11n
135
(K11n
135
)
A knot diagram
1
Linearized knot diagam
7 1 9 11 10 2 3 11 5 3 5
Solving Sequence
8,11 5,9
1 4 3 2 7 6 10
c
8
c
11
c
4
c
3
c
2
c
7
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
9
− 24u
8
− 34u
7
+ 183u
6
− 48u
5
− 359u
4
+ 366u
3
− 130u
2
+ b + 40u − 13,
− 13u
9
+ 45u
8
+ 63u
7
− 343u
6
+ 90u
5
+ 672u
4
− 681u
3
+ 245u
2
+ a − 78u + 25,
u
10
− 4u
9
− 3u
8
+ 29u
7
− 21u
6
− 48u
5
+ 80u
4
− 47u
3
+ 16u
2
− 5u + 1i
I
u
2
= h−2u
7
− 11u
6
− 18u
5
− 6u
4
+ u
3
− 5u
2
+ b − u − 2, 2u
7
+ 12u
6
+ 23u
5
+ 12u
4
− 4u
3
+ u
2
+ a + 5u + 3,
u
8
+ 7u
7
+ 17u
6
+ 15u
5
+ u
4
+ 5u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 18 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
= h7u
9
−24u
8
+· · ·+ b −13, −13u
9
+45u
8
+· · ·+ a + 25, u
10
−4u
9
+· · ·−5u+1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
5
=
13u
9
− 45u
8
+ ··· + 78u − 25
−7u
9
+ 24u
8
+ ··· − 40u + 13
a
9
=
1
−u
2
a
1
=
−u
8
+ u
7
+ 8u
6
− 7u
5
− 16u
4
+ 14u
3
− 6u
2
+ u − 1
u
9
− u
8
− 8u
7
+ 7u
6
+ 16u
5
− 14u
4
+ 6u
3
− u
2
+ 2u
a
4
=
13u
9
− 45u
8
+ ··· + 78u − 25
−11u
9
+ 37u
8
+ ··· − 62u + 20
a
3
=
6u
9
− 21u
8
+ ··· + 38u − 12
−8u
9
+ 29u
8
+ ··· − 49u + 16
a
2
=
8u
9
− 27u
8
+ ··· + 42u − 11
−10u
9
+ 36u
8
+ ··· − 56u + 16
a
7
=
u
9
− 2u
8
− 7u
7
+ 15u
6
+ 9u
5
− 30u
4
+ 20u
3
− 7u
2
+ 3u + 1
−u
9
+ u
8
+ 8u
7
− 7u
6
− 16u
5
+ 14u
4
− 7u
3
− u
a
6
=
−14u
9
+ 49u
8
+ ··· − 84u + 27
5u
9
− 21u
8
+ ··· + 36u − 12
a
10
=
u
9
− 2u
8
− 7u
7
+ 15u
6
+ 9u
5
− 30u
4
+ 20u
3
− 7u
2
+ 2u
−2u
9
+ 4u
8
+ 14u
7
− 30u
6
− 18u
5
+ 60u
4
− 40u
3
+ 13u
2
− 4u + 1
a
10
=
u
9
− 2u
8
− 7u
7
+ 15u
6
+ 9u
5
− 30u
4
+ 20u
3
− 7u
2
+ 2u
−2u
9
+ 4u
8
+ 14u
7
− 30u
6
− 18u
5
+ 60u
4
− 40u
3
+ 13u
2
− 4u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −34u
9
+ 113u
8
+ 180u
7
− 868u
6
+ 117u
5
+ 1739u
4
− 1537u
3
+ 504u
2
− 157u + 39
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
10
+ 6u
9
+ ··· + 14u + 4
c
2
u
10
+ 4u
9
+ ··· + 36u + 16
c
3
, c
4
, c
11
u
10
+ u
9
− 10u
8
− 30u
7
+ 42u
6
+ 25u
5
− 18u
4
− 7u
3
− 10u
2
− 2u − 1
c
5
, c
9
, c
10
u
10
− 2u
9
− 12u
8
+ 41u
7
+ 4u
6
+ 65u
5
+ 12u
4
− 18u
3
− 8u
2
+ u + 1
c
7
u
10
− 6u
9
+ ··· − 42u + 180
c
8
u
10
+ 4u
9
− 3u
8
− 29u
7
− 21u
6
+ 48u
5
+ 80u
4
+ 47u
3
+ 16u
2
+ 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
10
+ 4y
9
+ ··· + 36y + 16
c
2
y
10
+ 4y
9
+ ··· − 1648y + 256
c
3
, c
4
, c
11
y
10
− 21y
9
+ ··· + 16y + 1
c
5
, c
9
, c
10
y
10
− 28y
9
+ ··· − 17y + 1
c
7
y
10
− 56y
9
+ ··· + 244836y + 32400
c
8
y
10
− 22y
9
+ ··· + 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.636857 + 0.087196I
a = 1.74049 + 0.14257I
b = −1.096010 − 0.242560I
−4.86323 − 4.26845I −7.00188 + 6.39401I
u = 0.636857 − 0.087196I
a = 1.74049 − 0.14257I
b = −1.096010 + 0.242560I
−4.86323 + 4.26845I −7.00188 − 6.39401I
u = 0.517366
a = −1.64352
b = 0.850302
−1.55504 −5.61010
u = −0.028208 + 0.344442I
a = −0.952320 + 0.624641I
b = 0.188290 + 0.345639I
−0.422559 − 0.990373I −6.71540 + 6.78739I
u = −0.028208 − 0.344442I
a = −0.952320 − 0.624641I
b = 0.188290 − 0.345639I
−0.422559 + 0.990373I −6.71540 − 6.78739I
u = −2.02110 + 0.32502I
a = −0.034489 + 0.196626I
b = −0.005798 + 0.408612I
5.72141 − 3.10928I −8.09311 + 4.32692I
u = −2.02110 − 0.32502I
a = −0.034489 − 0.196626I
b = −0.005798 − 0.408612I
5.72141 + 3.10928I −8.09311 − 4.32692I
u = 2.07184 + 0.16391I
a = −1.21022 + 1.00746I
b = 2.67252 − 1.88893I
−17.7391 + 8.0399I −7.30663 − 2.83159I
u = 2.07184 − 0.16391I
a = −1.21022 − 1.00746I
b = 2.67252 + 1.88893I
−17.7391 − 8.0399I −7.30663 + 2.83159I
u = 2.16387
a = 1.55661
b = −3.36830
−13.1860 −6.15590
5
II. I
u
2
= h−2u
7
− 11u
6
+ · · · + b − 2, 2u
7
+ 12u
6
+ · · · + a + 3, u
8
+ 7u
7
+
17u
6
+ 15u
5
+ u
4
+ 5u
2
+ 2u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
5
=
−2u
7
− 12u
6
− 23u
5
− 12u
4
+ 4u
3
− u
2
− 5u − 3
2u
7
+ 11u
6
+ 18u
5
+ 6u
4
− u
3
+ 5u
2
+ u + 2
a
9
=
1
−u
2
a
1
=
u
6
+ 6u
5
+ 11u
4
+ 4u
3
− 3u
2
+ 3u + 2
u
7
+ 6u
6
+ 11u
5
+ 4u
4
− 3u
3
+ 3u
2
+ 3u
a
4
=
−2u
7
− 12u
6
− 23u
5
− 12u
4
+ 4u
3
− u
2
− 5u − 3
5u
7
+ 27u
6
+ 42u
5
+ 9u
4
− 6u
3
+ 14u
2
+ 3u + 4
a
3
=
−u
6
− 5u
5
− 6u
4
+ 3u
3
+ 4u
2
− 4u − 1
u
4
+ 3u
3
+ u
2
− u + 1
a
2
=
−u
6
− 4u
5
− u
4
+ 9u
3
+ 2u
2
− 6u + 2
u
7
+ 7u
6
+ 16u
5
+ 12u
4
+ u
2
+ 4u + 2
a
7
=
u
7
+ 7u
6
+ 17u
5
+ 15u
4
+ u
3
+ 6u + 4
u
7
+ 6u
6
+ 11u
5
+ 4u
4
− 4u
3
+ 2u
2
+ 4u
a
6
=
−u
7
− 6u
6
− 12u
5
− 9u
4
− 3u
3
− u
2
− 2
−u
5
− 3u
4
− u
3
+ u
2
− u
a
10
=
−u
7
− 7u
6
− 17u
5
− 15u
4
− u
3
− 5u − 1
−u
2
+ 1
a
10
=
−u
7
− 7u
6
− 17u
5
− 15u
4
− u
3
− 5u − 1
−u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
7
+ 3u
6
− 4u
5
− 18u
4
− 9u
3
+ 7u
2
− 3u − 14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 2u
6
+ 3u
4
− u
3
+ 2u
2
− u + 1
c
2
u
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 19u
4
+ 15u
3
+ 8u
2
+ 3u + 1
c
3
, c
11
u
8
+ u
6
+ u
5
− 2u
4
− u + 1
c
4
u
8
+ u
6
− u
5
− 2u
4
+ u + 1
c
5
, c
10
u
8
− u
7
− 2u
4
+ u
3
+ u
2
+ 1
c
6
u
8
+ 2u
6
+ 3u
4
+ u
3
+ 2u
2
+ u + 1
c
7
u
8
+ 2u
6
− 5u
5
+ u
4
+ u
3
+ 5u
2
+ 3u + 1
c
8
u
8
+ 7u
7
+ 17u
6
+ 15u
5
+ u
4
+ 5u
2
+ 2u + 1
c
9
u
8
+ u
7
− 2u
4
− u
3
+ u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ 4y
7
+ 10y
6
+ 16y
5
+ 19y
4
+ 15y
3
+ 8y
2
+ 3y + 1
c
2
y
8
+ 4y
7
+ 10y
6
+ 20y
5
+ 19y
4
+ 3y
3
+ 12y
2
+ 7y + 1
c
3
, c
4
, c
11
y
8
+ 2y
7
− 3y
6
− 5y
5
+ 6y
4
+ 4y
3
− 4y
2
− y + 1
c
5
, c
9
, c
10
y
8
− y
7
− 4y
6
+ 4y
5
+ 6y
4
− 5y
3
− 3y
2
+ 2y + 1
c
7
y
8
+ 4y
7
+ 6y
6
− 11y
5
+ 33y
4
+ 43y
3
+ 21y
2
+ y + 1
c
8
y
8
− 15y
7
+ 81y
6
− 181y
5
+ 145y
4
− 16y
3
+ 27y
2
+ 6y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500771 + 0.460860I
a = −0.735484 + 0.913410I
b = −0.789263 + 0.118455I
−3.52853 − 0.48963I −9.23600 − 1.05814I
u = 0.500771 − 0.460860I
a = −0.735484 − 0.913410I
b = −0.789263 − 0.118455I
−3.52853 + 0.48963I −9.23600 + 1.05814I
u = −1.50739 + 0.11112I
a = 0.165592 − 0.902942I
b = −0.149281 + 1.379480I
2.76707 + 1.04226I −7.14108 + 0.01449I
u = −1.50739 − 0.11112I
a = 0.165592 + 0.902942I
b = −0.149281 − 1.379480I
2.76707 − 1.04226I −7.14108 − 0.01449I
u = −0.172493 + 0.378694I
a = −1.50843 − 2.01752I
b = 1.024220 − 0.223225I
−5.60402 + 3.77609I −14.7696 − 2.3802I
u = −0.172493 − 0.378694I
a = −1.50843 + 2.01752I
b = 1.024220 + 0.223225I
−5.60402 − 3.77609I −14.7696 + 2.3802I
u = −2.32089 + 0.26670I
a = 0.078321 − 0.360330I
b = −0.085673 + 0.857175I
6.36547 − 2.93267I 4.14670 + 1.68828I
u = −2.32089 − 0.26670I
a = 0.078321 + 0.360330I
b = −0.085673 − 0.857175I
6.36547 + 2.93267I 4.14670 − 1.68828I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
+ 2u
6
+ 3u
4
− u
3
+ 2u
2
− u + 1)(u
10
+ 6u
9
+ ··· + 14u + 4)
c
2
(u
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 19u
4
+ 15u
3
+ 8u
2
+ 3u + 1)
· (u
10
+ 4u
9
+ ··· + 36u + 16)
c
3
, c
11
(u
8
+ u
6
+ u
5
− 2u
4
− u + 1)
· (u
10
+ u
9
− 10u
8
− 30u
7
+ 42u
6
+ 25u
5
− 18u
4
− 7u
3
− 10u
2
− 2u − 1)
c
4
(u
8
+ u
6
− u
5
− 2u
4
+ u + 1)
· (u
10
+ u
9
− 10u
8
− 30u
7
+ 42u
6
+ 25u
5
− 18u
4
− 7u
3
− 10u
2
− 2u − 1)
c
5
, c
10
(u
8
− u
7
− 2u
4
+ u
3
+ u
2
+ 1)
· (u
10
− 2u
9
− 12u
8
+ 41u
7
+ 4u
6
+ 65u
5
+ 12u
4
− 18u
3
− 8u
2
+ u + 1)
c
6
(u
8
+ 2u
6
+ 3u
4
+ u
3
+ 2u
2
+ u + 1)(u
10
+ 6u
9
+ ··· + 14u + 4)
c
7
(u
8
+ 2u
6
+ ··· + 3u + 1)(u
10
− 6u
9
+ ··· − 42u + 180)
c
8
(u
8
+ 7u
7
+ 17u
6
+ 15u
5
+ u
4
+ 5u
2
+ 2u + 1)
· (u
10
+ 4u
9
− 3u
8
− 29u
7
− 21u
6
+ 48u
5
+ 80u
4
+ 47u
3
+ 16u
2
+ 5u + 1)
c
9
(u
8
+ u
7
− 2u
4
− u
3
+ u
2
+ 1)
· (u
10
− 2u
9
− 12u
8
+ 41u
7
+ 4u
6
+ 65u
5
+ 12u
4
− 18u
3
− 8u
2
+ u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
8
+ 4y
7
+ 10y
6
+ 16y
5
+ 19y
4
+ 15y
3
+ 8y
2
+ 3y + 1)
· (y
10
+ 4y
9
+ ··· + 36y + 16)
c
2
(y
8
+ 4y
7
+ 10y
6
+ 20y
5
+ 19y
4
+ 3y
3
+ 12y
2
+ 7y + 1)
· (y
10
+ 4y
9
+ ··· − 1648y + 256)
c
3
, c
4
, c
11
(y
8
+ 2y
7
− 3y
6
− 5y
5
+ 6y
4
+ 4y
3
− 4y
2
− y + 1)
· (y
10
− 21y
9
+ ··· + 16y + 1)
c
5
, c
9
, c
10
(y
8
− y
7
− 4y
6
+ 4y
5
+ 6y
4
− 5y
3
− 3y
2
+ 2y + 1)
· (y
10
− 28y
9
+ ··· − 17y + 1)
c
7
(y
8
+ 4y
7
+ 6y
6
− 11y
5
+ 33y
4
+ 43y
3
+ 21y
2
+ y + 1)
· (y
10
− 56y
9
+ ··· + 244836y + 32400)
c
8
(y
8
− 15y
7
+ 81y
6
− 181y
5
+ 145y
4
− 16y
3
+ 27y
2
+ 6y + 1)
· (y
10
− 22y
9
+ ··· + 7y + 1)
11
