11n
142
(K11n
142
)
A knot diagram
1
Linearized knot diagam
8 1 9 11 8 10 2 5 1 6 4
Solving Sequence
5,11
4
1,9
3 2 8 6 7 10
c
4
c
11
c
3
c
2
c
8
c
5
c
7
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
13
+ 6u
12
+ ··· + b + 3,
u
13
− u
12
+ 15u
10
− 48u
9
+ 106u
8
− 178u
7
+ 222u
6
− 230u
5
+ 175u
4
− 107u
3
+ 57u
2
+ 2a − 19u + 10,
u
14
− 5u
13
+ ··· − 6u + 2i
I
u
2
= hu
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ b + 2u + 1, −u
7
− 2u
5
+ 2u
4
+ 3u
3
+ 4u
2
+ 2a + 4u + 1,
u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 11u
4
+ 10u
3
+ 8u
2
+ 5u + 2i
I
u
3
= hu
4
a + 2u
2
a − u
3
− au + b + a − u + 1, u
3
a + 2u
4
+ 2u
2
a + 3u
3
+ a
2
+ 2au + 5u
2
+ 2a + u − 1,
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 32 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
+6u
12
+· · ·+b+3, u
13
−u
12
+· · ·+2a+10, u
14
−5u
13
+· · ·−6u+2i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
1
=
−u
u
3
+ u
a
9
=
−
1
2
u
13
+
1
2
u
12
+ ··· +
19
2
u − 5
u
13
− 6u
12
+ ··· + 11u − 3
a
3
=
1
2
u
13
−
5
2
u
12
+ ··· −
11
2
u
2
+
3
2
u
u
13
− 4u
12
+ ··· + 3u − 1
a
2
=
1
2
u
13
−
3
2
u
12
+ ··· −
7
2
u
2
+
1
2
u
−u
13
+ 4u
12
+ ··· − 2u + 1
a
8
=
−
3
2
u
13
+
13
2
u
12
+ ··· −
3
2
u − 2
u
13
− 6u
12
+ ··· + 11u − 3
a
6
=
3
2
u
13
−
13
2
u
12
+ ··· −
31
2
u
2
+
13
2
u
−u
13
+ 5u
12
+ ··· − 8u + 3
a
7
=
−2u
13
+ 9u
12
+ ··· + 19u
2
− 6u
u
13
− 4u
12
+ ··· + 11u − 4
a
10
=
−
3
2
u
13
+
13
2
u
12
+ ··· −
3
2
u − 1
u
13
− 5u
12
+ ··· + 14u − 5
a
10
=
−
3
2
u
13
+
13
2
u
12
+ ··· −
3
2
u − 1
u
13
− 5u
12
+ ··· + 14u − 5
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
13
+ 11u
12
− 40u
11
+ 102u
10
− 203u
9
+ 326u
8
− 425u
7
+
456u
6
− 399u
5
+ 283u
4
− 172u
3
+ 85u
2
− 36u + 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
u
14
+ 11u
12
+ ··· − u + 1
c
2
u
14
+ 22u
13
+ ··· + u + 1
c
4
, c
11
u
14
− 5u
13
+ ··· − 6u + 2
c
5
, c
8
u
14
+ 8u
12
+ ··· − 4u + 1
c
6
, c
10
u
14
+ 11u
13
+ ··· + 208u + 32
c
9
u
14
+ u
13
+ ··· − 42u + 43
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
14
+ 22y
13
+ ··· + y + 1
c
2
y
14
− 66y
13
+ ··· + 57y + 1
c
4
, c
11
y
14
+ 11y
13
+ ··· + 48y + 4
c
5
, c
8
y
14
+ 16y
13
+ ··· + 6y + 1
c
6
, c
10
y
14
+ 5y
13
+ ··· + 4352y + 1024
c
9
y
14
+ 29y
13
+ ··· − 6924y + 1849
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.287050 + 0.917286I
a = 1.72431 − 0.08692I
b = 0.75441 − 1.21287I
0.82198 − 3.62125I 2.13881 + 1.61924I
u = 0.287050 − 0.917286I
a = 1.72431 + 0.08692I
b = 0.75441 + 1.21287I
0.82198 + 3.62125I 2.13881 − 1.61924I
u = 1.148320 + 0.063656I
a = 0.028321 − 0.233596I
b = −0.38125 − 1.63279I
−14.6114 + 5.0048I −3.11103 − 2.22395I
u = 1.148320 − 0.063656I
a = 0.028321 + 0.233596I
b = −0.38125 + 1.63279I
−14.6114 − 5.0048I −3.11103 + 2.22395I
u = 0.151463 + 0.669236I
a = −1.172520 + 0.541864I
b = −0.010117 + 0.820058I
0.100921 + 1.074380I 3.38569 − 3.60575I
u = 0.151463 − 0.669236I
a = −1.172520 − 0.541864I
b = −0.010117 − 0.820058I
0.100921 − 1.074380I 3.38569 + 3.60575I
u = −0.137919 + 0.533558I
a = −0.824651 + 0.595460I
b = −0.029422 + 0.445046I
0.158278 + 1.072210I 2.34747 − 5.95960I
u = −0.137919 − 0.533558I
a = −0.824651 − 0.595460I
b = −0.029422 − 0.445046I
0.158278 − 1.072210I 2.34747 + 5.95960I
u = −0.12127 + 1.46215I
a = 0.495610 + 0.295894I
b = 0.459360 − 0.015268I
6.08599 + 2.02171I 9.26276 − 3.22644I
u = −0.12127 − 1.46215I
a = 0.495610 − 0.295894I
b = 0.459360 + 0.015268I
6.08599 − 2.02171I 9.26276 + 3.22644I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.60561 + 1.35177I
a = −1.56827 + 0.69249I
b = −0.78424 + 1.62391I
−10.6271 − 11.1808I −0.33111 + 5.29605I
u = 0.60561 − 1.35177I
a = −1.56827 − 0.69249I
b = −0.78424 − 1.62391I
−10.6271 + 11.1808I −0.33111 − 5.29605I
u = 0.56676 + 1.45000I
a = 0.817208 − 1.060380I
b = −0.008750 − 1.377190I
−9.89254 − 1.11324I −1.192579 + 0.716159I
u = 0.56676 − 1.45000I
a = 0.817208 + 1.060380I
b = −0.008750 + 1.377190I
−9.89254 + 1.11324I −1.192579 − 0.716159I
6
II. I
u
2
= hu
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ b + 2u + 1, −u
7
− 2u
5
+ 2u
4
+ 3u
3
+
4u
2
+ 2a + 4u + 1, u
8
+ 2u
7
+ · · · + 5u + 2i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
1
=
−u
u
3
+ u
a
9
=
1
2
u
7
+ u
5
− u
4
−
3
2
u
3
− 2u
2
− 2u −
1
2
−u
6
− u
5
− 4u
4
− 3u
3
− 4u
2
− 2u − 1
a
3
=
−
1
2
u
7
− 2u
6
+ ··· − 2u +
1
2
−u
7
− 2u
6
− 5u
5
− 6u
4
− 6u
3
− 5u
2
− 2u − 1
a
2
=
1
2
u
7
+ u
5
− u
4
−
1
2
u
3
− u
2
− u +
1
2
−u
7
− 2u
6
− 5u
5
− 6u
4
− 5u
3
− 4u
2
− u − 1
a
8
=
1
2
u
7
+ u
6
+ 2u
5
+ 3u
4
+
3
2
u
3
+ 2u
2
+
1
2
−u
6
− u
5
− 4u
4
− 3u
3
− 4u
2
− 2u − 1
a
6
=
−
1
2
u
7
− u
6
+ ··· − 2u −
1
2
u
5
+ u
4
+ 3u
3
+ 2u
2
+ u + 1
a
7
=
u
2
+ u + 1
u
6
+ 2u
5
+ 4u
4
+ 5u
3
+ 4u
2
+ 2u
a
10
=
1
2
u
7
+ u
6
+ ··· + 2u +
3
2
−u
6
− u
5
− 4u
4
− 3u
3
− 4u
2
− u − 1
a
10
=
1
2
u
7
+ u
6
+ ··· + 2u +
3
2
−u
6
− u
5
− 4u
4
− 3u
3
− 4u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
7
− 6u
6
− 15u
5
− 22u
4
− 22u
3
− 22u
2
− 13u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 4u
6
+ u
5
+ 4u
4
+ 3u
3
+ 3u
2
+ u + 1
c
2
u
8
+ 8u
7
+ 24u
6
+ 37u
5
+ 36u
4
+ 21u
3
+ 11u
2
+ 5u + 1
c
3
, c
7
u
8
+ 4u
6
− u
5
+ 4u
4
− 3u
3
+ 3u
2
− u + 1
c
4
u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 11u
4
+ 10u
3
+ 8u
2
+ 5u + 2
c
5
u
8
+ u
6
− 3u
5
+ u
4
− 2u
3
+ 3u
2
+ 1
c
6
u
8
+ 3u
6
+ 2u
5
+ u
4
+ 3u
3
+ u
2
+ 1
c
8
u
8
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ 3u
2
+ 1
c
9
u
8
+ u
7
+ 4u
6
+ u
4
− 2u
2
+ 1
c
10
u
8
+ 3u
6
− 2u
5
+ u
4
− 3u
3
+ u
2
+ 1
c
11
u
8
− 2u
7
+ 6u
6
− 8u
5
+ 11u
4
− 10u
3
+ 8u
2
− 5u + 2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
8
+ 8y
7
+ 24y
6
+ 37y
5
+ 36y
4
+ 21y
3
+ 11y
2
+ 5y + 1
c
2
y
8
− 16y
7
+ 56y
6
+ 45y
5
+ 192y
4
+ 29y
3
− 17y
2
− 3y + 1
c
4
, c
11
y
8
+ 8y
7
+ 26y
6
+ 44y
5
+ 41y
4
+ 20y
3
+ 8y
2
+ 7y + 4
c
5
, c
8
y
8
+ 2y
7
+ 3y
6
− y
5
− 3y
4
+ 4y
3
+ 11y
2
+ 6y + 1
c
6
, c
10
y
8
+ 6y
7
+ 11y
6
+ 4y
5
− 3y
4
− y
3
+ 3y
2
+ 2y + 1
c
9
y
8
+ 7y
7
+ 18y
6
+ 4y
5
− 13y
4
+ 4y
3
+ 6y
2
− 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.255307 + 0.956150I
a = 1.69644 − 0.66169I
b = 1.095290 + 0.323314I
−5.22098 − 1.00599I 2.77337 + 0.09808I
u = 0.255307 − 0.956150I
a = 1.69644 + 0.66169I
b = 1.095290 − 0.323314I
−5.22098 + 1.00599I 2.77337 − 0.09808I
u = −0.420429 + 1.128350I
a = −1.43682 − 0.24968I
b = −0.744211 − 1.167310I
1.09366 + 5.02764I 3.89133 − 6.50935I
u = −0.420429 − 1.128350I
a = −1.43682 + 0.24968I
b = −0.744211 + 1.167310I
1.09366 − 5.02764I 3.89133 + 6.50935I
u = −0.669415 + 0.364330I
a = 0.671643 + 0.022513I
b = −0.279662 + 1.002820I
−1.22874 − 0.94773I −2.78542 + 1.04891I
u = −0.669415 − 0.364330I
a = 0.671643 − 0.022513I
b = −0.279662 − 1.002820I
−1.22874 + 0.94773I −2.78542 − 1.04891I
u = −0.16546 + 1.54832I
a = 0.318738 + 0.607785I
b = −0.071417 + 0.603353I
5.35605 + 1.96927I −2.37928 − 1.80892I
u = −0.16546 − 1.54832I
a = 0.318738 − 0.607785I
b = −0.071417 − 0.603353I
5.35605 − 1.96927I −2.37928 + 1.80892I
10
III. I
u
3
= hu
4
a + 2u
2
a − u
3
− au + b + a − u + 1, u
3
a + 2u
4
+ · · · + 2a −
1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
1
=
−u
u
3
+ u
a
9
=
a
−u
4
a − 2u
2
a + u
3
+ au − a + u − 1
a
3
=
2u
3
+ 4u
2
+ a + 4u + 4
u
4
a + u
2
a + u
3
− au + a + u + 1
a
2
=
u
4
a + u
4
+ 2u
2
a + u
3
+ 4u
2
+ 2a + u + 3
−u
4
a − u
3
a − u
4
− 2u
2
a + 2u
3
− au − a + 3u
a
8
=
u
4
a + 2u
2
a − u
3
− au + 2a − u + 1
−u
4
a − 2u
2
a + u
3
+ au − a + u − 1
a
6
=
−u
4
− u
3
− 2u
2
− u − 1
−u
4
a + u
4
− u
2
a + 2u
3
+ au + 2u
2
a
7
=
2u
4
+ 2u
3
+ 4u
2
+ 2u + 2
2u
4
a − 2u
4
+ 2u
2
a − 4u
3
− 2au − 4u
2
− u
a
10
=
−u
4
− u
3
− 2u
2
− u − 1
−u
4
a + u
4
− u
2
a + 2u
3
+ au + 2u
2
+ u
a
10
=
−u
4
− u
3
− 2u
2
− u − 1
−u
4
a + u
4
− u
2
a + 2u
3
+ au + 2u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u
2
+ 4u − 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
u
10
− u
9
+ ··· − 20u + 23
c
2
u
10
+ 15u
9
+ ··· + 2268u + 529
c
4
, c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
5
, c
8
u
10
+ 5u
9
+ ··· + 20u + 7
c
6
, c
10
(u − 1)
10
c
9
u
10
+ u
9
+ 10u
8
− 8u
7
+ 42u
6
+ 2u
5
+ 29u
4
+ 43u
3
+ 28u
2
− 12u + 67
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
10
+ 15y
9
+ ··· + 2268y + 529
c
2
y
10
− 33y
9
+ ··· − 245284y + 279841
c
4
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
5
, c
8
y
10
+ 3y
9
+ ··· + 468y + 49
c
6
, c
10
(y −1)
10
c
9
y
10
+ 19y
9
+ ··· + 3608y + 4489
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 1.56543 − 1.34638I
b = −0.144990 + 0.454920I
−6.25064 − 1.53058I −5.48489 + 4.43065I
u = 0.339110 + 0.822375I
a = −2.47201 − 1.14141I
b = −2.06136 − 0.79577I
−6.25064 − 1.53058I −5.48489 + 4.43065I
u = 0.339110 − 0.822375I
a = 1.56543 + 1.34638I
b = −0.144990 − 0.454920I
−6.25064 + 1.53058I −5.48489 − 4.43065I
u = 0.339110 − 0.822375I
a = −2.47201 + 1.14141I
b = −2.06136 + 0.79577I
−6.25064 + 1.53058I −5.48489 − 4.43065I
u = −0.766826
a = −0.595741 + 0.396465I
b = −0.258559 − 1.303830I
−4.17865 −4.51890
u = −0.766826
a = −0.595741 − 0.396465I
b = −0.258559 + 1.303830I
−4.17865 −4.51890
u = −0.455697 + 1.200150I
a = 1.04040 + 1.01526I
b = 0.43147 + 1.63522I
−0.70717 + 4.40083I −1.25569 − 3.49859I
u = −0.455697 + 1.200150I
a = −1.53808 − 0.24695I
b = −0.466561 − 1.013320I
−0.70717 + 4.40083I −1.25569 − 3.49859I
u = −0.455697 − 1.200150I
a = 1.04040 − 1.01526I
b = 0.43147 − 1.63522I
−0.70717 − 4.40083I −1.25569 + 3.49859I
u = −0.455697 − 1.200150I
a = −1.53808 + 0.24695I
b = −0.466561 + 1.013320I
−0.70717 − 4.40083I −1.25569 + 3.49859I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
+ 4u
6
+ ··· + u + 1)(u
10
− u
9
+ ··· − 20u + 23)
· (u
14
+ 11u
12
+ ··· − u + 1)
c
2
(u
8
+ 8u
7
+ 24u
6
+ 37u
5
+ 36u
4
+ 21u
3
+ 11u
2
+ 5u + 1)
· (u
10
+ 15u
9
+ ··· + 2268u + 529)(u
14
+ 22u
13
+ ··· + u + 1)
c
3
, c
7
(u
8
+ 4u
6
+ ··· − u + 1)(u
10
− u
9
+ ··· − 20u + 23)
· (u
14
+ 11u
12
+ ··· − u + 1)
c
4
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
· (u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 11u
4
+ 10u
3
+ 8u
2
+ 5u + 2)
· (u
14
− 5u
13
+ ··· − 6u + 2)
c
5
(u
8
+ u
6
− 3u
5
+ u
4
− 2u
3
+ 3u
2
+ 1)(u
10
+ 5u
9
+ ··· + 20u + 7)
· (u
14
+ 8u
12
+ ··· − 4u + 1)
c
6
(u − 1)
10
(u
8
+ 3u
6
+ 2u
5
+ u
4
+ 3u
3
+ u
2
+ 1)
· (u
14
+ 11u
13
+ ··· + 208u + 32)
c
8
(u
8
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ 3u
2
+ 1)(u
10
+ 5u
9
+ ··· + 20u + 7)
· (u
14
+ 8u
12
+ ··· − 4u + 1)
c
9
(u
8
+ u
7
+ 4u
6
+ u
4
− 2u
2
+ 1)
· (u
10
+ u
9
+ 10u
8
− 8u
7
+ 42u
6
+ 2u
5
+ 29u
4
+ 43u
3
+ 28u
2
− 12u + 67)
· (u
14
+ u
13
+ ··· − 42u + 43)
c
10
(u − 1)
10
(u
8
+ 3u
6
− 2u
5
+ u
4
− 3u
3
+ u
2
+ 1)
· (u
14
+ 11u
13
+ ··· + 208u + 32)
c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
· (u
8
− 2u
7
+ 6u
6
− 8u
5
+ 11u
4
− 10u
3
+ 8u
2
− 5u + 2)
· (u
14
− 5u
13
+ ··· − 6u + 2)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
(y
8
+ 8y
7
+ 24y
6
+ 37y
5
+ 36y
4
+ 21y
3
+ 11y
2
+ 5y + 1)
· (y
10
+ 15y
9
+ ··· + 2268y + 529)(y
14
+ 22y
13
+ ··· + y + 1)
c
2
(y
8
− 16y
7
+ 56y
6
+ 45y
5
+ 192y
4
+ 29y
3
− 17y
2
− 3y + 1)
· (y
10
− 33y
9
+ ··· − 245284y + 279841)(y
14
− 66y
13
+ ··· + 57y + 1)
c
4
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
· (y
8
+ 8y
7
+ 26y
6
+ 44y
5
+ 41y
4
+ 20y
3
+ 8y
2
+ 7y + 4)
· (y
14
+ 11y
13
+ ··· + 48y + 4)
c
5
, c
8
(y
8
+ 2y
7
+ 3y
6
− y
5
− 3y
4
+ 4y
3
+ 11y
2
+ 6y + 1)
· (y
10
+ 3y
9
+ ··· + 468y + 49)(y
14
+ 16y
13
+ ··· + 6y + 1)
c
6
, c
10
(y −1)
10
(y
8
+ 6y
7
+ 11y
6
+ 4y
5
− 3y
4
− y
3
+ 3y
2
+ 2y + 1)
· (y
14
+ 5y
13
+ ··· + 4352y + 1024)
c
9
(y
8
+ 7y
7
+ 18y
6
+ 4y
5
− 13y
4
+ 4y
3
+ 6y
2
− 4y + 1)
· (y
10
+ 19y
9
+ ··· + 3608y + 4489)
· (y
14
+ 29y
13
+ ··· − 6924y + 1849)
16
