11n
12
(K11n
12
)
A knot diagram
1
Linearized knot diagam
5 1 8 2 3 10 3 1 11 7 9
Solving Sequence
1,8 4,9
3 2 5 6 7 11 10
c
8
c
3
c
2
c
4
c
5
c
7
c
11
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
11
− 85u
10
+ ··· + 8858b − 4180, −5021u
11
+ 9565u
10
+ ··· + 17716a + 23565,
u
12
− u
11
+ 12u
10
− 11u
9
+ 47u
8
− 37u
7
+ 56u
6
− 30u
5
− 12u
4
+ 12u
3
− u + 1i
I
u
2
= hb, u
2
a + a
2
− au + 2u
2
+ 2a − u + 3, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 18 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−4u
11
− 85u
10
+ · · · + 8858b − 4180, −5021u
11
+ 9565u
10
+ · · · +
17716a + 23565, u
12
− u
11
+ · · · − u + 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
0.283416u
11
− 0.539907u
10
+ ··· + 0.840596u − 1.33015
0.000451569u
11
+ 0.00959585u
10
+ ··· + 0.903251u + 0.471890
a
9
=
1
u
2
a
3
=
0.282965u
11
− 0.549503u
10
+ ··· − 0.0626552u − 1.80204
0.000451569u
11
+ 0.00959585u
10
+ ··· + 0.903251u + 0.471890
a
2
=
0.282965u
11
− 0.549503u
10
+ ··· − 0.0626552u − 1.80204
0.0559381u
11
+ 0.00118537u
10
+ ··· + 1.45275u + 0.205351
a
5
=
−0.658106u
11
+ 0.452755u
10
+ ··· − 1.93836u − 0.720422
u
a
6
=
0.00203206u
11
− 0.206819u
10
+ ··· − 2.18537u + 1.12350
0.207496u
11
− 0.340709u
10
+ ··· − 1.20603u + 0.333371
a
7
=
−0.128584u
11
+ 0.142583u
10
+ ··· + 1.17419u − 0.870625
−0.204787u
11
+ 0.398284u
10
+ ··· + 1.12554u − 0.00203206
a
11
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
10
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
56005
17716
u
11
−
11673
4429
u
10
+
166280
4429
u
9
−
127032
4429
u
8
+
2558331
17716
u
7
−
418885
4429
u
6
+
726025
4429
u
5
−
1302141
17716
u
4
−
815229
17716
u
3
+
405337
17716
u
2
+
46603
8858
u −
28579
8858
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
12
+ 4u
11
+ ··· + 2u + 1
c
2
u
12
+ 14u
10
+ ··· + 10u + 1
c
3
, c
7
u
12
+ u
11
+ ··· + 288u + 64
c
5
u
12
− 4u
11
+ ··· + 532u + 193
c
6
, c
10
u
12
+ 3u
11
+ 4u
10
+ u
9
+ u
8
+ 7u
7
+ 12u
6
+ 6u
5
+ u + 1
c
8
, c
9
, c
11
u
12
+ u
11
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
12
+ 14y
10
+ ··· + 10y + 1
c
2
y
12
+ 28y
11
+ ··· − 66y + 1
c
3
, c
7
y
12
− 35y
11
+ ··· − 9216y + 4096
c
5
y
12
+ 56y
11
+ ··· + 602074y + 37249
c
6
, c
10
y
12
− y
11
+ ··· − y + 1
c
8
, c
9
, c
11
y
12
+ 23y
11
+ ··· − y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.625204 + 0.231089I
a = 0.161222 + 0.490070I
b = 0.412005 + 0.431173I
−1.40636 − 0.34980I −7.54487 + 0.48017I
u = 0.625204 − 0.231089I
a = 0.161222 − 0.490070I
b = 0.412005 − 0.431173I
−1.40636 + 0.34980I −7.54487 − 0.48017I
u = −0.449650 + 0.155107I
a = −1.84569 + 2.79630I
b = −0.674003 + 1.032060I
1.31906 − 1.56861I 1.73907 + 2.71444I
u = −0.449650 − 0.155107I
a = −1.84569 − 2.79630I
b = −0.674003 − 1.032060I
1.31906 + 1.56861I 1.73907 − 2.71444I
u = 0.170188 + 0.372008I
a = −1.72486 + 0.94221I
b = 0.737368 − 0.073970I
−0.55164 − 2.71818I −0.33339 + 6.77292I
u = 0.170188 − 0.372008I
a = −1.72486 − 0.94221I
b = 0.737368 + 0.073970I
−0.55164 + 2.71818I −0.33339 − 6.77292I
u = 0.18845 + 1.62161I
a = 0.231111 + 0.902812I
b = 0.359686 + 1.355750I
4.35182 − 3.22757I 0.42641 + 2.31513I
u = 0.18845 − 1.62161I
a = 0.231111 − 0.902812I
b = 0.359686 − 1.355750I
4.35182 + 3.22757I 0.42641 − 2.31513I
u = −0.19909 + 2.16177I
a = −2.12976 − 0.49437I
b = −3.33668 − 0.28827I
−18.9549 + 0.4085I 0.320898 + 0.107074I
u = −0.19909 − 2.16177I
a = −2.12976 + 0.49437I
b = −3.33668 + 0.28827I
−18.9549 − 0.4085I 0.320898 − 0.107074I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.16491 + 2.16925I
a = 1.80797 − 0.87122I
b = 3.00162 − 0.87325I
−19.3015 − 8.0703I −0.10811 + 3.87488I
u = 0.16491 − 2.16925I
a = 1.80797 + 0.87122I
b = 3.00162 + 0.87325I
−19.3015 + 8.0703I −0.10811 − 3.87488I
6
II. I
u
2
= hb, u
2
a + a
2
− au + 2u
2
+ 2a − u + 3, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
a
0
a
9
=
1
u
2
a
3
=
a
0
a
2
=
a
−u
2
a
a
5
=
u
2
+ a − u + 2
−u
a
6
=
0
−u
a
7
=
1
0
a
11
=
u
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
a
10
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3au − 2u
2
+ a + 3u − 7
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
7
u
6
c
4
(u
2
− u + 1)
3
c
6
(u
3
+ u
2
− 1)
2
c
8
, c
9
(u
3
− u
2
+ 2u − 1)
2
c
10
(u
3
− u
2
+ 1)
2
c
11
(u
3
+ u
2
+ 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
7
y
6
c
6
, c
10
(y
3
− y
2
+ 2y −1)
2
c
8
, c
9
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.706350 + 0.266290I
b = 0
3.02413 − 4.85801I −2.23639 + 5.66123I
u = 0.215080 + 1.307140I
a = 0.583789 + 0.478572I
b = 0
3.02413 − 0.79824I −0.946254 + 0.677361I
u = 0.215080 − 1.307140I
a = −0.706350 − 0.266290I
b = 0
3.02413 + 4.85801I −2.23639 − 5.66123I
u = 0.215080 − 1.307140I
a = 0.583789 − 0.478572I
b = 0
3.02413 + 0.79824I −0.946254 − 0.677361I
u = 0.569840
a = −0.87744 + 1.51977I
b = 0
−1.11345 + 2.02988I −5.31735 − 1.07831I
u = 0.569840
a = −0.87744 − 1.51977I
b = 0
−1.11345 − 2.02988I −5.31735 + 1.07831I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
12
+ 4u
11
+ ··· + 2u + 1)
c
2
((u
2
+ u + 1)
3
)(u
12
+ 14u
10
+ ··· + 10u + 1)
c
3
, c
7
u
6
(u
12
+ u
11
+ ··· + 288u + 64)
c
4
((u
2
− u + 1)
3
)(u
12
+ 4u
11
+ ··· + 2u + 1)
c
5
((u
2
+ u + 1)
3
)(u
12
− 4u
11
+ ··· + 532u + 193)
c
6
(u
3
+ u
2
− 1)
2
(u
12
+ 3u
11
+ 4u
10
+ u
9
+ u
8
+ 7u
7
+ 12u
6
+ 6u
5
+ u + 1)
c
8
, c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
12
+ u
11
+ ··· + u + 1)
c
10
(u
3
− u
2
+ 1)
2
(u
12
+ 3u
11
+ 4u
10
+ u
9
+ u
8
+ 7u
7
+ 12u
6
+ 6u
5
+ u + 1)
c
11
((u
3
+ u
2
+ 2u + 1)
2
)(u
12
+ u
11
+ ··· + u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
12
+ 14y
10
+ ··· + 10y + 1)
c
2
((y
2
+ y + 1)
3
)(y
12
+ 28y
11
+ ··· − 66y + 1)
c
3
, c
7
y
6
(y
12
− 35y
11
+ ··· − 9216y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
12
+ 56y
11
+ ··· + 602074y + 37249)
c
6
, c
10
((y
3
− y
2
+ 2y −1)
2
)(y
12
− y
11
+ ··· − y + 1)
c
8
, c
9
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
12
+ 23y
11
+ ··· − y + 1)
12
