12n
0010
(K12n
0010
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 9 11 4 12 8 9 10
Solving Sequence
5,8
4
9,11
12 7 6 3 2 1 10
c
4
c
8
c
11
c
7
c
6
c
3
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h373570485293358u
28
+ 1033333880231479u
27
+ ··· + 1737205994835979b − 46103598198659,
− 452869533551973u
28
− 703336104853671u
27
+ ··· + 1737205994835979a − 136865573763131,
u
29
+ 2u
28
+ ··· − u − 1i
I
u
2
= h−u
3
+ b − u − 1, a, u
4
+ u
2
+ u + 1i
I
u
3
= h−u
5
+ u
4
− 2u
3
+ 2u
2
+ b − 2u + 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
= h3.74×10
14
u
28
+1.03×10
15
u
27
+· · ·+1.74×10
15
b−4.61×10
13
, −4.53×
10
14
u
28
−7.03×10
14
u
27
+· · ·+1.74×10
15
a−1.37×10
14
, u
29
+2u
28
+· · ·−u−1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0.260688u
28
+ 0.404866u
27
+ ··· + 1.65378u + 0.0787849
−0.215041u
28
− 0.594825u
27
+ ··· + 2.17523u + 0.0265389
a
12
=
u
−0.574992u
28
− 1.22877u
27
+ ··· + 1.37727u + 0.0642647
a
7
=
0.281422u
28
+ 0.665341u
27
+ ··· + 0.425718u − 0.328077
−0.300941u
28
− 0.245733u
27
+ ··· + 0.372268u − 0.413702
a
6
=
0.0992626u
28
+ 0.229078u
27
+ ··· + 0.144178u − 0.116511
−0.422114u
28
− 0.580655u
27
+ ··· − 0.163376u − 0.274080
a
3
=
0.0599722u
28
+ 0.00259463u
27
+ ··· − 0.0401153u + 1.59959
0.138580u
28
+ 0.0401668u
27
+ ··· − 0.139073u + 1.22534
a
2
=
−0.0786083u
28
− 0.0375722u
27
+ ··· + 0.0989573u + 0.374257
0.138580u
28
+ 0.0401668u
27
+ ··· − 0.139073u + 1.22534
a
1
=
0.493107u
28
+ 0.774647u
27
+ ··· + 0.437369u + 0.188122
0.393844u
28
+ 0.545569u
27
+ ··· + 0.293191u + 0.304633
a
10
=
0.260688u
28
+ 0.404866u
27
+ ··· + 1.65378u + 0.0787849
−0.314304u
28
− 0.823903u
27
+ ··· + 2.03105u + 0.143050
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
7883435024310839
1737205994835979
u
28
−
14219216887312602
1737205994835979
u
27
+ ···+
21942781332157203
1737205994835979
u +
8811379112410107
1737205994835979
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
29
+ 12u
28
+ ··· + u − 1
c
2
, c
5
u
29
+ 2u
28
+ ··· + u − 1
c
3
u
29
− 2u
28
+ ··· + 120u − 36
c
4
, c
8
u
29
− 2u
28
+ ··· − u + 1
c
6
u
29
− 10u
28
+ ··· − 469083u + 52489
c
7
, c
10
u
29
− 5u
28
+ ··· − 3072u − 1024
c
9
, c
11
, c
12
u
29
+ 11u
28
+ ··· − 30u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
29
+ 12y
28
+ ··· + 85y −1
c
2
, c
5
y
29
+ 12y
28
+ ··· + y −1
c
3
y
29
+ 12y
28
+ ··· − 12456y −1296
c
4
, c
8
y
29
+ 30y
27
+ ··· + y −1
c
6
y
29
+ 92y
28
+ ··· − 44517246691y −2755095121
c
7
, c
10
y
29
+ 63y
28
+ ··· + 3670016y −1048576
c
9
, c
11
, c
12
y
29
− 51y
28
+ ··· − 60y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.269747 + 0.997117I
a = −0.080093 + 0.634791I
b = −0.459723 − 0.451799I
−3.76543 − 0.40137I −7.74784 + 0.84155I
u = 0.269747 − 0.997117I
a = −0.080093 − 0.634791I
b = −0.459723 + 0.451799I
−3.76543 + 0.40137I −7.74784 − 0.84155I
u = −0.520591 + 0.795499I
a = 0.040524 + 0.941049I
b = 1.201380 + 0.062565I
0.03144 − 1.92773I 1.38434 + 3.11728I
u = −0.520591 − 0.795499I
a = 0.040524 − 0.941049I
b = 1.201380 − 0.062565I
0.03144 + 1.92773I 1.38434 − 3.11728I
u = −0.896581 + 0.315451I
a = −1.90204 + 1.76626I
b = −0.36922 + 3.53096I
4.34061 − 5.06790I 7.76076 + 6.40955I
u = −0.896581 − 0.315451I
a = −1.90204 − 1.76626I
b = −0.36922 − 3.53096I
4.34061 + 5.06790I 7.76076 − 6.40955I
u = 0.888576 + 0.223844I
a = 2.22962 + 1.30457I
b = 1.34311 + 2.77938I
4.83420 − 0.14491I 9.33140 − 0.02437I
u = 0.888576 − 0.223844I
a = 2.22962 − 1.30457I
b = 1.34311 − 2.77938I
4.83420 + 0.14491I 9.33140 + 0.02437I
u = 0.586489 + 0.957504I
a = −0.302708 + 0.858093I
b = −1.34326 − 0.63754I
−1.89724 + 6.62062I −1.77451 − 5.81131I
u = 0.586489 − 0.957504I
a = −0.302708 − 0.858093I
b = −1.34326 + 0.63754I
−1.89724 − 6.62062I −1.77451 + 5.81131I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.538341 + 0.454787I
a = −0.661454 + 0.909012I
b = 0.67192 + 1.39992I
0.49621 − 1.44403I 2.32169 + 4.35214I
u = −0.538341 − 0.454787I
a = −0.661454 − 0.909012I
b = 0.67192 − 1.39992I
0.49621 + 1.44403I 2.32169 − 4.35214I
u = −0.454661 + 0.440048I
a = −0.603679 + 0.541804I
b = −0.128589 + 1.154160I
0.61760 − 1.38123I 3.89803 + 4.67424I
u = −0.454661 − 0.440048I
a = −0.603679 − 0.541804I
b = −0.128589 − 1.154160I
0.61760 + 1.38123I 3.89803 − 4.67424I
u = 0.571853 + 0.250420I
a = 0.590811 + 0.162782I
b = 0.832609 + 0.563271I
−0.23394 − 2.60554I 1.51395 + 2.58658I
u = 0.571853 − 0.250420I
a = 0.590811 − 0.162782I
b = 0.832609 − 0.563271I
−0.23394 + 2.60554I 1.51395 − 2.58658I
u = 0.600458
a = 1.34216
b = −0.231076
2.41354 4.07200
u = −0.092427 + 0.519848I
a = −0.242741 + 1.091430I
b = 0.12625 + 2.93959I
2.03975 + 2.27000I −9.41057 + 5.26980I
u = −0.092427 − 0.519848I
a = −0.242741 − 1.091430I
b = 0.12625 − 2.93959I
2.03975 − 2.27000I −9.41057 − 5.26980I
u = −1.10611 + 1.07911I
a = 0.66217 − 1.55771I
b = −4.13956 − 3.94336I
15.6166 − 12.3453I 4.17843 + 6.31172I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10611 − 1.07911I
a = 0.66217 + 1.55771I
b = −4.13956 + 3.94336I
15.6166 + 12.3453I 4.17843 − 6.31172I
u = 1.10589 + 1.08681I
a = −0.81738 − 1.52507I
b = 3.91452 − 4.33465I
17.5116 + 6.4339I 6.30123 − 2.11610I
u = 1.10589 − 1.08681I
a = −0.81738 + 1.52507I
b = 3.91452 + 4.33465I
17.5116 − 6.4339I 6.30123 + 2.11610I
u = 1.10403 + 1.10245I
a = −1.17352 − 1.22596I
b = 2.76455 − 5.02114I
17.4749 + 1.6883I 6.38533 − 1.83848I
u = 1.10403 − 1.10245I
a = −1.17352 + 1.22596I
b = 2.76455 + 5.02114I
17.4749 − 1.6883I 6.38533 + 1.83848I
u = −1.10116 + 1.10760I
a = 1.23857 − 1.06154I
b = −2.25205 − 5.03203I
15.5469 + 4.2345I 4.31422 − 2.32649I
u = −1.10116 − 1.10760I
a = 1.23857 + 1.06154I
b = −2.25205 + 5.03203I
15.5469 − 4.2345I 4.31422 + 2.32649I
u = −1.11694 + 1.09740I
a = 0.85084 − 1.21443I
b = −3.04641 − 4.13857I
10.89410 − 4.09225I 1.50754 + 2.07565I
u = −1.11694 − 1.09740I
a = 0.85084 + 1.21443I
b = −3.04641 + 4.13857I
10.89410 + 4.09225I 1.50754 − 2.07565I
7
II. I
u
2
= h−u
3
+ b − u − 1, a, u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
u
3
+ u + 1
a
12
=
−u
1
a
7
=
0
u
a
6
=
u
3
−u
2
a
3
=
u
3
+ u
2
+ 1
−u
a
2
=
u
3
+ u
2
+ u + 1
−u
a
1
=
−u
−u
3
− u
a
10
=
0
u
3
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
− 5u
2
− u + 3
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
4
− 2u
3
+ 3u
2
− u + 1
c
2
, c
4
u
4
+ u
2
+ u + 1
c
3
u
4
+ 3u
3
+ 4u
2
+ 3u + 2
c
5
, c
8
u
4
+ u
2
− u + 1
c
7
, c
10
u
4
c
9
(u + 1)
4
c
11
, c
12
(u − 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
2
, c
4
, c
5
c
8
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
3
y
4
− y
3
+ 2y
2
+ 7y + 4
c
7
, c
10
y
4
c
9
, c
11
, c
12
(y −1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = 0
b = 0.851808 + 0.911292I
2.62503 − 1.39709I 4.96170 + 3.59727I
u = −0.547424 − 0.585652I
a = 0
b = 0.851808 − 0.911292I
2.62503 + 1.39709I 4.96170 − 3.59727I
u = 0.547424 + 1.120870I
a = 0
b = −0.351808 + 0.720342I
−0.98010 + 7.64338I 1.53830 − 8.45840I
u = 0.547424 − 1.120870I
a = 0
b = −0.351808 − 0.720342I
−0.98010 − 7.64338I 1.53830 + 8.45840I
11
III.
I
u
3
= h−u
5
+ u
4
− 2u
3
+ 2u
2
+ b − 2u + 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
u
5
− u
4
+ 2u
3
− 2u
2
+ 2u − 2
a
12
=
−u
u
5
− u
4
+ u
3
− 2u
2
+ u − 2
a
7
=
0
u
a
6
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
−u
5
− 2u
3
+ u
2
− u + 1
a
2
=
u
4
+ u
2
− u + 1
−u
5
− 2u
3
+ u
2
− u + 1
a
1
=
−u
−u
3
− u
a
10
=
0
u
5
− u
4
+ 2u
3
− 2u
2
+ 2u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
5
+ u
4
+ u
2
+ u + 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
c
2
, c
4
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
3
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
7
, c
10
u
6
c
9
(u + 1)
6
c
11
, c
12
(u − 1)
6
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
2
, c
4
, c
5
c
8
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
3
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
y
6
c
9
, c
11
, c
12
(y −1)
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = 0
b = 0.398606 + 0.800120I
1.37919 − 2.82812I 4.90478 + 3.87141I
u = −0.498832 − 1.001300I
a = 0
b = 0.398606 − 0.800120I
1.37919 + 2.82812I 4.90478 − 3.87141I
u = 0.284920 + 1.115140I
a = 0
b = −0.215080 + 0.841795I
−2.75839 0.235367 − 0.997558I
u = 0.284920 − 1.115140I
a = 0
b = −0.215080 − 0.841795I
−2.75839 0.235367 + 0.997558I
u = 0.713912 + 0.305839I
a = 0
b = −1.183530 + 0.507021I
1.37919 − 2.82812I 5.35985 + 0.59776I
u = 0.713912 − 0.305839I
a = 0
b = −1.183530 − 0.507021I
1.37919 + 2.82812I 5.35985 − 0.59776I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
29
+ 12u
28
+ ··· + u − 1)
c
2
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
29
+ 2u
28
+ ··· + u − 1)
c
3
((u
3
− u
2
+ 1)
2
)(u
4
+ 3u
3
+ ··· + 3u + 2)(u
29
− 2u
28
+ ··· + 120u − 36)
c
4
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
29
− 2u
28
+ ··· − u + 1)
c
5
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
29
+ 2u
28
+ ··· + u − 1)
c
6
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
29
− 10u
28
+ ··· − 469083u + 52489)
c
7
, c
10
u
10
(u
29
− 5u
28
+ ··· − 3072u − 1024)
c
8
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
29
− 2u
28
+ ··· − u + 1)
c
9
((u + 1)
10
)(u
29
+ 11u
28
+ ··· − 30u
2
− 1)
c
11
, c
12
((u − 1)
10
)(u
29
+ 11u
28
+ ··· − 30u
2
− 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
29
+ 12y
28
+ ··· + 85y −1)
c
2
, c
5
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
29
+ 12y
28
+ ··· + y −1)
c
3
(y
3
− y
2
+ 2y −1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
29
+ 12y
28
+ ··· − 12456y −1296)
c
4
, c
8
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
29
+ 30y
27
+ ··· + y −1)
c
6
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
29
+ 92y
28
+ ··· − 44517246691y −2755095121)
c
7
, c
10
y
10
(y
29
+ 63y
28
+ ··· + 3670016y −1048576)
c
9
, c
11
, c
12
((y −1)
10
)(y
29
− 51y
28
+ ··· − 60y −1)
17
