12n
0474
(K12n
0474
)
A knot diagram
1
Linearized knot diagam
3 6 10 9 2 11 12 3 4 6 7 8
Solving Sequence
6,10
11 7
4,12
3 2 1 5 9 8
c
10
c
6
c
11
c
3
c
2
c
1
c
5
c
9
c
8
c
4
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−23u
13
+ 80u
12
+ ··· + 26b − 109, 31u
13
− 92u
12
+ ··· + 78a + 269,
u
14
− 2u
13
− 10u
12
+ 20u
11
+ 38u
10
− 77u
9
− 66u
8
+ 147u
7
+ 36u
6
− 140u
5
+ 27u
4
+ 57u
3
− 26u
2
− u + 3i
I
u
2
= hb, a − u + 1, u
2
− u − 1i
I
u
3
= hb + a + u + 1, a
2
+ 2au + 2a + u + 4, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
= h−23u
13
+ 80u
12
+ · · · + 26b − 109, 31u
13
− 92u
12
+ · · · + 78a +
269, u
14
− 2u
13
+ · · · − u + 3i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−u
−u
3
+ u
a
4
=
−0.397436u
13
+ 1.17949u
12
+ ··· − 0.858974u − 3.44872
0.884615u
13
− 3.07692u
12
+ ··· − 2.34615u + 4.19231
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
0.487179u
13
− 1.89744u
12
+ ··· − 3.20513u + 0.743590
0.884615u
13
− 3.07692u
12
+ ··· − 2.34615u + 4.19231
a
2
=
0.487179u
13
− 1.89744u
12
+ ··· − 3.20513u + 0.743590
0.346154u
13
− 0.769231u
12
+ ··· + 0.0384615u + 1.42308
a
1
=
−u
4
+ 3u
2
− 1
−u
6
+ 4u
4
− 3u
2
a
5
=
−1.10256u
13
+ 2.82051u
12
+ ··· + 4.35897u − 2.05128
0.576923u
13
− 0.615385u
12
+ ··· − 0.269231u − 0.961538
a
9
=
−0.474359u
13
+ 1.29487u
12
+ ··· − 1.08974u + 0.512821
−0.461538u
13
+ 1.69231u
12
+ ··· + 4.61538u − 1.23077
a
8
=
u
3
− 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
63
13
u
13
−
179
13
u
12
−
466
13
u
11
+ 122u
10
+
977
13
u
9
−
5058
13
u
8
+
190
13
u
7
+
7016
13
u
6
−
3311
13
u
5
−
2929
13
u
4
+
2999
13
u
3
−
571
13
u
2
+
20
13
u −
105
13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 25u
13
+ ··· + 2882u + 121
c
2
, c
5
u
14
+ 3u
13
+ ··· − 22u − 11
c
3
, c
4
, c
9
u
14
− u
13
+ ··· − 8u − 4
c
6
, c
7
, c
10
c
11
, c
12
u
14
− 2u
13
+ ··· − u + 3
c
8
u
14
+ u
13
+ ··· − 560u − 100
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 65y
13
+ ··· − 4823786y + 14641
c
2
, c
5
y
14
− 25y
13
+ ··· − 2882y + 121
c
3
, c
4
, c
9
y
14
+ 9y
13
+ ··· − 96y + 16
c
6
, c
7
, c
10
c
11
, c
12
y
14
− 24y
13
+ ··· − 157y + 9
c
8
y
14
− 51y
13
+ ··· − 79200y + 10000
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.099960 + 0.007934I
a = 0.373480 + 0.438522I
b = 0.255413 − 1.041350I
−1.42237 + 2.30354I −14.5680 − 3.7918I
u = −1.099960 − 0.007934I
a = 0.373480 − 0.438522I
b = 0.255413 + 1.041350I
−1.42237 − 2.30354I −14.5680 + 3.7918I
u = 0.698450 + 0.454350I
a = 0.59750 − 1.75057I
b = 0.525011 + 0.607663I
−3.03395 − 1.06008I −15.9574 + 4.7668I
u = 0.698450 − 0.454350I
a = 0.59750 + 1.75057I
b = 0.525011 − 0.607663I
−3.03395 + 1.06008I −15.9574 − 4.7668I
u = 0.374314 + 0.322623I
a = −1.30508 + 1.63203I
b = −0.001324 − 1.295280I
3.32563 − 1.11189I −8.52416 + 6.18288I
u = 0.374314 − 0.322623I
a = −1.30508 − 1.63203I
b = −0.001324 + 1.295280I
3.32563 + 1.11189I −8.52416 − 6.18288I
u = −1.45521 + 0.41134I
a = −0.39517 − 1.54570I
b = −0.742220 + 1.181770I
−10.11530 + 4.52944I −16.6083 − 3.1417I
u = −1.45521 − 0.41134I
a = −0.39517 + 1.54570I
b = −0.742220 − 1.181770I
−10.11530 − 4.52944I −16.6083 + 3.1417I
u = −0.298678
a = −0.459084
b = −0.324104
−0.507849 −19.4340
u = 1.73659 + 0.13204I
a = 0.081098 − 0.265527I
b = −0.786012 + 0.736054I
−11.60050 + 1.36524I −17.2554 − 2.6511I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73659 − 0.13204I
a = 0.081098 + 0.265527I
b = −0.786012 − 0.736054I
−11.60050 − 1.36524I −17.2554 + 2.6511I
u = 1.88375 + 0.14746I
a = 0.37986 − 1.43767I
b = 0.67581 + 1.59079I
17.0667 − 7.6448I −16.3310 + 2.7970I
u = 1.88375 − 0.14746I
a = 0.37986 + 1.43767I
b = 0.67581 − 1.59079I
17.0667 + 7.6448I −16.3310 − 2.7970I
u = −1.97719
a = −0.337623
b = 1.47076
12.0675 −18.0770
6
II. I
u
2
= hb, a − u + 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u + 1
a
7
=
−u
−u − 1
a
4
=
u − 1
0
a
12
=
−u
−u
a
3
=
u − 1
0
a
2
=
u − 1
−u
a
1
=
0
−u
a
5
=
u − 1
0
a
9
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −14
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
u
2
+ u − 1
c
10
, c
11
, c
12
u
2
− u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.61803
b = 0
−2.63189 −14.0000
u = 1.61803
a = 0.618034
b = 0
−10.5276 −14.0000
10
III. I
u
3
= hb + a + u + 1, a
2
+ 2au + 2a + u + 4, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u + 1
a
7
=
−u
−u + 1
a
4
=
a
−a − u − 1
a
12
=
u
u
a
3
=
−u − 1
−a − u − 1
a
2
=
−u − 1
−a − 1
a
1
=
0
u
a
5
=
u + 1
a + u + 1
a
9
=
−au − a − u − 3
2
a
8
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
2
c
6
, c
7
(u
2
− u − 1)
2
c
10
, c
11
, c
12
(u
2
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
8
c
9
(y + 2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.61803 + 1.41421I
b = − 1.414210I
2.30291 −16.0000
u = 0.618034
a = −1.61803 − 1.41421I
b = 1.414210I
2.30291 −16.0000
u = −1.61803
a = 0.61803 + 1.41421I
b = − 1.414210I
−5.59278 −16.0000
u = −1.61803
a = 0.61803 − 1.41421I
b = 1.414210I
−5.59278 −16.0000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
14
+ 25u
13
+ ··· + 2882u + 121)
c
2
((u − 1)
2
)(u + 1)
4
(u
14
+ 3u
13
+ ··· − 22u − 11)
c
3
, c
4
, c
9
u
2
(u
2
+ 2)
2
(u
14
− u
13
+ ··· − 8u − 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
14
+ 3u
13
+ ··· − 22u − 11)
c
6
, c
7
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
14
− 2u
13
+ ··· − u + 3)
c
8
u
2
(u
2
+ 2)
2
(u
14
+ u
13
+ ··· − 560u − 100)
c
10
, c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
14
− 2u
13
+ ··· − u + 3)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
14
− 65y
13
+ ··· − 4823786y + 14641)
c
2
, c
5
((y −1)
6
)(y
14
− 25y
13
+ ··· − 2882y + 121)
c
3
, c
4
, c
9
y
2
(y + 2)
4
(y
14
+ 9y
13
+ ··· − 96y + 16)
c
6
, c
7
, c
10
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
14
− 24y
13
+ ··· − 157y + 9)
c
8
y
2
(y + 2)
4
(y
14
− 51y
13
+ ··· − 79200y + 10000)
16
