11n
61
(K11n
61
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 10 1 4 11 7 6
Solving Sequence
6,10
7 11
1,4
2 3 9 5 8
c
6
c
10
c
11
c
1
c
2
c
9
c
5
c
8
c
3
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
13
− u
12
+ 2u
11
+ 3u
10
− 2u
9
− 4u
8
+ 4u
6
+ 2u
5
− 2u
4
− u
3
+ 2u
2
+ b + u,
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
− 4u
4
− u
3
+ 2u
2
+ a − 2,
u
14
+ 2u
13
− u
12
− 6u
11
− 2u
10
+ 8u
9
+ 7u
8
− 6u
7
− 10u
6
+ 6u
4
− 4u
2
− u + 1i
I
u
2
= hb + 1, u
4
− u
2
+ a + u, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
* 2 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−u
13
− u
12
+ · · · + b + u, u
11
+ u
10
+ · · · + a − 2, u
14
+ 2u
13
+ · · · − u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
11
=
u
−u
3
+ u
a
1
=
u
3
−u
3
+ u
a
4
=
−u
11
− u
10
+ 2u
9
+ 3u
8
− 2u
7
− 4u
6
+ 4u
4
+ u
3
− 2u
2
+ 2
u
13
+ u
12
− 2u
11
− 3u
10
+ 2u
9
+ 4u
8
− 4u
6
− 2u
5
+ 2u
4
+ u
3
− 2u
2
− u
a
2
=
−u
13
− u
12
+ ··· + u − 2
−u
13
− u
12
+ 2u
11
+ 4u
10
− 2u
9
− 6u
8
+ 7u
6
+ 2u
5
− 4u
4
− u
3
+ 3u
2
+ u
a
3
=
−2u
13
− 2u
12
+ ··· + 3u − 3
−u
13
− u
12
+ ··· − u
3
+ 5u
2
a
9
=
−u
3
u
5
− u
3
+ u
a
5
=
u
8
− u
6
+ u
4
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
8
=
u
8
− u
6
+ u
4
+ 1
−u
8
+ 2u
6
− 2u
4
a
8
=
u
8
− u
6
+ u
4
+ 1
−u
8
+ 2u
6
− 2u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes =
8u
13
+10u
12
−16u
11
−35u
10
+13u
9
+55u
8
+10u
7
−57u
6
−34u
5
+28u
4
+22u
3
−19u
2
−16u−2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
− 7u
13
+ ··· + 4u − 1
c
2
u
14
+ 29u
13
+ ··· + 2u + 1
c
3
, c
8
u
14
− u
13
+ ··· − 64u − 64
c
5
, c
7
u
14
+ 2u
13
+ ··· + 3u + 1
c
6
, c
10
u
14
− 2u
13
+ ··· + u + 1
c
9
u
14
− 6u
13
+ ··· − 9u + 1
c
11
u
14
− 6u
13
+ ··· − u − 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
14
− 29y
13
+ ··· − 2y + 1
c
2
y
14
− 129y
13
+ ··· + 462y + 1
c
3
, c
8
y
14
− 39y
13
+ ··· + 8192y + 4096
c
5
, c
7
y
14
− 30y
13
+ ··· − 9y + 1
c
6
, c
10
y
14
− 6y
13
+ ··· − 9y + 1
c
9
y
14
+ 6y
13
+ ··· − 25y + 1
c
11
y
14
− 6y
13
+ ··· − 301y + 25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.959410 + 0.328783I
a = 0.495533 − 0.463828I
b = 0.191801 + 0.163474I
1.63965 − 1.19495I 1.59955 + 1.11588I
u = −0.959410 − 0.328783I
a = 0.495533 + 0.463828I
b = 0.191801 − 0.163474I
1.63965 + 1.19495I 1.59955 − 1.11588I
u = −0.501889 + 0.920209I
a = −0.201970 + 0.008787I
b = 2.28288 − 0.17435I
19.1238 + 2.3664I −10.04321 − 0.09569I
u = −0.501889 − 0.920209I
a = −0.201970 − 0.008787I
b = 2.28288 + 0.17435I
19.1238 − 2.3664I −10.04321 + 0.09569I
u = −0.853744 + 0.641916I
a = −0.29441 + 1.45158I
b = −1.59669 − 0.17157I
−3.47956 − 2.50408I −8.95669 + 2.99860I
u = −0.853744 − 0.641916I
a = −0.29441 − 1.45158I
b = −1.59669 + 0.17157I
−3.47956 + 2.50408I −8.95669 − 2.99860I
u = 1.014210 + 0.562829I
a = −0.229267 − 0.800962I
b = 0.036725 + 0.627532I
−0.01563 + 4.65799I −4.40917 − 5.70687I
u = 1.014210 − 0.562829I
a = −0.229267 + 0.800962I
b = 0.036725 − 0.627532I
−0.01563 − 4.65799I −4.40917 + 5.70687I
u = 0.589347 + 0.525928I
a = 0.836757 + 0.496215I
b = −0.355616 − 0.529402I
−1.309150 − 0.137583I −8.56031 + 0.56305I
u = 0.589347 − 0.525928I
a = 0.836757 − 0.496215I
b = −0.355616 + 0.529402I
−1.309150 + 0.137583I −8.56031 − 0.56305I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.25934
a = −2.87186
b = 2.12501
−13.7717 −5.12960
u = −1.128420 + 0.686699I
a = −1.08197 − 2.42300I
b = 2.21915 + 0.28216I
−18.4364 − 8.2751I −7.93412 + 4.24282I
u = −1.128420 − 0.686699I
a = −1.08197 + 2.42300I
b = 2.21915 − 0.28216I
−18.4364 + 8.2751I −7.93412 − 4.24282I
u = 0.420479
a = 1.82253
b = −0.681509
−1.01289 −10.2630
6
II. I
u
2
= hb + 1, u
4
− u
2
+ a + u, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
11
=
u
−u
3
+ u
a
1
=
u
3
−u
3
+ u
a
4
=
−u
4
+ u
2
− u
−1
a
2
=
−u
4
+ u
3
+ u
2
− u
−u
3
+ u − 1
a
3
=
−u
4
+ u
2
− u
−1
a
9
=
−u
3
u
5
− u
3
+ u
a
5
=
−u
3
u
3
− u
a
8
=
−u
3
u
5
− u
3
+ u
a
8
=
−u
3
u
5
− u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
+ 3u
2
− 3u − 9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
8
u
6
c
5
, c
7
, c
10
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
6
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
9
, c
11
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
6
, c
7
c
10
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
9
, c
11
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.002190 + 0.295542I
a = 1.42918 + 0.19856I
b = −1.00000
0.245672 − 0.924305I −5.20252 + 1.68215I
u = −1.002190 − 0.295542I
a = 1.42918 − 0.19856I
b = −1.00000
0.245672 + 0.924305I −5.20252 − 1.68215I
u = 0.428243 + 0.664531I
a = −0.429179 + 0.198557I
b = −1.00000
−3.53554 − 0.92430I −10.03026 + 0.88960I
u = 0.428243 − 0.664531I
a = −0.429179 − 0.198557I
b = −1.00000
−3.53554 + 0.92430I −10.03026 − 0.88960I
u = 1.073950 + 0.558752I
a = 0.50000 − 1.37764I
b = −1.00000
−1.64493 + 5.69302I −6.76721 − 6.15196I
u = 1.073950 − 0.558752I
a = 0.50000 + 1.37764I
b = −1.00000
−1.64493 − 5.69302I −6.76721 + 6.15196I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
14
− 7u
13
+ ··· + 4u − 1)
c
2
((u + 1)
6
)(u
14
+ 29u
13
+ ··· + 2u + 1)
c
3
, c
8
u
6
(u
14
− u
13
+ ··· − 64u − 64)
c
4
((u + 1)
6
)(u
14
− 7u
13
+ ··· + 4u − 1)
c
5
, c
7
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
6
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
14
− 2u
13
+ ··· + u + 1)
c
9
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)(u
14
− 6u
13
+ ··· − 9u + 1)
c
10
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
14
− 2u
13
+ ··· + u + 1)
c
11
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)(u
14
− 6u
13
+ ··· − u − 5)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
14
− 29y
13
+ ··· − 2y + 1)
c
2
((y −1)
6
)(y
14
− 129y
13
+ ··· + 462y + 1)
c
3
, c
8
y
6
(y
14
− 39y
13
+ ··· + 8192y + 4096)
c
5
, c
7
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
14
− 30y
13
+ ··· − 9y + 1)
c
6
, c
10
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
14
− 6y
13
+ ··· − 9y + 1)
c
9
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
14
+ 6y
13
+ ··· − 25y + 1)
c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
14
− 6y
13
+ ··· − 301y + 25)
12
