12a
0731
(K12a
0731
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 12 11 2 7 4 1 6 5
Solving Sequence
1,5
12 6 11 7 10 4 9 3 2 8
c
12
c
5
c
11
c
6
c
10
c
4
c
9
c
3
c
1
c
8
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
52
− u
51
+ ··· − u
2
− 1i
* 1 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
52
− u
51
+ · · · − u
2
− 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
12
=
1
u
2
a
6
=
u
u
3
+ u
a
11
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
10
=
u
4
+ 3u
2
+ 1
u
4
+ 2u
2
a
4
=
u
9
+ 6u
7
+ 11u
5
+ 6u
3
+ u
u
9
+ 5u
7
+ 7u
5
+ 2u
3
+ u
a
9
=
−u
14
− 9u
12
− 30u
10
− 45u
8
− 30u
6
− 8u
4
+ 2u
2
+ 1
−u
14
− 8u
12
− 23u
10
− 28u
8
− 14u
6
− 4u
4
+ u
2
a
3
=
−u
19
− 12u
17
+ ··· + 11u
3
+ 2u
−u
19
− 11u
17
+ ··· + 3u
3
+ u
a
2
=
u
38
+ 23u
36
+ ··· + 2u
2
+ 1
u
38
+ 22u
36
+ ··· + 6u
4
+ u
2
a
8
=
u
22
+ 13u
20
+ ··· − 15u
4
+ 1
u
24
+ 14u
22
+ ··· − 30u
6
− 10u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
51
− 4u
50
+ ··· − 12u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
52
+ 19u
51
+ ··· − 2u + 1
c
2
, c
7
u
52
+ u
51
+ ··· − 2u − 1
c
3
, c
4
, c
9
u
52
− u
51
+ ··· − 8u − 4
c
5
, c
6
, c
11
c
12
u
52
− u
51
+ ··· − u
2
− 1
c
10
u
52
− 17u
51
+ ··· − 26192u + 2993
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
52
+ 29y
51
+ ··· + 18y + 1
c
2
, c
7
y
52
− 19y
51
+ ··· + 2y + 1
c
3
, c
4
, c
9
y
52
− 55y
51
+ ··· − 184y + 16
c
5
, c
6
, c
11
c
12
y
52
+ 61y
51
+ ··· + 2y + 1
c
10
y
52
− 31y
51
+ ··· − 170476614y + 8958049
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.417129 + 0.833683I
−6.57030 − 3.29491I −10.38217 + 1.00924I
u = 0.417129 − 0.833683I
−6.57030 + 3.29491I −10.38217 − 1.00924I
u = 0.447855 + 0.817464I
−10.54530 + 3.62754I −13.7524 − 4.1082I
u = 0.447855 − 0.817464I
−10.54530 − 3.62754I −13.7524 + 4.1082I
u = 0.468982 + 0.795969I
−6.17965 + 10.49310I −9.45880 − 8.85692I
u = 0.468982 − 0.795969I
−6.17965 − 10.49310I −9.45880 + 8.85692I
u = −0.415985 + 0.814109I
−4.96278 − 2.01233I −8.04647 + 3.92304I
u = −0.415985 − 0.814109I
−4.96278 + 2.01233I −8.04647 − 3.92304I
u = −0.456551 + 0.790170I
−4.67136 − 5.01095I −7.40961 + 4.34724I
u = −0.456551 − 0.790170I
−4.67136 + 5.01095I −7.40961 − 4.34724I
u = −0.307728 + 0.673902I
−3.04997 − 2.38050I −13.2356 + 6.5396I
u = −0.307728 − 0.673902I
−3.04997 + 2.38050I −13.2356 − 6.5396I
u = −0.431177 + 0.595940I
1.33174 − 6.67291I −4.55025 + 10.19057I
u = −0.431177 − 0.595940I
1.33174 + 6.67291I −4.55025 − 10.19057I
u = −0.075469 + 0.725881I
−0.97753 + 2.21512I −11.08908 − 2.69956I
u = −0.075469 − 0.725881I
−0.97753 − 2.21512I −11.08908 + 2.69956I
u = 0.417591 + 0.557999I
1.94823 + 1.38800I −2.50449 − 4.64036I
u = 0.417591 − 0.557999I
1.94823 − 1.38800I −2.50449 + 4.64036I
u = 0.636854
−8.09348 −9.74010
u = 0.633274 + 0.038957I
−3.92657 − 6.79204I −5.72913 + 4.81858I
u = 0.633274 − 0.038957I
−3.92657 + 6.79204I −5.72913 − 4.81858I
u = −0.614460 + 0.031464I
−2.42291 + 1.41178I −3.50849 − 0.14454I
u = −0.614460 − 0.031464I
−2.42291 − 1.41178I −3.50849 + 0.14454I
u = 0.241452 + 0.491984I
−0.158403 + 0.970828I −3.16514 − 6.89693I
u = 0.241452 − 0.491984I
−0.158403 − 0.970828I −3.16514 + 6.89693I
u = 0.435224 + 0.311244I
2.64348 + 1.66592I 0.51705 − 3.90838I
u = 0.435224 − 0.311244I
2.64348 − 1.66592I 0.51705 + 3.90838I
u = −0.459005 + 0.259005I
2.28374 + 3.52243I −0.68639 − 3.07419I
u = −0.459005 − 0.259005I
2.28374 − 3.52243I −0.68639 + 3.07419I
u = 0.01062 + 1.50422I
−3.13178 + 2.70355I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.01062 − 1.50422I
−3.13178 − 2.70355I 0
u = 0.09297 + 1.55850I
−5.18614 + 3.13538I 0
u = 0.09297 − 1.55850I
−5.18614 − 3.13538I 0
u = −0.10423 + 1.56695I
−5.96476 − 8.55636I 0
u = −0.10423 − 1.56695I
−5.96476 + 8.55636I 0
u = 0.04555 + 1.57078I
−7.31168 + 1.86434I 0
u = 0.04555 − 1.57078I
−7.31168 − 1.86434I 0
u = −0.07403 + 1.59817I
−10.82600 − 3.74121I 0
u = −0.07403 − 1.59817I
−10.82600 + 3.74121I 0
u = −0.02520 + 1.60559I
−8.95532 + 1.81146I 0
u = −0.02520 − 1.60559I
−8.95532 − 1.81146I 0
u = −0.386669
−1.22216 −7.16550
u = −0.13033 + 1.63602I
−12.9782 − 7.2412I 0
u = −0.13033 − 1.63602I
−12.9782 + 7.2412I 0
u = 0.13423 + 1.63834I
−14.5123 + 12.7892I 0
u = 0.13423 − 1.63834I
−14.5123 − 12.7892I 0
u = −0.11630 + 1.64091I
−13.39160 − 4.03746I 0
u = −0.11630 − 1.64091I
−13.39160 + 4.03746I 0
u = 0.12590 + 1.64441I
−18.9946 + 5.8156I 0
u = 0.12590 − 1.64441I
−18.9946 − 5.8156I 0
u = 0.11458 + 1.64704I
−15.1010 − 1.2710I 0
u = 0.11458 − 1.64704I
−15.1010 + 1.2710I 0
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
u
52
+ 19u
51
+ ··· − 2u + 1
c
2
, c
7
u
52
+ u
51
+ ··· − 2u − 1
c
3
, c
4
, c
9
u
52
− u
51
+ ··· − 8u − 4
c
5
, c
6
, c
11
c
12
u
52
− u
51
+ ··· − u
2
− 1
c
10
u
52
− 17u
51
+ ··· − 26192u + 2993
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
52
+ 29y
51
+ ··· + 18y + 1
c
2
, c
7
y
52
− 19y
51
+ ··· + 2y + 1
c
3
, c
4
, c
9
y
52
− 55y
51
+ ··· − 184y + 16
c
5
, c
6
, c
11
c
12
y
52
+ 61y
51
+ ··· + 2y + 1
c
10
y
52
− 31y
51
+ ··· − 170476614y + 8958049
8
