10
165
(K10n
37
)
A knot diagram
1
Linearized knot diagam
5 7 9 2 8 9 4 1 4 6
Solving Sequence
1,5 2,8
6 9 4 3 7 10
c
1
c
5
c
8
c
4
c
3
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
12
+ 5u
11
+ 16u
10
+ 34u
9
+ 53u
8
+ 61u
7
+ 48u
6
+ 20u
5
− 9u
4
− 24u
3
− 20u
2
+ b − 11u − 3,
− 3u
12
− 13u
11
− 41u
10
− 85u
9
− 136u
8
− 167u
7
− 148u
6
− 90u
5
− 5u
4
+ 54u
3
+ 63u
2
+ 2a + 50u + 14,
u
13
+ 5u
12
+ 17u
11
+ 39u
10
+ 68u
9
+ 91u
8
+ 90u
7
+ 62u
6
+ 15u
5
− 24u
4
− 37u
3
− 30u
2
− 12u − 2i
I
u
2
= h−u
3
− u
2
+ b − u − 1, u
5
+ 2u
4
+ 4u
3
+ 3u
2
+ 2a + 3u + 2, u
6
+ 2u
5
+ 4u
4
+ 5u
3
+ 5u
2
+ 4u + 2i
I
u
3
= h−u
5
+ u
4
− 2u
3
+ au + u
2
+ b − u + 1,
u
5
a − 2u
4
a − 2u
5
+ 4u
3
a + 3u
4
− 4u
2
a − 7u
3
+ a
2
+ 3au + 7u
2
− 2a − 6u + 4,
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 3 irreducible components of dim
C
= 0, with total 31 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
=
hu
12
+5u
11
+· · ·+b−3, −3u
12
−13u
11
+· · ·+2a+14, u
13
+5u
12
+· · ·−12u−2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
3
2
u
12
+
13
2
u
11
+ ··· − 25u − 7
−u
12
− 5u
11
+ ··· + 11u + 3
a
6
=
3
2
u
12
+
13
2
u
11
+ ··· − 20u − 4
−u
12
− 5u
11
+ ··· + 15u + 3
a
9
=
1
2
u
12
+
3
2
u
11
+ ··· − 14u − 4
−u
12
− 5u
11
+ ··· + 11u + 3
a
4
=
−u
u
3
+ u
a
3
=
1
2
u
12
+
5
2
u
11
+ ··· − 4u − 1
u
12
+ 4u
11
+ ··· − 5u − 1
a
7
=
1
2
u
12
+
5
2
u
11
+ ··· − 26u − 7
−u
12
− 4u
11
+ ··· + 2u + 1
a
10
=
1
2
u
12
+
3
2
u
11
+ ··· − 13u − 4
−u
12
− 5u
11
+ ··· + 10u + 3
(ii) Obstruction class = −1
(iii) Cusp Shapes =
5u
12
+24u
11
+78u
10
+170u
9
+277u
8
+342u
7
+296u
6
+161u
5
−15u
4
−125u
3
−126u
2
−82u−16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
13
+ 5u
12
+ ··· − 12u − 2
c
2
, c
3
, c
9
u
13
+ 10u
11
+ ··· + 2u − 1
c
5
, c
8
u
13
+ u
12
+ ··· + 5u − 1
c
6
u
13
− 8u
12
+ ··· + 18u − 10
c
7
u
13
− u
12
+ ··· + 20u − 7
c
10
u
13
+ 12u
12
+ ··· − 288u − 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
13
+ 9y
12
+ ··· + 24y −4
c
2
, c
3
, c
9
y
13
+ 20y
12
+ ··· − 10y −1
c
5
, c
8
y
13
+ 7y
12
+ ··· + 23y −1
c
6
y
13
− 16y
12
+ ··· + 1284y −100
c
7
y
13
+ 13y
12
+ ··· + 64y −49
c
10
y
13
− 2y
12
+ ··· + 1024y −4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.152860 + 0.170520I
a = 0.717142 − 0.770562I
b = −0.695367 + 1.010640I
−6.75019 − 5.87953I 3.71309 + 4.79533I
u = −1.152860 − 0.170520I
a = 0.717142 + 0.770562I
b = −0.695367 − 1.010640I
−6.75019 + 5.87953I 3.71309 − 4.79533I
u = −0.034812 + 1.171400I
a = 0.739139 + 0.284263I
b = −0.358718 + 0.855935I
−3.39029 + 0.96735I 2.31477 − 3.00161I
u = −0.034812 − 1.171400I
a = 0.739139 − 0.284263I
b = −0.358718 − 0.855935I
−3.39029 − 0.96735I 2.31477 + 3.00161I
u = −0.175701 + 1.175030I
a = −1.067580 − 0.632688I
b = 0.93100 − 1.14328I
−2.32319 − 3.89550I 2.16216 + 1.95849I
u = −0.175701 − 1.175030I
a = −1.067580 + 0.632688I
b = 0.93100 + 1.14328I
−2.32319 + 3.89550I 2.16216 − 1.95849I
u = 0.773330
a = 0.244870
b = 0.189365
1.09959 6.33360
u = −0.48596 + 1.43258I
a = 1.058960 + 0.295073I
b = −0.93732 + 1.37365I
−11.8216 − 11.6031I 1.77641 + 5.73851I
u = −0.48596 − 1.43258I
a = 1.058960 − 0.295073I
b = −0.93732 − 1.37365I
−11.8216 + 11.6031I 1.77641 − 5.73851I
u = −0.363253 + 0.187651I
a = −0.56911 − 2.04054I
b = 0.589641 + 0.634441I
0.57483 + 1.68891I 3.43240 − 5.42565I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.363253 − 0.187651I
a = −0.56911 + 2.04054I
b = 0.589641 − 0.634441I
0.57483 − 1.68891I 3.43240 + 5.42565I
u = −0.67408 + 1.45370I
a = −0.500985 + 0.317553I
b = −0.123919 − 0.942337I
−10.56050 − 0.87235I −1.56565 + 0.23907I
u = −0.67408 − 1.45370I
a = −0.500985 − 0.317553I
b = −0.123919 + 0.942337I
−10.56050 + 0.87235I −1.56565 − 0.23907I
6
II. I
u
2
= h−u
3
− u
2
+ b − u − 1, u
5
+ 2u
4
+ 4u
3
+ 3u
2
+ 2a + 3u + 2, u
6
+
2u
5
+ 4u
4
+ 5u
3
+ 5u
2
+ 4u + 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
−
1
2
u
5
− u
4
− 2u
3
−
3
2
u
2
−
3
2
u − 1
u
3
+ u
2
+ u + 1
a
6
=
1
2
u
5
+ u
4
+ u
3
+
3
2
u
2
+
1
2
u
−u
4
− u
3
− 2u
2
− u − 1
a
9
=
−
1
2
u
5
− u
4
− u
3
−
1
2
u
2
−
1
2
u
u
3
+ u
2
+ u + 1
a
4
=
−u
u
3
+ u
a
3
=
1
2
u
5
+ 2u
4
+ 3u
3
+
7
2
u
2
+
3
2
u + 1
u
5
+ 2u
4
+ 4u
3
+ 3u
2
+ 3u + 1
a
7
=
−
1
2
u
5
+
3
2
u
2
+
3
2
u + 1
u
3
+ u
2
+ 2u + 1
a
10
=
−
1
2
u
5
− u
4
− 2u
3
−
3
2
u
2
−
3
2
u
u
5
+ u
4
+ 3u
3
+ 2u
2
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 4u
3
+ 9u
2
+ 6u + 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 2u
5
+ 4u
4
+ 5u
3
+ 5u
2
+ 4u + 2
c
2
, c
3
, c
9
u
6
+ 2u
4
− 2u
2
+ 1
c
4
u
6
− 2u
5
+ 4u
4
− 5u
3
+ 5u
2
− 4u + 2
c
5
, c
8
, c
10
u
6
+ u
5
− 2u
3
+ u + 1
c
6
u
6
− 5u
5
+ 10u
4
− 12u
3
+ 11u
2
− 6u + 2
c
7
u
6
+ u
5
+ 3u
4
− u
3
+ u
2
− 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
6
+ 4y
5
+ 6y
4
+ 3y
3
+ y
2
+ 4y + 4
c
2
, c
3
, c
9
(y
3
+ 2y
2
− 2y + 1)
2
c
5
, c
8
, c
10
y
6
− y
5
+ 4y
4
− 4y
3
+ 4y
2
− y + 1
c
6
y
6
− 5y
5
+ 2y
4
+ 20y
3
+ 17y
2
+ 8y + 4
c
7
y
6
+ 5y
5
+ 13y
4
+ 11y
3
+ 3y
2
− 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.862023 + 0.412869I
a = −0.233003 − 0.750879I
b = 0.510869 + 0.551075I
1.44750 + 0.78507I 8.28869 − 4.60495I
u = −0.862023 − 0.412869I
a = −0.233003 + 0.750879I
b = 0.510869 − 0.551075I
1.44750 − 0.78507I 8.28869 + 4.60495I
u = 0.238984 + 1.138460I
a = 0.176605 + 0.841305I
b = −0.915589 + 0.402116I
−8.28528 + 1.18132I 2.81561 − 0.13577I
u = 0.238984 − 1.138460I
a = 0.176605 − 0.841305I
b = −0.915589 − 0.402116I
−8.28528 − 1.18132I 2.81561 + 0.13577I
u = −0.376961 + 1.214800I
a = −0.943602 − 0.451942I
b = 0.904720 − 0.975923I
−1.38689 − 5.20040I 6.89570 + 6.16090I
u = −0.376961 − 1.214800I
a = −0.943602 + 0.451942I
b = 0.904720 + 0.975923I
−1.38689 + 5.20040I 6.89570 − 6.16090I
10
III. I
u
3
= h−u
5
+ u
4
− 2u
3
+ au + u
2
+ b − u + 1, u
5
a − 2u
5
+ · · · − 2a +
4, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
a
u
5
− u
4
+ 2u
3
− au − u
2
+ u − 1
a
6
=
u
5
a − u
4
a − u
5
+ 2u
3
a + u
4
− u
2
a − 3u
3
+ au + 2u
2
− a − 2u + 2
1
a
9
=
u
5
− u
4
+ 2u
3
− au − u
2
+ a + u − 1
u
5
− u
4
+ 2u
3
− au − u
2
+ u − 1
a
4
=
−u
u
3
+ u
a
3
=
u
5
− 3u
4
+ 6u
3
− au − 7u
2
+ a + 5u − 3
u
3
a − 2u
4
− u
2
a + 3u
3
+ au − 3u
2
+ 2u + 1
a
7
=
−u
4
a + u
5
+ u
3
a − u
4
− u
2
a + u
3
+ a − u
−u
4
a + 2u
5
+ u
3
a − 2u
4
− u
2
a + 4u
3
− 2u
2
+ a + u − 1
a
10
=
−u
5
a + u
4
a + u
5
− 2u
3
a − u
4
+ u
2
a + 3u
3
− au − 2u
2
+ a + 2u − 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
− 4u
3
+ 8u
2
− 4u + 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
c
2
, c
3
, c
9
u
12
+ u
11
+ ··· + 2u + 13
c
5
, c
8
u
12
+ 5u
11
+ ··· + 6u
2
+ 1
c
6
(u
6
+ 5u
5
+ 7u
4
− 2u
2
+ 3u − 1)
2
c
7
u
12
− u
11
+ ··· − 18u + 23
c
10
(u − 1)
12
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
2
c
2
, c
3
, c
9
y
12
+ 15y
11
+ ··· + 360y + 169
c
5
, c
8
y
12
− y
11
+ ··· + 12y + 1
c
6
(y
6
− 11y
5
+ 45y
4
− 60y
3
− 10y
2
− 5y + 1)
2
c
7
y
12
+ 11y
11
+ ··· − 416y + 529
c
10
(y −1)
12
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
a = 0.211090 + 0.348879I
b = 0.184327 − 0.304646I
1.08035 4.26950
u = 0.873214
a = 0.211090 − 0.348879I
b = 0.184327 + 0.304646I
1.08035 4.26950
u = −0.138835 + 1.234450I
a = 0.576096 + 0.033593I
b = −1.96628 − 1.27394I
−9.53998 − 1.97241I −3.42428 + 3.68478I
u = −0.138835 + 1.234450I
a = −0.84220 + 1.68756I
b = −0.121451 + 0.706495I
−9.53998 − 1.97241I −3.42428 + 3.68478I
u = −0.138835 − 1.234450I
a = 0.576096 − 0.033593I
b = −1.96628 + 1.27394I
−9.53998 + 1.97241I −3.42428 − 3.68478I
u = −0.138835 − 1.234450I
a = −0.84220 − 1.68756I
b = −0.121451 − 0.706495I
−9.53998 + 1.97241I −3.42428 − 3.68478I
u = 0.408802 + 1.276380I
a = 1.089440 − 0.275882I
b = −0.511061 − 0.781659I
−2.88416 + 4.59213I 0.58114 − 3.20482I
u = 0.408802 + 1.276380I
a = −0.671738 + 0.185253I
b = 0.79749 + 1.27775I
−2.88416 + 4.59213I 0.58114 − 3.20482I
u = 0.408802 − 1.276380I
a = 1.089440 + 0.275882I
b = −0.511061 + 0.781659I
−2.88416 − 4.59213I 0.58114 + 3.20482I
u = 0.408802 − 1.276380I
a = −0.671738 − 0.185253I
b = 0.79749 − 1.27775I
−2.88416 − 4.59213I 0.58114 + 3.20482I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.413150
a = 2.13731 + 1.92634I
b = −0.883031 + 0.795869I
−5.84089 5.41680
u = −0.413150
a = 2.13731 − 1.92634I
b = −0.883031 − 0.795869I
−5.84089 5.41680
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
· (u
6
+ 2u
5
+ ··· + 4u + 2)(u
13
+ 5u
12
+ ··· − 12u − 2)
c
2
, c
3
, c
9
(u
6
+ 2u
4
− 2u
2
+ 1)(u
12
+ u
11
+ ··· + 2u + 13)
· (u
13
+ 10u
11
+ ··· + 2u − 1)
c
4
(u
6
− 2u
5
+ 4u
4
− 5u
3
+ 5u
2
− 4u + 2)
· ((u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
)(u
13
+ 5u
12
+ ··· − 12u − 2)
c
5
, c
8
(u
6
+ u
5
− 2u
3
+ u + 1)(u
12
+ 5u
11
+ ··· + 6u
2
+ 1)
· (u
13
+ u
12
+ ··· + 5u − 1)
c
6
(u
6
− 5u
5
+ 10u
4
− 12u
3
+ 11u
2
− 6u + 2)
· ((u
6
+ 5u
5
+ 7u
4
− 2u
2
+ 3u − 1)
2
)(u
13
− 8u
12
+ ··· + 18u − 10)
c
7
(u
6
+ u
5
+ 3u
4
− u
3
+ u
2
− 2u + 1)(u
12
− u
11
+ ··· − 18u + 23)
· (u
13
− u
12
+ ··· + 20u − 7)
c
10
((u − 1)
12
)(u
6
+ u
5
− 2u
3
+ u + 1)(u
13
+ 12u
12
+ ··· − 288u − 64)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
6
+ 4y
5
+ 6y
4
+ 3y
3
+ y
2
+ 4y + 4)
· ((y
6
+ 5y
5
+ ··· − 5y + 1)
2
)(y
13
+ 9y
12
+ ··· + 24y −4)
c
2
, c
3
, c
9
((y
3
+ 2y
2
− 2y + 1)
2
)(y
12
+ 15y
11
+ ··· + 360y + 169)
· (y
13
+ 20y
12
+ ··· − 10y −1)
c
5
, c
8
(y
6
− y
5
+ 4y
4
− 4y
3
+ 4y
2
− y + 1)(y
12
− y
11
+ ··· + 12y + 1)
· (y
13
+ 7y
12
+ ··· + 23y −1)
c
6
(y
6
− 11y
5
+ 45y
4
− 60y
3
− 10y
2
− 5y + 1)
2
· (y
6
− 5y
5
+ 2y
4
+ 20y
3
+ 17y
2
+ 8y + 4)
· (y
13
− 16y
12
+ ··· + 1284y −100)
c
7
(y
6
+ 5y
5
+ 13y
4
+ 11y
3
+ 3y
2
− 2y + 1)
· (y
12
+ 11y
11
+ ··· − 416y + 529)(y
13
+ 13y
12
+ ··· + 64y −49)
c
10
(y −1)
12
(y
6
− y
5
+ 4y
4
− 4y
3
+ 4y
2
− y + 1)
· (y
13
− 2y
12
+ ··· + 1024y −4096)
17
