12n
0230
(K12n
0230
)
A knot diagram
1
Linearized knot diagam
3 5 8 6 2 9 4 11 6 12 8 10
Solving Sequence
8,11
9
4,12
3 7 6 5 2 10 1
c
8
c
11
c
3
c
7
c
6
c
4
c
2
c
10
c
12
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
+ 2u
6
− 3u
5
+ 2u
4
− 2u
3
+ 2u
2
+ b − u, u
5
− 2u
4
+ 2u
3
+ a − u,
u
9
− 3u
8
+ 6u
7
− 7u
6
+ 7u
5
− 7u
4
+ 6u
3
− 4u
2
+ u − 1i
I
u
2
= h130u
15
− 449u
14
+ ··· + 1816b − 497, −1016u
15
+ 4012u
14
+ ··· + 1816a − 9397,
u
16
− 4u
15
+ ··· + 2u + 1i
I
u
3
= hb, −u
3
a + 2u
2
a − u
3
+ a
2
− 2au − u
2
+ 3u − 4, u
4
− u
3
+ u
2
+ 1i
I
u
4
= h−a
3
u − 2a
3
− 3a
2
− au + 3b + a + u + 5, a
4
− a
3
u + 2a
3
− a
2
u − 4a − u − 4, u
2
+ u + 1i
* 4 irreducible components of dim
C
= 0, with total 41 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
7
+ 2u
6
− 3u
5
+ 2u
4
− 2u
3
+ 2u
2
+ b − u, u
5
− 2u
4
+ 2u
3
+ a −
u, u
9
− 3u
8
+ · · · + u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
4
=
−u
5
+ 2u
4
− 2u
3
+ u
u
7
− 2u
6
+ 3u
5
− 2u
4
+ 2u
3
− 2u
2
+ u
a
12
=
u
u
a
3
=
u
7
− 2u
6
+ 2u
5
− 2u
2
+ 2u
u
7
− 2u
6
+ 3u
5
− 2u
4
+ 2u
3
− 2u
2
+ u
a
7
=
−u
5
+ 2u
4
− 2u
3
+ 2u
2
− u + 2
−u
5
− u
3
− u
a
6
=
−u
7
+ 2u
6
− 4u
5
+ 4u
4
− 4u
3
+ 4u
2
− 2u + 2
u
8
− 3u
7
+ 5u
6
− 6u
5
+ 5u
4
− 6u
3
+ 4u
2
− 2u + 1
a
5
=
−u
8
+ 2u
7
− 3u
6
+ 2u
5
− u
4
+ 2u
3
− 2u
2
+ 2u − 1
−u
8
+ 2u
7
− 4u
6
+ 4u
5
− 4u
4
+ 4u
3
− 2u
2
+ 2u
a
2
=
−u
8
+ 4u
7
− 7u
6
+ 8u
5
− 5u
4
+ 4u
3
− 4u
2
+ 2u − 1
−u
8
+ 3u
7
− 5u
6
+ 6u
5
− 5u
4
+ 6u
3
− 4u
2
+ 2u − 1
a
10
=
u
3
u
3
+ u
a
1
=
u
5
+ u
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
8
− 24u
7
+ 48u
6
− 56u
5
+ 48u
4
− 32u
3
+ 8u
2
− 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
c
12
u
9
+ 3u
8
+ 8u
7
+ 5u
6
+ u
5
− 15u
4
− 20u
3
− 18u
2
− 7u − 1
c
2
, c
5
, c
8
c
11
u
9
+ 3u
8
+ 6u
7
+ 7u
6
+ 7u
5
+ 7u
4
+ 6u
3
+ 4u
2
+ u + 1
c
3
, c
6
, c
7
c
9
u
9
− u
8
− 2u
7
+ 9u
6
+ 3u
5
+ 17u
4
+ 6u
3
+ 4u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
c
12
y
9
+ 7y
8
+ 36y
7
+ 41y
6
− 75y
5
− 191y
4
− 144y
3
− 74y
2
+ 13y −1
c
2
, c
5
, c
8
c
11
y
9
+ 3y
8
+ 8y
7
+ 5y
6
+ y
5
− 15y
4
− 20y
3
− 18y
2
− 7y −1
c
3
, c
6
, c
7
c
9
y
9
− 5y
8
+ 28y
7
− 47y
6
− 315y
5
− 319y
4
− 124y
3
− 62y
2
− 7y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.461481 + 0.837544I
a = −2.45813 + 3.10206I
b = −0.213712 − 0.318134I
−0.13903 − 3.77297I 10.5564 + 43.0949I
u = −0.461481 − 0.837544I
a = −2.45813 − 3.10206I
b = −0.213712 + 0.318134I
−0.13903 + 3.77297I 10.5564 − 43.0949I
u = 0.736616 + 0.869782I
a = 0.787377 + 0.049850I
b = 0.073895 − 1.113510I
7.66695 + 5.59873I 4.90357 − 6.20498I
u = 0.736616 − 0.869782I
a = 0.787377 − 0.049850I
b = 0.073895 + 1.113510I
7.66695 − 5.59873I 4.90357 + 6.20498I
u = 1.15634
a = −0.427626
b = 2.18402
−4.53774 −0.758800
u = −0.102202 + 0.554352I
a = −0.092950 + 0.960014I
b = 0.439047 + 0.496789I
−0.75640 − 1.26978I −6.33576 + 4.10506I
u = −0.102202 − 0.554352I
a = −0.092950 − 0.960014I
b = 0.439047 − 0.496789I
−0.75640 + 1.26978I −6.33576 − 4.10506I
u = 0.74890 + 1.31534I
a = −1.52248 − 1.01662I
b = −1.89124 + 1.45525I
−11.9049 + 13.3161I −3.74481 − 5.95110I
u = 0.74890 − 1.31534I
a = −1.52248 + 1.01662I
b = −1.89124 − 1.45525I
−11.9049 − 13.3161I −3.74481 + 5.95110I
5
II. I
u
2
= h130u
15
− 449u
14
+ · · · + 1816b − 497, −1016u
15
+ 4012u
14
+ · · · +
1816a − 9397, u
16
− 4u
15
+ · · · + 2u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
4
=
0.559471u
15
− 2.20925u
14
+ ··· + 18.6828u + 5.17456
−0.0715859u
15
+ 0.247247u
14
+ ··· − 3.57654u + 0.273678
a
12
=
u
u
a
3
=
0.487885u
15
− 1.96200u
14
+ ··· + 15.1063u + 5.44824
−0.0715859u
15
+ 0.247247u
14
+ ··· − 3.57654u + 0.273678
a
7
=
0.198789u
15
− 0.833700u
14
+ ··· + 12.0231u − 1.85518
0.106278u
15
− 0.327643u
14
+ ··· + 2.14152u + 0.192731
a
6
=
0.226322u
15
− 0.976872u
14
+ ··· + 14.0430u − 1.62390
0.0666300u
15
− 0.271476u
14
+ ··· + 2.10297u + 0.159692
a
5
=
0.318282u
15
− 1.18007u
14
+ ··· + 4.34416u + 4.72357
−0.0440529u
15
+ 0.229075u
14
+ ··· − 3.93172u + 0.00495595
a
2
=
−0.101872u
15
+ 0.529736u
14
+ ··· − 12.1233u + 4.14427
−0.0666300u
15
+ 0.271476u
14
+ ··· − 2.10297u − 0.159692
a
10
=
u
3
u
3
+ u
a
1
=
u
5
+ u
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1189
1816
u
15
+
4957
1816
u
14
+ ··· −
61683
1816
u −
2179
908
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
c
12
u
16
+ 14u
15
+ ··· + 88u + 1
c
2
, c
5
, c
8
c
11
u
16
+ 4u
15
+ ··· − 2u + 1
c
3
, c
6
, c
7
c
9
u
16
− 2u
15
+ ··· − 128u + 256
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
c
12
y
16
− 18y
15
+ ··· − 2472y + 1
c
2
, c
5
, c
8
c
11
y
16
+ 14y
15
+ ··· + 88y + 1
c
3
, c
6
, c
7
c
9
y
16
− 30y
15
+ ··· + 540672y + 65536
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.363037 + 0.817564I
a = −0.689592 + 0.163353I
b = −0.232606 + 0.296439I
−0.31180 − 1.54577I −2.35937 + 4.98634I
u = −0.363037 − 0.817564I
a = −0.689592 − 0.163353I
b = −0.232606 − 0.296439I
−0.31180 + 1.54577I −2.35937 − 4.98634I
u = −0.479632 + 1.036130I
a = 1.70605 − 1.00375I
b = 0.266035 + 0.849001I
−0.679161 −6 − 0.644221 + 0.10I
u = −0.479632 − 1.036130I
a = 1.70605 + 1.00375I
b = 0.266035 − 0.849001I
−0.679161 −6 − 0.644221 + 0.10I
u = 0.735167 + 1.044790I
a = −0.615383 + 0.536646I
b = 0.72302 + 1.24109I
7.14404 −6 − 0.483738 + 0.10I
u = 0.735167 − 1.044790I
a = −0.615383 − 0.536646I
b = 0.72302 − 1.24109I
7.14404 −6 − 0.483738 + 0.10I
u = 1.264520 + 0.320297I
a = 0.438859 + 0.167437I
b = −2.28152 − 0.82827I
−8.81126 − 6.26912I −2.84932 + 2.54582I
u = 1.264520 − 0.320297I
a = 0.438859 − 0.167437I
b = −2.28152 + 0.82827I
−8.81126 + 6.26912I −2.84932 − 2.54582I
u = 0.61669 + 1.39802I
a = 1.61861 + 0.89803I
b = 2.33600 − 1.15509I
−8.81126 + 6.26912I −2.84932 − 2.54582I
u = 0.61669 − 1.39802I
a = 1.61861 − 0.89803I
b = 2.33600 + 1.15509I
−8.81126 − 6.26912I −2.84932 + 2.54582I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14859 + 1.53192I
a = 0.687769 − 1.077410I
b = 1.29707 − 2.19154I
−5.68794 −5.90825 + 0.I
u = −0.14859 − 1.53192I
a = 0.687769 + 1.077410I
b = 1.29707 + 2.19154I
−5.68794 −5.90825 + 0.I
u = 0.41065 + 1.67828I
a = −1.61714 − 0.65436I
b = −3.40684 + 0.49631I
−15.4295 −5.54640 + 0.I
u = 0.41065 − 1.67828I
a = −1.61714 + 0.65436I
b = −3.40684 − 0.49631I
−15.4295 −5.54640 + 0.I
u = −0.035772 + 0.140099I
a = 4.97083 + 2.67001I
b = 0.298852 − 0.519319I
−0.31180 + 1.54577I −2.35937 − 4.98634I
u = −0.035772 − 0.140099I
a = 4.97083 − 2.67001I
b = 0.298852 + 0.519319I
−0.31180 − 1.54577I −2.35937 + 4.98634I
10
III. I
u
3
= hb, −u
3
a + 2u
2
a − u
3
+ a
2
− 2au − u
2
+ 3u − 4, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
4
=
a
0
a
12
=
u
u
a
3
=
a
0
a
7
=
1
0
a
6
=
u
2
+ 1
−u
3
+ u
2
+ 1
a
5
=
2u
2
a + au + 2a
−2u
3
a + 2u
2
a + au + 2a
a
2
=
−u
3
+ u
2
+ a − 2u − 1
u
3
− u
2
− 1
a
10
=
u
3
u
3
+ u
a
1
=
−u
2
− 1
u
3
− u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
a − 3u
2
a + u
3
− au − 3u
2
− a + u − 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
, c
7
u
8
c
6
, c
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
8
(u
4
− u
3
+ u
2
+ 1)
2
c
9
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
(u
4
+ u
3
+ u
2
+ 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
4
c
3
, c
7
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
8
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = −1.73811 + 1.68562I
b = 0
−0.211005 + 0.614778I 1.30302 + 4.44028I
u = −0.351808 + 0.720342I
a = 2.32885 + 0.66243I
b = 0
−0.21101 − 3.44499I −3.64182 + 2.68374I
u = −0.351808 − 0.720342I
a = −1.73811 − 1.68562I
b = 0
−0.211005 − 0.614778I 1.30302 − 4.44028I
u = −0.351808 − 0.720342I
a = 2.32885 − 0.66243I
b = 0
−0.21101 + 3.44499I −3.64182 − 2.68374I
u = 0.851808 + 0.911292I
a = 0.156525 − 0.382204I
b = 0
6.79074 + 1.13408I −1.68800 − 4.61015I
u = 0.851808 + 0.911292I
a = 0.252736 + 0.326656I
b = 0
6.79074 + 5.19385I −4.47320 − 2.03656I
u = 0.851808 − 0.911292I
a = 0.156525 + 0.382204I
b = 0
6.79074 − 1.13408I −1.68800 + 4.61015I
u = 0.851808 − 0.911292I
a = 0.252736 − 0.326656I
b = 0
6.79074 − 5.19385I −4.47320 + 2.03656I
14
IV. I
u
4
=
h−a
3
u−2a
3
−3a
2
−au+3b+a+u+5, a
4
−a
3
u+2a
3
−a
2
u−4a−u−4, u
2
+u+1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
u + 1
a
4
=
a
1
3
a
3
u +
1
3
au + ··· −
1
3
a −
5
3
a
12
=
u
u
a
3
=
1
3
a
3
u +
1
3
au + ··· +
2
3
a −
5
3
1
3
a
3
u +
1
3
au + ··· −
1
3
a −
5
3
a
7
=
1
3
a
3
u −
2
3
a
2
u + ··· − a −
4
3
2
3
a
3
u +
2
3
a
2
u + ··· +
4
3
a
2
+
1
3
a
6
=
1
3
a
3
u −
2
3
a
2
u + ··· − a −
4
3
2
3
a
3
u +
2
3
a
2
u + ··· +
4
3
a
2
+
1
3
a
5
=
2
3
a
3
u +
4
3
a
2
u + ··· −
1
3
a + 1
−
1
3
a
3
u −
1
3
a
2
u + ··· −
2
3
a
2
+
4
3
a
2
=
a
3
u + a
3
+ 2a
2
− 2au − a − 3u − 3
2
3
a
3
u +
2
3
a
2
u + ··· +
4
3
a
2
+
1
3
a
10
=
1
u + 1
a
1
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
5
3
a
3
u −
7
3
a
3
− a
2
u − 5a
2
+
4
3
au +
8
3
a +
5
3
u +
13
3
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
(u
4
− u
3
+ u
2
+ 1)
2
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
c
6
, c
9
u
8
c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
8
, c
12
(u
2
+ u + 1)
4
c
10
, c
11
(u
2
− u + 1)
4
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
6
, c
9
y
8
c
8
, c
10
, c
11
c
12
(y
2
+ y + 1)
4
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.715307 − 0.631577I
b = −0.395123 + 0.506844I
−0.211005 − 0.614778I 1.30302 − 4.44028I
u = −0.500000 + 0.866025I
a = 1.248740 + 0.225872I
b = −0.10488 + 1.55249I
6.79074 + 1.13408I −1.68800 − 4.61015I
u = −0.500000 + 0.866025I
a = −1.44025 − 0.04422I
b = −0.10488 − 1.55249I
6.79074 − 5.19385I −4.47320 + 2.03656I
u = −0.500000 + 0.866025I
a = −1.59319 + 1.31595I
b = −0.395123 − 0.506844I
−0.21101 − 3.44499I −3.64182 + 2.68374I
u = −0.500000 − 0.866025I
a = −0.715307 + 0.631577I
b = −0.395123 − 0.506844I
−0.211005 + 0.614778I 1.30302 + 4.44028I
u = −0.500000 − 0.866025I
a = 1.248740 − 0.225872I
b = −0.10488 − 1.55249I
6.79074 − 1.13408I −1.68800 + 4.61015I
u = −0.500000 − 0.866025I
a = −1.44025 + 0.04422I
b = −0.10488 + 1.55249I
6.79074 + 5.19385I −4.47320 − 2.03656I
u = −0.500000 − 0.866025I
a = −1.59319 − 1.31595I
b = −0.395123 + 0.506844I
−0.21101 + 3.44499I −3.64182 − 2.68374I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
(u
2
− u + 1)
4
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
9
+ 3u
8
+ 8u
7
+ 5u
6
+ u
5
− 15u
4
− 20u
3
− 18u
2
− 7u − 1)
· (u
16
+ 14u
15
+ ··· + 88u + 1)
c
2
, c
8
(u
2
+ u + 1)
4
(u
4
− u
3
+ u
2
+ 1)
2
· (u
9
+ 3u
8
+ 6u
7
+ 7u
6
+ 7u
5
+ 7u
4
+ 6u
3
+ 4u
2
+ u + 1)
· (u
16
+ 4u
15
+ ··· − 2u + 1)
c
3
, c
6
u
8
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
9
− u
8
− 2u
7
+ 9u
6
+ 3u
5
+ 17u
4
+ 6u
3
+ 4u
2
− u + 1)
· (u
16
− 2u
15
+ ··· − 128u + 256)
c
5
, c
11
(u
2
− u + 1)
4
(u
4
+ u
3
+ u
2
+ 1)
2
· (u
9
+ 3u
8
+ 6u
7
+ 7u
6
+ 7u
5
+ 7u
4
+ 6u
3
+ 4u
2
+ u + 1)
· (u
16
+ 4u
15
+ ··· − 2u + 1)
c
7
, c
9
u
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
9
− u
8
− 2u
7
+ 9u
6
+ 3u
5
+ 17u
4
+ 6u
3
+ 4u
2
− u + 1)
· (u
16
− 2u
15
+ ··· − 128u + 256)
c
12
(u
2
+ u + 1)
4
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
9
+ 3u
8
+ 8u
7
+ 5u
6
+ u
5
− 15u
4
− 20u
3
− 18u
2
− 7u − 1)
· (u
16
+ 14u
15
+ ··· + 88u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
c
12
(y
2
+ y + 1)
4
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
9
+ 7y
8
+ 36y
7
+ 41y
6
− 75y
5
− 191y
4
− 144y
3
− 74y
2
+ 13y −1)
· (y
16
− 18y
15
+ ··· − 2472y + 1)
c
2
, c
5
, c
8
c
11
(y
2
+ y + 1)
4
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
9
+ 3y
8
+ 8y
7
+ 5y
6
+ y
5
− 15y
4
− 20y
3
− 18y
2
− 7y −1)
· (y
16
+ 14y
15
+ ··· + 88y + 1)
c
3
, c
6
, c
7
c
9
y
8
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
9
− 5y
8
+ 28y
7
− 47y
6
− 315y
5
− 319y
4
− 124y
3
− 62y
2
− 7y −1)
· (y
16
− 30y
15
+ ··· + 540672y + 65536)
20
