12a
0537
(K12a
0537
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 11 9 2 1 12 6 5 4
Solving Sequence
2,8
7 3 4 1 9 6 12 10 5 11
c
7
c
2
c
3
c
1
c
8
c
6
c
12
c
9
c
4
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
89
− u
88
+ ··· + u − 1i
* 1 irreducible components of dim
C
= 0, with total 89 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
89
− u
88
+ · · · + u − 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
9
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
6
=
u
16
+ 3u
14
+ 5u
12
+ 4u
10
+ 3u
8
+ 2u
6
+ 2u
4
+ 1
u
18
+ 4u
16
+ 9u
14
+ 12u
12
+ 11u
10
+ 6u
8
+ 2u
6
+ u
2
a
12
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
10
=
−u
32
− 7u
30
+ ··· + 2u
4
+ 1
u
32
+ 8u
30
+ ··· + 4u
4
+ 2u
2
a
5
=
u
61
+ 14u
59
+ ··· − 2u
3
− u
−u
61
− 15u
59
+ ··· − u
3
+ u
a
11
=
−u
66
− 15u
64
+ ··· − u
2
+ 1
−u
68
− 16u
66
+ ··· + 3u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
87
− 4u
86
+ ··· − 4u
2
− 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
89
+ 43u
88
+ ··· + u − 1
c
2
, c
7
u
89
+ u
88
+ ··· + u + 1
c
3
u
89
− u
88
+ ··· + 1497u + 457
c
4
u
89
− u
88
+ ··· − 5u + 1
c
5
, c
10
, c
11
u
89
+ u
88
+ ··· + 3u + 1
c
6
, c
12
u
89
+ 7u
88
+ ··· + 161u + 5
c
8
u
89
+ 5u
88
+ ··· + u + 1
c
9
u
89
− 21u
88
+ ··· − 4023u + 187
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
89
+ 7y
88
+ ··· + 13y −1
c
2
, c
7
y
89
+ 43y
88
+ ··· + y −1
c
3
y
89
− 29y
88
+ ··· + 6081637y −208849
c
4
y
89
− 5y
88
+ ··· + 17y −1
c
5
, c
10
, c
11
y
89
+ 79y
88
+ ··· + y −1
c
6
, c
12
y
89
+ 67y
88
+ ··· + 201y −25
c
8
y
89
− y
88
+ ··· + 61y −1
c
9
y
89
+ 11y
88
+ ··· − 1504923y −34969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.549845 + 0.852652I
3.32155 + 4.57387I 0
u = −0.549845 − 0.852652I
3.32155 − 4.57387I 0
u = 0.526296 + 0.826367I
−1.86086 − 1.10863I 0
u = 0.526296 − 0.826367I
−1.86086 + 1.10863I 0
u = −0.445113 + 0.947902I
−0.40489 − 1.97636I 0
u = −0.445113 − 0.947902I
−0.40489 + 1.97636I 0
u = −0.130520 + 0.933837I
3.46999 + 3.77804I −4.00000 − 3.55964I
u = −0.130520 − 0.933837I
3.46999 − 3.77804I −4.00000 + 3.55964I
u = 0.237242 + 0.899093I
−1.63382 − 0.87283I −8.59634 + 3.63079I
u = 0.237242 − 0.899093I
−1.63382 + 0.87283I −8.59634 − 3.63079I
u = −0.621956 + 0.688948I
3.81774 − 9.21070I 0. + 8.07606I
u = −0.621956 − 0.688948I
3.81774 + 9.21070I 0. − 8.07606I
u = 0.605630 + 0.694228I
−1.43820 + 5.63420I −4.00000 − 7.71170I
u = 0.605630 − 0.694228I
−1.43820 − 5.63420I −4.00000 + 7.71170I
u = 0.523965 + 0.945138I
5.28882 + 3.53448I 0
u = 0.523965 − 0.945138I
5.28882 − 3.53448I 0
u = −0.544235 + 0.732626I
0.19552 − 2.15749I −2.95032 + 4.14794I
u = −0.544235 − 0.732626I
0.19552 + 2.15749I −2.95032 − 4.14794I
u = −0.556700 + 0.687523I
0.25637 − 2.05649I −1.44246 + 3.13624I
u = −0.556700 − 0.687523I
0.25637 + 2.05649I −1.44246 − 3.13624I
u = 0.601306 + 0.620005I
6.23656 + 0.95324I 3.59693 − 2.99187I
u = 0.601306 − 0.620005I
6.23656 − 0.95324I 3.59693 + 2.99187I
u = −0.389004 + 1.075220I
−0.038564 − 0.742399I 0
u = −0.389004 − 1.075220I
−0.038564 + 0.742399I 0
u = −0.270428 + 1.114400I
0.573665 − 0.094345I 0
u = −0.270428 − 1.114400I
0.573665 + 0.094345I 0
u = 0.443236 + 1.076980I
−3.58294 + 3.54004I 0
u = 0.443236 − 1.076980I
−3.58294 − 3.54004I 0
u = 0.773116 + 0.297168I
1.90632 − 11.11310I −1.08984 + 6.87770I
u = 0.773116 − 0.297168I
1.90632 + 11.11310I −1.08984 − 6.87770I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.555744 + 1.035790I
6.42783 + 1.42029I 0
u = 0.555744 − 1.035790I
6.42783 − 1.42029I 0
u = −0.537606 + 1.047480I
0.71771 − 3.82158I 0
u = −0.537606 − 1.047480I
0.71771 + 3.82158I 0
u = −0.767527 + 0.290372I
−3.36844 + 7.41224I −5.77885 − 6.28864I
u = −0.767527 − 0.290372I
−3.36844 − 7.41224I −5.77885 + 6.28864I
u = 0.654390 + 0.489201I
8.03046 + 3.30236I 5.19136 − 3.35930I
u = 0.654390 − 0.489201I
8.03046 − 3.30236I 5.19136 + 3.35930I
u = 0.280299 + 1.149490I
−5.92026 − 0.49034I 0
u = 0.280299 − 1.149490I
−5.92026 + 0.49034I 0
u = 0.260330 + 1.156600I
−2.58902 − 8.08324I 0
u = 0.260330 − 1.156600I
−2.58902 + 8.08324I 0
u = −0.267783 + 1.155220I
−7.80687 + 4.35790I 0
u = −0.267783 − 1.155220I
−7.80687 − 4.35790I 0
u = 0.293939 + 1.151340I
−6.07434 − 0.28887I 0
u = 0.293939 − 1.151340I
−6.07434 + 0.28887I 0
u = −0.683893 + 0.428051I
7.74822 + 5.29418I 4.36161 − 4.33354I
u = −0.683893 − 0.428051I
7.74822 − 5.29418I 4.36161 + 4.33354I
u = −0.308826 + 1.154160I
−8.28930 − 3.48798I 0
u = −0.308826 − 1.154160I
−8.28930 + 3.48798I 0
u = −0.471075 + 1.098060I
0.51569 − 6.51976I 0
u = −0.471075 − 1.098060I
0.51569 + 6.51976I 0
u = 0.752387 + 0.281402I
−1.59860 − 3.55841I −3.12371 + 1.57420I
u = 0.752387 − 0.281402I
−1.59860 + 3.55841I −3.12371 − 1.57420I
u = 0.318069 + 1.155530I
−3.26936 + 7.19305I 0
u = 0.318069 − 1.155530I
−3.26936 − 7.19305I 0
u = −0.734105 + 0.315100I
4.84491 + 2.69309I 1.74893 − 2.46850I
u = −0.734105 − 0.315100I
4.84491 − 2.69309I 1.74893 + 2.46850I
u = 0.548293 + 1.068850I
0.30757 + 7.15974I 0
u = 0.548293 − 1.068850I
0.30757 − 7.15974I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.561612 + 1.069280I
5.87439 − 10.10820I 0
u = −0.561612 − 1.069280I
5.87439 + 10.10820I 0
u = 0.746796 + 0.262051I
−1.84029 − 3.44425I −4.01435 + 3.41599I
u = 0.746796 − 0.262051I
−1.84029 + 3.44425I −4.01435 − 3.41599I
u = −0.621550 + 0.465042I
2.42382 − 0.75571I 1.71757 + 3.84156I
u = −0.621550 − 0.465042I
2.42382 + 0.75571I 1.71757 − 3.84156I
u = 0.653966 + 0.417174I
2.20293 − 2.46586I 0.44156 + 5.21162I
u = 0.653966 − 0.417174I
2.20293 + 2.46586I 0.44156 − 5.21162I
u = −0.737893 + 0.238500I
−4.15809 − 0.23287I −7.53192 + 1.38224I
u = −0.737893 − 0.238500I
−4.15809 + 0.23287I −7.53192 − 1.38224I
u = 0.733536 + 0.221566I
0.79810 + 3.87164I −2.73189 − 2.63327I
u = 0.733536 − 0.221566I
0.79810 − 3.87164I −2.73189 + 2.63327I
u = −0.553875 + 1.124130I
2.48561 − 7.58332I 0
u = −0.553875 − 1.124130I
2.48561 + 7.58332I 0
u = 0.526277 + 1.142710I
−1.85636 + 0.86225I 0
u = 0.526277 − 1.142710I
−1.85636 − 0.86225I 0
u = −0.532862 + 1.141630I
−6.76881 − 4.54935I 0
u = −0.532862 − 1.141630I
−6.76881 + 4.54935I 0
u = 0.541979 + 1.140190I
−4.39315 + 8.29738I 0
u = 0.541979 − 1.140190I
−4.39315 − 8.29738I 0
u = 0.549916 + 1.137700I
−4.09560 + 8.46534I 0
u = 0.549916 − 1.137700I
−4.09560 − 8.46534I 0
u = −0.556243 + 1.140170I
−5.85930 − 12.38420I 0
u = −0.556243 − 1.140170I
−5.85930 + 12.38420I 0
u = 0.559891 + 1.140150I
−0.5695 + 16.1155I 0
u = 0.559891 − 1.140150I
−0.5695 − 16.1155I 0
u = −0.570709 + 0.131663I
3.08005 + 2.48479I −2.33160 − 3.10830I
u = −0.570709 − 0.131663I
3.08005 − 2.48479I −2.33160 + 3.10830I
u = 0.453510
−1.01886 −9.58030
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
89
+ 43u
88
+ ··· + u − 1
c
2
, c
7
u
89
+ u
88
+ ··· + u + 1
c
3
u
89
− u
88
+ ··· + 1497u + 457
c
4
u
89
− u
88
+ ··· − 5u + 1
c
5
, c
10
, c
11
u
89
+ u
88
+ ··· + 3u + 1
c
6
, c
12
u
89
+ 7u
88
+ ··· + 161u + 5
c
8
u
89
+ 5u
88
+ ··· + u + 1
c
9
u
89
− 21u
88
+ ··· − 4023u + 187
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
89
+ 7y
88
+ ··· + 13y −1
c
2
, c
7
y
89
+ 43y
88
+ ··· + y −1
c
3
y
89
− 29y
88
+ ··· + 6081637y −208849
c
4
y
89
− 5y
88
+ ··· + 17y −1
c
5
, c
10
, c
11
y
89
+ 79y
88
+ ··· + y −1
c
6
, c
12
y
89
+ 67y
88
+ ··· + 201y −25
c
8
y
89
− y
88
+ ··· + 61y −1
c
9
y
89
+ 11y
88
+ ··· − 1504923y −34969
9
