12n
0289
(K12n
0289
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 10 12 5 1 8 11
Solving Sequence
2,6
3
1,10
11 5 7 8 4 12 9
c
2
c
1
c
10
c
5
c
6
c
7
c
4
c
12
c
8
c
3
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− 2u
6
+ u
5
+ 2u
4
− u
3
− u
2
+ b + u, −u
11
+ 2u
9
− 4u
7
+ 4u
5
− 2u
4
− 3u
3
+ 2u
2
+ a + 2u − 2,
u
12
− u
11
− 2u
10
+ 3u
9
+ 3u
8
− 5u
7
− 2u
6
+ 6u
5
− 4u
3
+ 3u − 1i
I
u
2
= h−3u
41
+ 7u
40
+ ··· + 2b + 7, 3u
41
− 5u
40
+ ··· + 2a + 5, u
42
− 3u
41
+ ··· − 2u + 1i
I
u
3
= hb + u, a + u, u
3
+ u
2
− 1i
I
u
4
= hb − a, u
2
a + a
2
+ u
2
+ 2u + 1, u
3
+ u
2
− 1i
* 4 irreducible components of dim
C
= 0, with total 63 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
8
− 2u
6
+ u
5
+ 2u
4
− u
3
− u
2
+ b + u, −u
11
+ 2u
9
+ · · · + a −
2, u
12
− u
11
+ · · · + 3u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
10
=
u
11
− 2u
9
+ 4u
7
− 4u
5
+ 2u
4
+ 3u
3
− 2u
2
− 2u + 2
−u
8
+ 2u
6
− u
5
− 2u
4
+ u
3
+ u
2
− u
a
11
=
u
11
− 2u
9
+ 4u
7
− 4u
5
+ u
4
+ 3u
3
− u
2
− 2u + 2
−u
8
+ u
6
− u
5
− 2u
4
+ u
3
+ u
2
− u
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
8
=
−u
9
+ 2u
7
− u
6
− 3u
5
+ u
4
+ 2u
3
− u
2
− u
−u
9
+ 2u
7
− u
6
− 3u
5
+ 2u
4
+ 2u
3
− 2u
2
− u + 1
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
12
=
u
11
− u
10
− 2u
9
+ 2u
8
+ 3u
7
− 3u
6
− 3u
5
+ 3u
4
+ 2u
3
− 2u
2
− 2u + 2
−u
10
+ u
8
− u
7
− 2u
6
+ u
5
− u
3
a
9
=
−u
4
+ u
2
− 1
u
11
− 2u
9
+ u
8
+ 4u
7
− 2u
6
− 3u
5
+ 3u
4
+ 2u
3
− 2u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
11
− 6u
9
+ 12u
7
+ 2u
6
− 8u
5
+ 2u
4
+ 8u
3
+ 2u
2
− 8u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
10
c
12
u
12
+ 5u
11
+ ··· + 9u + 1
c
2
, c
5
, c
8
c
11
u
12
+ u
11
− 2u
10
− 3u
9
+ 3u
8
+ 5u
7
− 2u
6
− 6u
5
+ 4u
3
− 3u − 1
c
3
, c
7
u
12
− u
11
+ ··· − 5u − 1
c
4
, c
9
u
12
+ 7u
11
+ ··· + 32u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
10
c
12
y
12
+ 7y
11
+ ··· − 33y + 1
c
2
, c
5
, c
8
c
11
y
12
− 5y
11
+ ··· − 9y + 1
c
3
, c
7
y
12
− 17y
11
+ ··· − 9y + 1
c
4
, c
9
y
12
− 7y
11
+ ··· − 192y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.921925 + 0.343588I
a = −0.611717 + 0.476706I
b = 0.056760 − 0.351806I
−2.36994 − 2.76031I −17.1687 + 5.8086I
u = 0.921925 − 0.343588I
a = −0.611717 − 0.476706I
b = 0.056760 + 0.351806I
−2.36994 + 2.76031I −17.1687 − 5.8086I
u = 0.588705 + 0.829892I
a = 1.69413 − 0.79218I
b = 1.295740 + 0.564858I
−0.17368 + 3.06646I −8.92631 − 0.43083I
u = 0.588705 − 0.829892I
a = 1.69413 + 0.79218I
b = 1.295740 − 0.564858I
−0.17368 − 3.06646I −8.92631 + 0.43083I
u = −0.700347 + 0.661080I
a = 0.49057 + 1.67492I
b = 1.37850 + 1.19320I
3.36282 + 2.18981I −7.17700 − 3.81343I
u = −0.700347 − 0.661080I
a = 0.49057 − 1.67492I
b = 1.37850 − 1.19320I
3.36282 − 2.18981I −7.17700 + 3.81343I
u = −0.993915 + 0.611197I
a = −1.53521 − 1.43733I
b = −2.02283 − 0.51409I
1.49384 + 7.77925I −11.7273 − 7.9652I
u = −0.993915 − 0.611197I
a = −1.53521 + 1.43733I
b = −2.02283 + 0.51409I
1.49384 − 7.77925I −11.7273 + 7.9652I
u = −1.18481
a = −0.513967
b = 0.968302
−12.5188 −20.4260
u = 1.073430 + 0.702670I
a = −0.31962 + 2.22050I
b = −1.57673 + 2.45459I
−3.0728 − 14.5878I −12.5949 + 9.1386I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.073430 − 0.702670I
a = −0.31962 − 2.22050I
b = −1.57673 − 2.45459I
−3.0728 + 14.5878I −12.5949 − 9.1386I
u = 0.405199
a = 1.07767
b = −0.231197
−0.765991 −12.3860
6
II. I
u
2
=
h−3u
41
+7u
40
+· · ·+2b+7, 3u
41
−5u
40
+· · ·+2a+5, u
42
−3u
41
+· · ·−2u+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
10
=
−
3
2
u
41
+
5
2
u
40
+ ··· +
21
2
u −
5
2
3
2
u
41
−
7
2
u
40
+ ··· +
15
2
u −
7
2
a
11
=
11
2
u
41
− 15u
40
+ ··· + 17u −
17
2
7u
41
−
35
2
u
40
+ ··· +
31
2
u −
21
2
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
8
=
1
2
u
39
− u
38
+ ··· + u −
1
2
1
2
u
39
− u
38
+ ··· + u +
1
2
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
12
=
1
2
u
41
−
3
2
u
40
+ ··· −
1
2
u + 1
−
1
2
u
40
+
9
2
u
38
+ ··· +
1
2
u − 1
a
9
=
−
3
2
u
41
+
9
2
u
40
+ ··· −
27
2
u +
11
2
−
9
2
u
41
+
21
2
u
40
+ ··· −
21
2
u +
13
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
41
−
9
2
u
40
+ ··· −
45
2
u −
21
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
10
c
12
u
42
+ 15u
41
+ ··· + 20u + 1
c
2
, c
5
, c
8
c
11
u
42
+ 3u
41
+ ··· + 2u + 1
c
3
, c
7
u
42
− 3u
41
+ ··· + 24u + 1
c
4
, c
9
(u
21
− 3u
20
+ ··· − 4u + 8)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
10
c
12
y
42
+ 25y
41
+ ··· − 12y + 1
c
2
, c
5
, c
8
c
11
y
42
− 15y
41
+ ··· − 20y + 1
c
3
, c
7
y
42
− 35y
41
+ ··· − 372y + 1
c
4
, c
9
(y
21
− 21y
20
+ ··· + 80y −64)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.001590 + 0.071446I
a = −0.247034 + 0.982841I
b = −0.0703049 − 0.0213694I
−1.68246 + 2.02701I −16.1395 − 3.2075I
u = 1.001590 − 0.071446I
a = −0.247034 − 0.982841I
b = −0.0703049 + 0.0213694I
−1.68246 − 2.02701I −16.1395 + 3.2075I
u = −0.884746 + 0.507709I
a = −1.00430 − 1.65860I
b = −1.70041 − 0.93550I
−1.68246 + 2.02701I −16.1395 − 3.2075I
u = −0.884746 − 0.507709I
a = −1.00430 + 1.65860I
b = −1.70041 + 0.93550I
−1.68246 − 2.02701I −16.1395 + 3.2075I
u = 0.498112 + 0.843586I
a = −1.51048 + 0.33957I
b = −0.92205 − 1.13548I
−6.44569 + 1.90498I −15.1767 − 0.6933I
u = 0.498112 − 0.843586I
a = −1.51048 − 0.33957I
b = −0.92205 + 1.13548I
−6.44569 − 1.90498I −15.1767 + 0.6933I
u = 0.833041 + 0.624453I
a = −0.244467 − 1.048030I
b = −0.795422 − 0.433395I
4.76367 + 0.56948I −11.53430 − 0.71170I
u = 0.833041 − 0.624453I
a = −0.244467 + 1.048030I
b = −0.795422 + 0.433395I
4.76367 − 0.56948I −11.53430 + 0.71170I
u = 0.589823 + 0.864603I
a = −1.89138 + 0.71030I
b = −1.53680 − 0.72844I
−1.59942 + 8.75882I −10.82911 − 4.89320I
u = 0.589823 − 0.864603I
a = −1.89138 − 0.71030I
b = −1.53680 + 0.72844I
−1.59942 − 8.75882I −10.82911 + 4.89320I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.865106 + 0.622456I
a = 0.478460 + 0.785204I
b = 1.077610 + 0.227526I
4.66319 − 5.46111I −12.12408 + 5.29794I
u = 0.865106 − 0.622456I
a = 0.478460 − 0.785204I
b = 1.077610 − 0.227526I
4.66319 + 5.46111I −12.12408 − 5.29794I
u = 0.370195 + 0.797477I
a = −0.975137 + 0.163149I
b = −0.109542 − 1.285730I
−2.89368 − 5.09092I −11.85705 + 4.85512I
u = 0.370195 − 0.797477I
a = −0.975137 − 0.163149I
b = −0.109542 + 1.285730I
−2.89368 + 5.09092I −11.85705 − 4.85512I
u = −0.877054 + 0.709305I
a = 0.621738 + 0.723259I
b = 0.924556 + 0.344996I
2.39962 + 2.72155I −2.38517 − 1.80674I
u = −0.877054 − 0.709305I
a = 0.621738 − 0.723259I
b = 0.924556 − 0.344996I
2.39962 − 2.72155I −2.38517 + 1.80674I
u = −1.140080 + 0.053957I
a = 0.121069 − 0.443029I
b = −1.234320 − 0.298114I
−6.44569 + 1.90498I −15.1767 − 0.6933I
u = −1.140080 − 0.053957I
a = 0.121069 + 0.443029I
b = −1.234320 + 0.298114I
−6.44569 − 1.90498I −15.1767 + 0.6933I
u = −0.948500 + 0.657394I
a = 1.35145 + 1.05024I
b = 1.70425 + 0.27722I
2.62978 + 2.94639I −8.38979 − 1.94831I
u = −0.948500 − 0.657394I
a = 1.35145 − 1.05024I
b = 1.70425 − 0.27722I
2.62978 − 2.94639I −8.38979 + 1.94831I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.440380 + 0.720566I
a = 1.022170 − 0.478261I
b = 0.238333 + 0.804288I
−1.18994 −9.46820 + 0.I
u = 0.440380 − 0.720566I
a = 1.022170 + 0.478261I
b = 0.238333 − 0.804288I
−1.18994 −9.46820 + 0.I
u = −0.602886 + 0.586036I
a = −0.56399 − 1.78814I
b = −1.51211 − 1.10890I
2.62978 − 2.94639I −8.38979 + 1.94831I
u = −0.602886 − 0.586036I
a = −0.56399 + 1.78814I
b = −1.51211 + 1.10890I
2.62978 + 2.94639I −8.38979 − 1.94831I
u = −1.169360 + 0.079358I
a = −0.342452 + 0.671662I
b = 1.087440 + 0.452577I
−8.23029 + 7.48200I −17.0704 − 5.2473I
u = −1.169360 − 0.079358I
a = −0.342452 − 0.671662I
b = 1.087440 − 0.452577I
−8.23029 − 7.48200I −17.0704 + 5.2473I
u = −0.869430 + 0.810182I
a = −0.90284 + 1.14540I
b = −0.28767 + 1.52547I
4.76367 + 0.56948I −11.53430 − 0.71170I
u = −0.869430 − 0.810182I
a = −0.90284 − 1.14540I
b = −0.28767 − 1.52547I
4.76367 − 0.56948I −11.53430 + 0.71170I
u = −0.903021 + 0.803753I
a = 1.23931 − 0.82374I
b = 0.76478 − 1.36737I
4.66319 + 5.46111I −12.00000 − 5.29794I
u = −0.903021 − 0.803753I
a = 1.23931 + 0.82374I
b = 0.76478 + 1.36737I
4.66319 − 5.46111I −12.00000 + 5.29794I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.054680 + 0.618331I
a = −0.151148 − 1.199150I
b = 0.95395 − 1.70109I
−2.89368 − 5.09092I −12.00000 + 4.85512I
u = 1.054680 − 0.618331I
a = −0.151148 + 1.199150I
b = 0.95395 + 1.70109I
−2.89368 + 5.09092I −12.00000 − 4.85512I
u = 1.086360 + 0.586494I
a = 0.636112 + 1.121460I
b = −0.52511 + 1.74837I
−5.00675 −15.0273 + 0.I
u = 1.086360 − 0.586494I
a = 0.636112 − 1.121460I
b = −0.52511 − 1.74837I
−5.00675 −15.0273 + 0.I
u = 1.060980 + 0.690081I
a = 0.35900 − 1.97361I
b = 1.55493 − 2.24059I
−1.59942 − 8.75882I −10.82911 + 4.89320I
u = 1.060980 − 0.690081I
a = 0.35900 + 1.97361I
b = 1.55493 + 2.24059I
−1.59942 + 8.75882I −10.82911 − 4.89320I
u = 1.091260 + 0.656406I
a = 0.23548 + 1.83960I
b = −1.02758 + 2.26060I
−8.23029 − 7.48200I −17.0704 + 5.2473I
u = 1.091260 − 0.656406I
a = 0.23548 − 1.83960I
b = −1.02758 − 2.26060I
−8.23029 + 7.48200I −17.0704 − 5.2473I
u = 0.448134 + 0.218946I
a = 0.896777 − 0.436761I
b = −0.220693 + 0.109203I
−0.751959 −11.49246 + 0.I
u = 0.448134 − 0.218946I
a = 0.896777 + 0.436761I
b = −0.220693 − 0.109203I
−0.751959 −11.49246 + 0.I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.444576 + 0.086145I
a = −0.12834 − 2.59332I
b = −0.36382 − 1.40958I
2.39962 − 2.72155I −2.38517 + 1.80674I
u = −0.444576 − 0.086145I
a = −0.12834 + 2.59332I
b = −0.36382 + 1.40958I
2.39962 + 2.72155I −2.38517 − 1.80674I
14
III. I
u
3
= hb + u, a + u, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
10
=
−u
−u
a
11
=
−2u + 1
u
2
− 2u
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
8
=
2u
2
− 2
2u
2
+ u − 2
a
4
=
u
u
a
12
=
2u
2
− 1
2u
2
− 2
a
9
=
−u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u − 12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
u
3
− u
2
+ 2u − 1
c
2
, c
8
u
3
+ u
2
− 1
c
4
, c
9
u
3
c
5
, c
11
u
3
− u
2
+ 1
c
6
, c
12
u
3
+ u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
8
c
11
y
3
− y
2
+ 2y −1
c
4
, c
9
y
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.877439 − 0.744862I
b = 0.877439 − 0.744862I
6.04826 + 5.65624I −4.98049 − 5.95889I
u = −0.877439 − 0.744862I
a = 0.877439 + 0.744862I
b = 0.877439 + 0.744862I
6.04826 − 5.65624I −4.98049 + 5.95889I
u = 0.754878
a = −0.754878
b = −0.754878
−2.22691 −18.0390
18
IV. I
u
4
= hb − a, u
2
a + a
2
+ u
2
+ 2u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
10
=
a
a
a
11
=
−u
2
a − au + 2a
−au + 2a
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
8
=
−u
2
a + 2u
2
+ a + u
−u
2
a + 2u
2
+ a + 2u
a
4
=
u
u
a
12
=
−3u
2
a + 2a + 2
−3u
2
a − au + u
2
+ 3a + 2
a
9
=
a
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
2
a + au − u
2
− 8a − 19
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
8
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
, c
11
(u
3
− u
2
+ 1)
2
c
6
, c
12
(u
3
+ u
2
+ 2u + 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
8
c
11
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.592519 + 0.986732I
b = −0.592519 + 0.986732I
6.04826 −5.39114 + 0.I
u = −0.877439 + 0.744862I
a = 0.377439 + 0.320410I
b = 0.377439 + 0.320410I
1.91067 + 2.82812I −18.8044 − 4.6518I
u = −0.877439 − 0.744862I
a = −0.592519 − 0.986732I
b = −0.592519 − 0.986732I
6.04826 −5.39114 + 0.I
u = −0.877439 − 0.744862I
a = 0.377439 − 0.320410I
b = 0.377439 − 0.320410I
1.91067 − 2.82812I −18.8044 + 4.6518I
u = 0.754878
a = −0.28492 + 1.73159I
b = −0.28492 + 1.73159I
1.91067 + 2.82812I −18.8044 − 4.6518I
u = 0.754878
a = −0.28492 − 1.73159I
b = −0.28492 − 1.73159I
1.91067 − 2.82812I −18.8044 + 4.6518I
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
12
+ 5u
11
+ ··· + 9u + 1)
· (u
42
+ 15u
41
+ ··· + 20u + 1)
c
2
, c
8
(u
3
+ u
2
− 1)
3
· (u
12
+ u
11
− 2u
10
− 3u
9
+ 3u
8
+ 5u
7
− 2u
6
− 6u
5
+ 4u
3
− 3u − 1)
· (u
42
+ 3u
41
+ ··· + 2u + 1)
c
3
, c
7
((u
3
− u
2
+ 2u − 1)
3
)(u
12
− u
11
+ ··· − 5u − 1)
· (u
42
− 3u
41
+ ··· + 24u + 1)
c
4
, c
9
u
9
(u
12
+ 7u
11
+ ··· + 32u + 8)(u
21
− 3u
20
+ ··· − 4u + 8)
2
c
5
, c
11
(u
3
− u
2
+ 1)
3
· (u
12
+ u
11
− 2u
10
− 3u
9
+ 3u
8
+ 5u
7
− 2u
6
− 6u
5
+ 4u
3
− 3u − 1)
· (u
42
+ 3u
41
+ ··· + 2u + 1)
c
6
, c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
12
+ 5u
11
+ ··· + 9u + 1)
· (u
42
+ 15u
41
+ ··· + 20u + 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
10
c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
12
+ 7y
11
+ ··· − 33y + 1)
· (y
42
+ 25y
41
+ ··· − 12y + 1)
c
2
, c
5
, c
8
c
11
((y
3
− y
2
+ 2y −1)
3
)(y
12
− 5y
11
+ ··· − 9y + 1)
· (y
42
− 15y
41
+ ··· − 20y + 1)
c
3
, c
7
((y
3
+ 3y
2
+ 2y −1)
3
)(y
12
− 17y
11
+ ··· − 9y + 1)
· (y
42
− 35y
41
+ ··· − 372y + 1)
c
4
, c
9
y
9
(y
12
− 7y
11
+ ··· − 192y + 64)(y
21
− 21y
20
+ ··· + 80y −64)
2
24
