2.97651I
7
II. I
u
2
= h−u
5
b − u
5
+ · · · + b
2
+ 2b, −u
4
+ a − 1, u
6
+ u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
u
4
+ 1
b
a
12
=
u
2
+ 1
u
2
a
4
=
−u
5
− u
3
− 2u
−u
5
b − bu + u
a
3
=
−u
5
b − u
5
− u
3
− bu − 2u
u
3
b + u
a
6
=
u
4
+ b + 1
b
a
2
=
−u
5
b − u
5
− u
4
− u
3
− bu − 2u − 1
−u
5
+ u
3
b − u
3
+ b − u
a
1
=
0
−u
4
− u
2
− 1
a
9
=
u
4
+ u
2
+ 1
u
4
a
8
=
2u
4
+ u
2
+ b + 2
u
2
b − u
2
+ b − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
5
+ 4u
4
− 4bu + 4u
2
− 4u
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
6
c
2
, c
5
(u
4
− u
2
+ 1)
3
c
3
u
12
+ 6u
10
+ ··· − 4u + 1
c
4
, c
8
(u
2
+ 1)
6
c
6
u
12
− 6u
10
+ ··· + 2u + 1
c
7
u
12
− 4u
11
+ ··· − 70u + 37
c
9
(u
3
+ u
2
+ 2u + 1)
4
c
10
(u
6
+ u
4
+ 2u
2
+ 1)
2
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
6
c
2
, c
5
(y
2
− y + 1)
6
c
3
y
12
+ 12y
11
+ ··· − 6y + 1
c
4
, c
8
(y + 1)
12
c
6
y
12
− 12y
11
+ ··· + 6y + 1
c
7
y
12
+ 8y
11
+ ··· + 1094y + 1369
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
4
c
10
(y
3
+ y
2
+ 2y + 1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744862 + 0.877439I
a = −0.662359 − 0.562280I
b = −0.192400 + 0.406511I
−4.66906 − 4.85801I −5.50976 + 6.44355I
u = 0.744862 + 0.877439I
a = −0.662359 − 0.562280I
b = 0.95484 + 1.38041I
−4.66906 − 0.79824I −5.50976 − 0.48465I
u = 0.744862 − 0.877439I
a = −0.662359 + 0.562280I
b = −0.192400 − 0.406511I
−4.66906 + 4.85801I −5.50976 − 6.44355I
u = 0.744862 − 0.877439I
a = −0.662359 + 0.562280I
b = 0.95484 − 1.38041I
−4.66906 + 0.79824I −5.50976 + 0.48465I
u = −0.744862 + 0.877439I
a = −0.662359 + 0.562280I
b = 0.369879 + 0.255848I
−4.66906 + 0.79824I −5.50976 + 0.48465I
u = −0.744862 + 0.877439I
a = −0.662359 + 0.562280I
b = 1.51712 − 0.71805I
−4.66906 + 4.85801I −5.50976 − 6.44355I
u = −0.744862 − 0.877439I
a = −0.662359 − 0.562280I
b = 0.369879 − 0.255848I
−4.66906 − 0.79824I −5.50976 − 0.48465I
u = −0.744862 − 0.877439I
a = −0.662359 − 0.562280I
b = 1.51712 + 0.71805I
−4.66906 − 4.85801I −5.50976 + 6.44355I
u = 0.754878I
a = 1.32472
b = −0.177479 + 0.662359I
−0.53148 + 2.02988I 1.01951 − 3.46410I
u = 0.754878I
a = 1.32472
b = −2.47196 + 0.66236I
−0.53148 − 2.02988I 1.01951 + 3.46410I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = − 0.754878I
a = 1.32472
b = −0.177479 − 0.662359I
−0.53148 − 2.02988I 1.01951 + 3.46410I
u = − 0.754878I
a = 1.32472
b = −2.47196 − 0.66236I
−0.53148 + 2.02988I 1.01951 − 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
23
+ 19u
22
+ ··· − 12u + 1)
c
2
, c
5
((u
4
− u
2
+ 1)
3
)(u
23
+ u
22
+ ··· + 6u + 1)
c
3
(u
12
+ 6u
10
+ ··· − 4u + 1)(u
23
+ 5u
22
+ ··· + 138708u + 29957)
c
4
, c
8
((u
2
+ 1)
6
)(u
23
+ u
22
+ ··· − 140u + 25)
c
6
(u
12
− 6u
10
+ ··· + 2u + 1)(u
23
+ 7u
22
+ ··· − 11462u − 5383)
c
7
(u
12
− 4u
11
+ ··· − 70u + 37)(u
23
+ u
22
+ ··· + 376u − 7)
c
9
((u
3
+ u
2
+ 2u + 1)
4
)(u
23
− 3u
22
+ ··· + 6u + 1)
c
10
((u
6
+ u
4
+ 2u
2
+ 1)
2
)(u
23
− u
22
+ ··· − 4u + 1)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
4
)(u
23
− 3u
22
+ ··· + 6u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
23
− 23y
22
+ ··· − 984y −1)
c
2
, c
5
((y
2
− y + 1)
6
)(y
23
− 19y
22
+ ··· − 12y −1)
c
3
(y
12
+ 12y
11
+ ··· − 6y + 1)
· (y
23
− 73y
22
+ ··· − 10832245444y −897421849)
c
4
, c
8
((y + 1)
12
)(y
23
+ 43y
22
+ ··· + 8150y −625)
c
6
(y
12
− 12y
11
+ ··· + 6y + 1)
· (y
23
− 41y
22
+ ··· + 193389604y −28976689)
c
7
(y
12
+ 8y
11
+ ··· + 1094y + 1369)(y
23
+ 43y
22
+ ··· + 163692y −49)
c
9
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
4
)(y
23
+ 39y
22
+ ··· + 30y −1)
c
10
((y
3
+ y
2
+ 2y + 1)
4
)(y
23
+ 3y
22
+ ··· + 6y −1)
14
