11n
69
(K11n
69
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 10 9 11 4 6 1 7
Solving Sequence
7,11 4,8
9 1 3 2 5 6 10
c
7
c
8
c
11
c
3
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h3951u
19
+ 6529u
18
+ ··· + 2424b + 6325, −115811u
19
− 169485u
18
+ ··· + 79992a − 272713,
u
20
+ 2u
19
+ ··· + 2u + 1i
I
u
2
= hu
3
+ 2b − u + 1, −u
3
+ 2u
2
+ 2a − 3u + 1, u
4
− u
3
+ u
2
+ 1i
I
u
3
= hu
5
+ u
4
+ u
3
− u
2
+ b − 2u, u
4
− u
3
+ u
2
+ a − 3u + 2, u
6
+ 2u
4
− 3u
3
+ u
2
− 3u + 1i
* 3 irreducible components of dim
C
= 0, with total 30 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
= h3951u
19
+ 6529u
18
+ · · · + 2424b + 6325, −1.16 × 10
5
u
19
− 1.69 ×
10
5
u
18
+ · · · + 8.00 × 10
4
a − 2.73 × 10
5
, u
20
+ 2u
19
+ · · · + 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
1.44778u
19
+ 2.11877u
18
+ ··· + 7.14120u + 3.40925
−1.62995u
19
− 2.69348u
18
+ ··· + 1.21988u − 2.60932
a
8
=
1
−u
2
a
9
=
−0.490249u
19
− 0.377738u
18
+ ··· + 1.43684u − 0.710171
2.45375u
19
+ 3.57816u
18
+ ··· + 2.48335u + 4.64226
a
1
=
u
u
a
3
=
0.315419u
19
+ 0.424955u
18
+ ··· + 8.46688u + 1.57672
−1.38148u
19
− 2.28739u
18
+ ··· + 1.22934u − 2.03842
a
2
=
−0.349372u
19
− 0.474610u
18
+ ··· + 8.80089u + 0.895277
−2.04627u
19
− 3.18696u
18
+ ··· + 1.56334u − 2.71986
a
5
=
3.68497u
19
+ 5.12691u
18
+ ··· − 2.11341u + 4.48245
−0.354535u
19
− 0.633363u
18
+ ··· − 0.327633u − 1.82848
a
6
=
−4.03950u
19
− 5.76028u
18
+ ··· + 1.78578u − 5.31093
−1.32933u
19
− 1.83018u
18
+ ··· − 0.265227u − 1.45375
a
10
=
u
3
u
3
+ u
a
10
=
u
3
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
189289
17776
u
19
+
857557
53328
u
18
+ ··· +
286735
53328
u +
706201
53328
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
20
− 2u
19
+ ··· − 35u + 4
c
2
u
20
+ 22u
19
+ ··· + 353u + 16
c
3
, c
8
u
20
+ 2u
19
+ ··· + 112u + 64
c
5
, c
6
, c
9
u
20
− 2u
19
+ ··· − 2u + 1
c
7
, c
11
u
20
− 2u
19
+ ··· − 2u + 1
c
10
u
20
+ 14u
19
+ ··· + 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
20
− 22y
19
+ ··· − 353y + 16
c
2
y
20
− 46y
19
+ ··· + 223775y + 256
c
3
, c
8
y
20
− 18y
19
+ ··· + 45824y + 4096
c
5
, c
6
, c
9
y
20
+ 14y
19
+ ··· + 4y + 1
c
7
, c
11
y
20
+ 14y
19
+ ··· + 4y + 1
c
10
y
20
− 14y
19
+ ··· + 56y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.131630 + 0.044257I
a = 0.0439473 + 0.1087160I
b = 1.40407 + 0.37862I
−4.62345 − 6.11364I −7.31633 + 3.70196I
u = −1.131630 − 0.044257I
a = 0.0439473 − 0.1087160I
b = 1.40407 − 0.37862I
−4.62345 + 6.11364I −7.31633 − 3.70196I
u = −0.280810 + 0.786452I
a = −0.395074 + 0.246972I
b = −0.197007 + 0.388860I
−0.440396 − 1.279690I −4.66436 + 4.97948I
u = −0.280810 − 0.786452I
a = −0.395074 − 0.246972I
b = −0.197007 − 0.388860I
−0.440396 + 1.279690I −4.66436 − 4.97948I
u = 0.769131 + 0.907087I
a = 0.304381 − 0.204058I
b = −0.165796 + 0.163987I
5.68828 + 2.93127I 2.45037 − 0.45578I
u = 0.769131 − 0.907087I
a = 0.304381 + 0.204058I
b = −0.165796 − 0.163987I
5.68828 − 2.93127I 2.45037 + 0.45578I
u = 0.167664 + 1.190790I
a = 1.75974 + 0.44692I
b = 1.243080 − 0.457519I
−4.30761 + 1.95796I −13.39097 − 1.55059I
u = 0.167664 − 1.190790I
a = 1.75974 − 0.44692I
b = 1.243080 + 0.457519I
−4.30761 − 1.95796I −13.39097 + 1.55059I
u = 0.050177 + 1.253190I
a = 0.722975 + 0.985298I
b = 0.495740 − 0.417798I
−5.41434 + 1.81549I −13.14101 − 3.54833I
u = 0.050177 − 1.253190I
a = 0.722975 − 0.985298I
b = 0.495740 + 0.417798I
−5.41434 − 1.81549I −13.14101 + 3.54833I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.261687 + 1.231270I
a = −2.05611 + 0.13028I
b = −1.122580 − 0.533009I
−2.03730 − 5.64317I −8.76092 + 6.80873I
u = −0.261687 − 1.231270I
a = −2.05611 − 0.13028I
b = −1.122580 + 0.533009I
−2.03730 + 5.64317I −8.76092 − 6.80873I
u = −0.553404 + 0.030516I
a = −0.394310 + 0.673104I
b = −0.895721 − 0.664676I
1.73140 − 2.57417I −2.05095 + 4.12677I
u = −0.553404 − 0.030516I
a = −0.394310 − 0.673104I
b = −0.895721 + 0.664676I
1.73140 + 2.57417I −2.05095 − 4.12677I
u = −0.52166 + 1.39992I
a = 1.63987 − 0.44598I
b = 1.74672 + 0.69883I
−9.1950 − 11.9560I −9.10352 + 6.09824I
u = −0.52166 − 1.39992I
a = 1.63987 + 0.44598I
b = 1.74672 − 0.69883I
−9.1950 + 11.9560I −9.10352 − 6.09824I
u = 0.54264 + 1.40344I
a = −1.38809 − 0.74127I
b = −1.65276 + 0.37354I
−13.3225 + 5.9895I −12.03384 − 3.05262I
u = 0.54264 − 1.40344I
a = −1.38809 + 0.74127I
b = −1.65276 − 0.37354I
−13.3225 − 5.9895I −12.03384 + 3.05262I
u = 0.219579 + 0.305083I
a = −0.48733 + 3.67126I
b = 0.894252 + 0.888486I
−0.977750 + 0.984957I −3.11347 + 0.07087I
u = 0.219579 − 0.305083I
a = −0.48733 − 3.67126I
b = 0.894252 − 0.888486I
−0.977750 − 0.984957I −3.11347 − 0.07087I
6
II. I
u
2
= hu
3
+ 2b − u + 1, −u
3
+ 2u
2
+ 2a − 3u + 1, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
1
2
u
3
− u
2
+
3
2
u −
1
2
−
1
2
u
3
+
1
2
u −
1
2
a
8
=
1
−u
2
a
9
=
1
−u
2
a
1
=
u
u
a
3
=
1
2
u
3
− u
2
+
3
2
u −
1
2
−
1
2
u
3
+
1
2
u −
1
2
a
2
=
1
2
u
3
− u
2
+
5
2
u −
1
2
−
1
2
u
3
+
3
2
u −
1
2
a
5
=
−u
−u
a
6
=
u
2
+ 1
−u
3
+ u
2
+ 1
a
10
=
u
3
u
3
+ u
a
10
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
1
4
u
3
−
9
2
u
2
+
9
4
u −
53
4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
8
u
4
c
5
, c
6
, c
10
u
4
− u
3
+ 3u
2
− 2u + 1
c
7
u
4
− u
3
+ u
2
+ 1
c
9
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
11
u
4
+ u
3
+ u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
9
c
10
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
7
, c
11
y
4
+ y
3
+ 3y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = −0.38053 + 1.53420I
b = −0.927958 + 0.413327I
−1.85594 − 1.41510I −12.38954 + 3.92814I
u = −0.351808 − 0.720342I
a = −0.38053 − 1.53420I
b = −0.927958 − 0.413327I
−1.85594 + 1.41510I −12.38954 − 3.92814I
u = 0.851808 + 0.911292I
a = 0.130534 + 0.427872I
b = 0.677958 − 0.157780I
5.14581 + 3.16396I −10.48546 − 5.24252I
u = 0.851808 − 0.911292I
a = 0.130534 − 0.427872I
b = 0.677958 + 0.157780I
5.14581 − 3.16396I −10.48546 + 5.24252I
10
III. I
u
3
=
hu
5
+u
4
+u
3
−u
2
+b−2u, u
4
−u
3
+u
2
+a−3u+2, u
6
+2u
4
−3u
3
+u
2
−3u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
−u
4
+ u
3
− u
2
+ 3u − 2
−u
5
− u
4
− u
3
+ u
2
+ 2u
a
8
=
1
−u
2
a
9
=
u
u
a
1
=
u
u
a
3
=
−u
4
− u
2
+ 2u − 1
−u
4
+ 2u
a
2
=
2u − 1
u
2
+ 2u
a
5
=
u
4
+ u
2
− 1
u
4
+ 2u
2
a
6
=
−u
2
+ 1
−u
2
a
10
=
u
3
u
3
+ u
a
10
=
u
3
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
(u
2
− u − 1)
3
c
2
(u
2
+ 3u + 1)
3
c
5
, c
6
, c
7
c
9
, c
11
u
6
+ 2u
4
+ 3u
3
+ u
2
+ 3u + 1
c
10
u
6
+ 4u
5
+ 6u
4
− 3u
3
− 13u
2
− 7u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
(y
2
− 3y + 1)
3
c
2
(y
2
− 7y + 1)
3
c
5
, c
6
, c
7
c
9
, c
11
y
6
+ 4y
5
+ 6y
4
− 3y
3
− 13y
2
− 7y + 1
c
10
y
6
− 4y
5
+ 34y
4
− 107y
3
+ 139y
2
− 75y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.170987 + 1.042930I
a = −1.89470 + 1.68750I
b = −1.98931 + 1.11042I
−0.986960 −10.0000
u = −0.170987 − 1.042930I
a = −1.89470 − 1.68750I
b = −1.98931 − 1.11042I
−0.986960 −10.0000
u = 1.13928
a = −0.0860817
b = −1.50630
−8.88264 −10.0000
u = −0.56964 + 1.40480I
a = 0.970092 − 0.868217I
b = 1.371190 + 0.120928I
−8.88264 −10.0000
u = −0.56964 − 1.40480I
a = 0.970092 + 0.868217I
b = 1.371190 − 0.120928I
−8.88264 −10.0000
u = 0.341974
a = −1.06471
b = 0.742547
−0.986960 −10.0000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
2
− u − 1)
3
(u
20
− 2u
19
+ ··· − 35u + 4)
c
2
((u + 1)
4
)(u
2
+ 3u + 1)
3
(u
20
+ 22u
19
+ ··· + 353u + 16)
c
3
, c
8
u
4
(u
2
− u − 1)
3
(u
20
+ 2u
19
+ ··· + 112u + 64)
c
4
((u + 1)
4
)(u
2
− u − 1)
3
(u
20
− 2u
19
+ ··· − 35u + 4)
c
5
, c
6
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
6
+ 2u
4
+ 3u
3
+ u
2
+ 3u + 1)
· (u
20
− 2u
19
+ ··· − 2u + 1)
c
7
(u
4
− u
3
+ u
2
+ 1)(u
6
+ 2u
4
+ ··· + 3u + 1)(u
20
− 2u
19
+ ··· − 2u + 1)
c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
6
+ 2u
4
+ 3u
3
+ u
2
+ 3u + 1)
· (u
20
− 2u
19
+ ··· − 2u + 1)
c
10
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
6
+ 4u
5
+ 6u
4
− 3u
3
− 13u
2
− 7u + 1)
· (u
20
+ 14u
19
+ ··· + 4u + 1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
6
+ 2u
4
+ ··· + 3u + 1)(u
20
− 2u
19
+ ··· − 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
2
− 3y + 1)
3
(y
20
− 22y
19
+ ··· − 353y + 16)
c
2
((y −1)
4
)(y
2
− 7y + 1)
3
(y
20
− 46y
19
+ ··· + 223775y + 256)
c
3
, c
8
y
4
(y
2
− 3y + 1)
3
(y
20
− 18y
19
+ ··· + 45824y + 4096)
c
5
, c
6
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
6
+ 4y
5
+ 6y
4
− 3y
3
− 13y
2
− 7y + 1)
· (y
20
+ 14y
19
+ ··· + 4y + 1)
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
6
+ 4y
5
+ 6y
4
− 3y
3
− 13y
2
− 7y + 1)
· (y
20
+ 14y
19
+ ··· + 4y + 1)
c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
6
− 4y
5
+ ··· − 75y + 1)
· (y
20
− 14y
19
+ ··· + 56y + 1)
16
