12a
0843
(K12a
0843
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 12 11 1 3 8 7 6
Solving Sequence
7,11
8 12 6 1 10
3,5
2 9 4
c
7
c
11
c
6
c
12
c
10
c
5
c
2
c
9
c
3
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
45
+ 26u
43
+ ··· + b − 1, −u
46
− 3u
45
+ ··· + a − 2, u
47
+ 2u
46
+ ··· + 2u + 1i
I
u
2
= hu
2
+ b − u + 1, u
4
− u
3
+ 3u
2
+ a − 2u + 1, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
* 2 irreducible components of dim
C
= 0, with total 52 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
45
+26u
43
+· · ·+b−1, −u
46
−3u
45
+· · ·+a−2, u
47
+2u
46
+· · ·+2u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
1
=
−u
3
− 2u
u
3
+ u
a
10
=
u
u
3
+ u
a
3
=
u
46
+ 3u
45
+ ··· − 5u + 2
−u
45
− 26u
43
+ ··· − u + 1
a
5
=
−u
6
− 3u
4
+ 1
−u
8
− 4u
6
− 4u
4
− 2u
2
a
2
=
u
46
+ 2u
45
+ ··· − 7u + 2
u
44
+ 2u
43
+ ··· − u
2
+ 1
a
9
=
−u
8
− 5u
6
− 7u
4
− 2u
2
+ 1
u
8
+ 4u
6
+ 4u
4
+ 2u
2
a
4
=
u
46
+ u
45
+ ··· − 7u + 1
u
45
+ 2u
44
+ ··· + 19u
3
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
46
+ 2u
45
+ ··· − 3u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
47
− 6u
46
+ ··· + 4u − 1
c
3
, c
9
u
47
− u
46
+ ··· − 64u − 32
c
5
, c
8
u
47
− 2u
46
+ ··· + 160u − 100
c
6
, c
7
, c
10
c
11
, c
12
u
47
− 2u
46
+ ··· + 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
47
− 48y
46
+ ··· + 28y −1
c
3
, c
9
y
47
+ 33y
46
+ ··· + 3584y −1024
c
5
, c
8
y
47
− 36y
46
+ ··· + 55800y −10000
c
6
, c
7
, c
10
c
11
, c
12
y
47
+ 60y
46
+ ··· + 18y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.413508 + 0.910925I
a = −1.83324 − 1.08728I
b = −0.378490 − 0.145011I
−8.75070 + 10.08960I −10.18236 − 7.10316I
u = −0.413508 − 0.910925I
a = −1.83324 + 1.08728I
b = −0.378490 + 0.145011I
−8.75070 − 10.08960I −10.18236 + 7.10316I
u = −0.381689 + 0.871051I
a = 2.02691 + 1.16470I
b = 0.497884 − 0.080589I
−2.05630 + 5.96873I −8.18293 − 7.13665I
u = −0.381689 − 0.871051I
a = 2.02691 − 1.16470I
b = 0.497884 + 0.080589I
−2.05630 − 5.96873I −8.18293 + 7.13665I
u = 0.091071 + 0.931262I
a = 0.711455 − 0.425911I
b = −0.488416 − 0.498587I
3.10179 − 1.86671I −0.52311 + 5.01743I
u = 0.091071 − 0.931262I
a = 0.711455 + 0.425911I
b = −0.488416 + 0.498587I
3.10179 + 1.86671I −0.52311 − 5.01743I
u = 0.192703 + 1.050590I
a = −0.491605 + 0.630434I
b = 0.819514 + 0.152813I
−1.81566 − 3.80529I −8.08850 + 4.35592I
u = 0.192703 − 1.050590I
a = −0.491605 − 0.630434I
b = 0.819514 − 0.152813I
−1.81566 + 3.80529I −8.08850 − 4.35592I
u = 0.386775 + 0.839886I
a = 0.354292 + 0.843743I
b = 0.287734 − 0.602113I
−4.41584 − 3.34895I −9.57199 + 4.15022I
u = 0.386775 − 0.839886I
a = 0.354292 − 0.843743I
b = 0.287734 + 0.602113I
−4.41584 + 3.34895I −9.57199 − 4.15022I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.275185 + 0.865499I
a = −0.224202 − 0.422215I
b = −0.175658 + 0.313273I
1.41842 − 2.55289I −1.19798 + 4.60383I
u = 0.275185 − 0.865499I
a = −0.224202 + 0.422215I
b = −0.175658 − 0.313273I
1.41842 + 2.55289I −1.19798 − 4.60383I
u = −0.376433 + 0.803567I
a = −2.17677 − 0.95906I
b = −0.372360 + 0.375511I
−2.47676 + 0.64455I −9.45745 − 1.20852I
u = −0.376433 − 0.803567I
a = −2.17677 + 0.95906I
b = −0.372360 − 0.375511I
−2.47676 − 0.64455I −9.45745 + 1.20852I
u = −0.441065 + 0.749468I
a = 2.04675 + 0.85956I
b = 0.100717 − 0.418593I
−9.72112 − 2.90941I −11.56361 − 0.62182I
u = −0.441065 − 0.749468I
a = 2.04675 − 0.85956I
b = 0.100717 + 0.418593I
−9.72112 + 2.90941I −11.56361 + 0.62182I
u = −0.064336 + 0.822622I
a = −1.48802 + 0.46619I
b = 0.238982 + 0.908240I
0.328476 + 0.959018I −5.90793 + 0.74635I
u = −0.064336 − 0.822622I
a = −1.48802 − 0.46619I
b = 0.238982 − 0.908240I
0.328476 − 0.959018I −5.90793 − 0.74635I
u = −0.638824 + 0.074996I
a = −0.421838 − 0.513599I
b = −0.04958 + 1.70077I
−11.75550 + 6.54260I −15.0022 − 4.1132I
u = −0.638824 − 0.074996I
a = −0.421838 + 0.513599I
b = −0.04958 − 1.70077I
−11.75550 − 6.54260I −15.0022 + 4.1132I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.602974
a = 1.18447
b = −0.503637
−6.95614 −14.4200
u = −0.596607 + 0.032355I
a = 0.220109 + 0.214041I
b = 0.05568 − 1.78658I
−4.79872 + 2.65777I −13.67344 − 3.52824I
u = −0.596607 − 0.032355I
a = 0.220109 − 0.214041I
b = 0.05568 + 1.78658I
−4.79872 − 2.65777I −13.67344 + 3.52824I
u = 0.474154 + 0.347428I
a = 1.099760 + 0.840634I
b = −0.249987 − 0.545071I
−6.24028 − 1.59922I −13.28615 + 4.03816I
u = 0.474154 − 0.347428I
a = 1.099760 − 0.840634I
b = −0.249987 + 0.545071I
−6.24028 + 1.59922I −13.28615 − 4.03816I
u = 0.473616
a = −0.668057
b = 0.235856
−1.20734 −8.21660
u = −0.09345 + 1.61990I
a = −1.84799 − 0.75382I
b = 3.72034 + 1.96860I
−1.65071 − 1.00207I 0
u = −0.09345 − 1.61990I
a = −1.84799 + 0.75382I
b = 3.72034 − 1.96860I
−1.65071 + 1.00207I 0
u = −0.08668 + 1.65706I
a = 2.61073 + 1.11521I
b = −4.96661 − 2.65420I
6.07762 + 2.32488I 0
u = −0.08668 − 1.65706I
a = 2.61073 − 1.11521I
b = −4.96661 + 2.65420I
6.07762 − 2.32488I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.09573 + 1.66580I
a = −1.009190 − 0.039969I
b = 1.63623 + 0.52270I
4.30233 − 5.15506I 0
u = 0.09573 − 1.66580I
a = −1.009190 + 0.039969I
b = 1.63623 − 0.52270I
4.30233 + 5.15506I 0
u = 0.238719 + 0.226721I
a = −1.01393 − 1.12454I
b = 0.038509 + 0.406266I
−0.387281 − 0.808831I −8.67312 + 8.42283I
u = 0.238719 − 0.226721I
a = −1.01393 + 1.12454I
b = 0.038509 − 0.406266I
−0.387281 + 0.808831I −8.67312 − 8.42283I
u = −0.01112 + 1.67312I
a = 1.51780 − 1.37183I
b = −3.19718 + 2.02255I
9.18863 + 1.20864I 0
u = −0.01112 − 1.67312I
a = 1.51780 + 1.37183I
b = −3.19718 − 2.02255I
9.18863 − 1.20864I 0
u = −0.09778 + 1.67602I
a = −2.52969 − 1.85960I
b = 4.67873 + 3.90951I
6.83164 + 7.79802I 0
u = −0.09778 − 1.67602I
a = −2.52969 + 1.85960I
b = 4.67873 − 3.90951I
6.83164 − 7.79802I 0
u = 0.06732 + 1.68247I
a = 0.599873 − 0.006463I
b = −1.007440 − 0.229685I
10.40890 − 3.84665I 0
u = 0.06732 − 1.68247I
a = 0.599873 + 0.006463I
b = −1.007440 + 0.229685I
10.40890 + 3.84665I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.11098 + 1.68676I
a = 2.12470 + 2.11421I
b = −3.92124 − 4.22596I
0.30937 + 12.13810I 0
u = −0.11098 − 1.68676I
a = 2.12470 − 2.11421I
b = −3.92124 + 4.22596I
0.30937 − 12.13810I 0
u = 0.02004 + 1.69490I
a = −0.497803 + 1.204650I
b = 1.25861 − 1.93145I
12.42340 − 2.28034I 0
u = 0.02004 − 1.69490I
a = −0.497803 − 1.204650I
b = 1.25861 + 1.93145I
12.42340 + 2.28034I 0
u = 0.04372 + 1.72277I
a = −0.332562 − 1.108820I
b = 0.20612 + 1.95580I
8.05569 − 4.72882I 0
u = 0.04372 − 1.72277I
a = −0.332562 + 1.108820I
b = 0.20612 − 1.95580I
8.05569 + 4.72882I 0
u = −0.222465
a = 2.59250
b = 0.803601
−2.01148 −1.29870
9
II.
I
u
2
= hu
2
+ b − u + 1, u
4
− u
3
+ 3u
2
+ a − 2u + 1, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
1
=
−u
3
− 2u
u
3
+ u
a
10
=
u
u
3
+ u
a
3
=
−u
4
+ u
3
− 3u
2
+ 2u − 1
−u
2
+ u − 1
a
5
=
u
3
+ 2u
−u
3
− u
a
2
=
−u
4
− 3u
2
− 1
u
3
− u
2
+ 2u − 1
a
9
=
u
u
3
+ u
a
4
=
−u
4
+ u
3
− 3u
2
+ 2u − 1
−u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
4
+ 5u
3
− 20u
2
+ 14u − 21
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
9
u
5
c
4
(u + 1)
5
c
5
, c
8
u
5
− u
4
+ u
2
+ u − 1
c
6
, c
7
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
10
, c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
9
y
5
c
5
, c
8
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
6
, c
7
, c
10
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.758138 + 0.584034I
b = −0.036717 + 0.471689I
0.17487 − 2.21397I −7.62657 + 4.39306I
u = 0.233677 − 0.885557I
a = 0.758138 − 0.584034I
b = −0.036717 − 0.471689I
0.17487 + 2.21397I −7.62657 − 4.39306I
u = 0.416284
a = −0.645200
b = −0.757008
−2.52712 −18.4270
u = 0.05818 + 1.69128I
a = −0.935538 − 0.903908I
b = 1.91522 + 1.49448I
9.31336 − 3.33174I −6.15976 + 1.26157I
u = 0.05818 − 1.69128I
a = −0.935538 + 0.903908I
b = 1.91522 − 1.49448I
9.31336 + 3.33174I −6.15976 − 1.26157I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
5
)(u
47
− 6u
46
+ ··· + 4u − 1)
c
3
, c
9
u
5
(u
47
− u
46
+ ··· − 64u − 32)
c
4
((u + 1)
5
)(u
47
− 6u
46
+ ··· + 4u − 1)
c
5
, c
8
(u
5
− u
4
+ u
2
+ u − 1)(u
47
− 2u
46
+ ··· + 160u − 100)
c
6
, c
7
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
47
− 2u
46
+ ··· + 2u − 1)
c
10
, c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
47
− 2u
46
+ ··· + 2u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
5
)(y
47
− 48y
46
+ ··· + 28y −1)
c
3
, c
9
y
5
(y
47
+ 33y
46
+ ··· + 3584y −1024)
c
5
, c
8
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
47
− 36y
46
+ ··· + 55800y −10000)
c
6
, c
7
, c
10
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
47
+ 60y
46
+ ··· + 18y −1)
15
