11n
59
(K11n
59
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 11 10 4 5 6 7 9
Solving Sequence
6,9
10 7 11
1,4
3 2 5 8
c
9
c
6
c
10
c
11
c
3
c
2
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
18
− 8u
16
+ 25u
14
− 36u
12
+ 19u
10
+ 4u
8
+ 2u
7
− 2u
6
− 6u
5
− 4u
4
+ 4u
3
+ u
2
+ b + 2u,
− u
29
− u
28
+ ··· + a − 1, u
31
+ 2u
30
+ ··· + 2u + 1i
I
u
2
= h−u
4
+ 2u
2
+ b, −u
3
+ a + u + 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 36 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
18
−8u
16
+· · ·+b+2u, −u
29
−u
28
+· · ·+a−1, u
31
+2u
30
+· · ·+2u+1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
7
=
−u
−u
3
+ u
a
11
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
4
=
u
29
+ u
28
+ ··· + u + 1
−u
18
+ 8u
16
+ ··· − u
2
− 2u
a
3
=
2u
30
+ 2u
29
+ ··· + 2u + 2
2u
30
+ u
29
+ ··· + u + 2
a
2
=
u
30
+ u
29
+ ··· + 2u + 2
u
30
− 13u
28
+ ··· + 7u
3
+ 1
a
5
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
u
12
− 5u
10
+ 9u
8
− 6u
6
+ u
2
+ 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
a
8
=
u
12
− 5u
10
+ 9u
8
− 6u
6
+ u
2
+ 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
30
− 26u
28
+ 5u
27
+ 147u
26
− 59u
25
− 460u
24
+ 296u
23
+ 817u
22
− 799u
21
−
658u
20
+ 1179u
19
− 269u
18
− 741u
17
+ 1010u
16
− 254u
15
− 570u
14
+ 536u
13
− 222u
12
−
2u
11
+ 186u
10
− 168u
9
+ 98u
8
− 68u
7
− u
6
+ 41u
5
− 40u
4
+ 35u
3
− 19u
2
+ 5u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
31
− 6u
30
+ ··· − 4u + 1
c
2
u
31
+ 8u
30
+ ··· + 12u + 1
c
3
, c
7
u
31
+ u
30
+ ··· + 64u + 32
c
5
u
31
− 6u
30
+ ··· − 18u + 5
c
6
, c
9
, c
10
u
31
+ 2u
30
+ ··· + 2u + 1
c
8
u
31
− 2u
30
+ ··· + 2u + 1
c
11
u
31
+ 8u
30
+ ··· + 30u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
31
− 8y
30
+ ··· + 12y −1
c
2
y
31
+ 36y
30
+ ··· + 48y −1
c
3
, c
7
y
31
+ 33y
30
+ ··· − 10752y −1024
c
5
y
31
+ 4y
30
+ ··· + 174y −25
c
6
, c
9
, c
10
y
31
− 28y
30
+ ··· + 2y −1
c
8
y
31
− 36y
30
+ ··· + 2y −1
c
11
y
31
+ 32y
29
+ ··· + 1054y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.953391 + 0.341205I
a = −0.060208 − 0.824217I
b = −0.154559 − 1.385810I
5.55958 − 3.20800I −6.18542 + 3.44031I
u = 0.953391 − 0.341205I
a = −0.060208 + 0.824217I
b = −0.154559 + 1.385810I
5.55958 + 3.20800I −6.18542 − 3.44031I
u = 0.825063 + 0.385979I
a = 0.618554 + 1.193580I
b = −0.63882 + 1.61785I
5.21409 + 3.77786I −6.77733 − 1.30179I
u = 0.825063 − 0.385979I
a = 0.618554 − 1.193580I
b = −0.63882 − 1.61785I
5.21409 − 3.77786I −6.77733 + 1.30179I
u = 0.256805 + 0.769307I
a = 1.018830 − 0.906649I
b = 0.90393 + 1.86214I
7.07782 − 7.98216I −4.46644 + 5.97470I
u = 0.256805 − 0.769307I
a = 1.018830 + 0.906649I
b = 0.90393 − 1.86214I
7.07782 + 7.98216I −4.46644 − 5.97470I
u = −1.191040 + 0.124902I
a = −0.150730 + 0.017487I
b = 0.447590 + 0.550052I
−1.84615 + 0.55694I −6.11373 + 0.01533I
u = −1.191040 − 0.124902I
a = −0.150730 − 0.017487I
b = 0.447590 − 0.550052I
−1.84615 − 0.55694I −6.11373 − 0.01533I
u = 0.195264 + 0.773377I
a = −1.255470 + 0.164780I
b = −0.177950 − 1.155760I
7.92090 − 0.90622I −3.00591 + 1.13607I
u = 0.195264 − 0.773377I
a = −1.255470 − 0.164780I
b = −0.177950 + 1.155760I
7.92090 + 0.90622I −3.00591 − 1.13607I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.371835 + 0.568068I
a = 0.797944 − 0.031326I
b = −0.214074 − 0.837766I
−0.87857 + 1.75044I −3.85072 − 4.54648I
u = −0.371835 − 0.568068I
a = 0.797944 + 0.031326I
b = −0.214074 + 0.837766I
−0.87857 − 1.75044I −3.85072 + 4.54648I
u = 1.325090 + 0.154351I
a = −2.42287 + 0.53159I
b = −1.338960 − 0.362468I
−5.25829 − 1.15681I −12.57201 + 1.48822I
u = 1.325090 − 0.154351I
a = −2.42287 − 0.53159I
b = −1.338960 + 0.362468I
−5.25829 + 1.15681I −12.57201 − 1.48822I
u = −0.156640 + 0.637516I
a = −0.041090 − 1.349110I
b = −0.896955 + 0.568672I
1.00966 + 2.22196I −3.40092 − 5.38737I
u = −0.156640 − 0.637516I
a = −0.041090 + 1.349110I
b = −0.896955 − 0.568672I
1.00966 − 2.22196I −3.40092 + 5.38737I
u = −1.351700 + 0.206762I
a = 0.051503 − 0.972430I
b = −0.62993 − 1.46088I
−6.05722 + 3.45238I −10.37888 − 2.19312I
u = −1.351700 − 0.206762I
a = 0.051503 + 0.972430I
b = −0.62993 + 1.46088I
−6.05722 − 3.45238I −10.37888 + 2.19312I
u = 1.355400 + 0.255032I
a = 1.82855 − 0.84093I
b = 1.180600 + 0.666054I
−3.77819 − 5.48065I −9.45984 + 5.94075I
u = 1.355400 − 0.255032I
a = 1.82855 + 0.84093I
b = 1.180600 − 0.666054I
−3.77819 + 5.48065I −9.45984 − 5.94075I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.374710 + 0.318270I
a = 1.72875 − 0.73774I
b = 0.359105 − 0.934571I
2.95558 + 4.85038I −7.33363 − 2.61184I
u = −1.374710 − 0.318270I
a = 1.72875 + 0.73774I
b = 0.359105 + 0.934571I
2.95558 − 4.85038I −7.33363 + 2.61184I
u = −1.40893 + 0.31052I
a = −2.64694 + 0.69829I
b = −1.12415 + 1.93346I
1.77560 + 11.89500I −8.90588 − 6.87931I
u = −1.40893 − 0.31052I
a = −2.64694 − 0.69829I
b = −1.12415 − 1.93346I
1.77560 − 11.89500I −8.90588 + 6.87931I
u = −1.45164 + 0.04136I
a = 0.77948 + 1.85121I
b = 0.915385 + 0.997789I
−1.95006 − 2.95334I −10.65956 + 2.73175I
u = −1.45164 − 0.04136I
a = 0.77948 − 1.85121I
b = 0.915385 − 0.997789I
−1.95006 + 2.95334I −10.65956 − 2.73175I
u = 1.43633 + 0.21708I
a = −0.458519 − 1.234670I
b = 0.376409 − 1.104670I
−6.67621 − 4.65354I −6.91647 + 4.60285I
u = 1.43633 − 0.21708I
a = −0.458519 + 1.234670I
b = 0.376409 + 1.104670I
−6.67621 + 4.65354I −6.91647 − 4.60285I
u = 0.127284 + 0.484686I
a = 0.09092 + 1.50930I
b = 0.622522 − 0.861672I
−1.33374 − 0.83076I −4.07951 − 0.75098I
u = 0.127284 − 0.484686I
a = 0.09092 − 1.50930I
b = 0.622522 + 0.861672I
−1.33374 + 0.83076I −4.07951 + 0.75098I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.336229
a = 1.24262
b = 0.739715
−0.889878 −11.7880
8
II. I
u
2
= h−u
4
+ 2u
2
+ b, −u
3
+ a + u + 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
7
=
−u
−u
3
+ u
a
11
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
4
=
u
3
− u − 1
u
4
− 2u
2
a
3
=
u
3
− u − 1
u
4
− 2u
2
a
2
=
−u
4
+ u
3
+ u
2
− u
0
a
5
=
u
4
− u
2
− 1
u
4
− 2u
2
a
8
=
−u
−u
3
+ u
a
8
=
−u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
+ u
2
− 8u − 15
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
6
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
8
, c
11
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
9
, c
10
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
6
, c
9
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
8
, c
11
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −1.58802
b = −0.766826
−4.04602 −9.19250
u = −0.309916 + 0.549911I
a = −0.438694 − 0.557752I
b = 0.339110 + 0.822375I
−1.97403 + 1.53058I −11.97286 − 4.76366I
u = −0.309916 − 0.549911I
a = −0.438694 + 0.557752I
b = 0.339110 − 0.822375I
−1.97403 − 1.53058I −11.97286 + 4.76366I
u = 1.41878 + 0.21917I
a = 0.232705 + 1.093810I
b = −0.455697 + 1.200150I
−7.51750 − 4.40083I −16.4309 + 2.8075I
u = 1.41878 − 0.21917I
a = 0.232705 − 1.093810I
b = −0.455697 − 1.200150I
−7.51750 + 4.40083I −16.4309 − 2.8075I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
31
− 6u
30
+ ··· − 4u + 1)
c
2
((u + 1)
5
)(u
31
+ 8u
30
+ ··· + 12u + 1)
c
3
, c
7
u
5
(u
31
+ u
30
+ ··· + 64u + 32)
c
4
((u + 1)
5
)(u
31
− 6u
30
+ ··· − 4u + 1)
c
5
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
31
− 6u
30
+ ··· − 18u + 5)
c
6
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
c
8
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
31
− 2u
30
+ ··· + 2u + 1)
c
9
, c
10
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
31
+ 8u
30
+ ··· + 30u + 7)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
31
− 8y
30
+ ··· + 12y −1)
c
2
((y −1)
5
)(y
31
+ 36y
30
+ ··· + 48y −1)
c
3
, c
7
y
5
(y
31
+ 33y
30
+ ··· − 10752y −1024)
c
5
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
31
+ 4y
30
+ ··· + 174y −25)
c
6
, c
9
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
31
− 28y
30
+ ··· + 2y −1)
c
8
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
31
− 36y
30
+ ··· + 2y −1)
c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
31
+ 32y
29
+ ··· + 1054y −49)
14
