12n
0749
(K12n
0749
)
A knot diagram
1
Linearized knot diagam
4 6 8 11 2 12 3 12 1 4 6 9
Solving Sequence
8,12
9 1
4,10
3 7 6 2 5 11
c
8
c
12
c
9
c
3
c
7
c
6
c
2
c
5
c
11
c
1
, c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
7
+ u
6
− 5u
5
− 5u
4
+ 6u
3
+ 8u
2
+ b + u − 1, −u
7
− u
6
+ 5u
5
+ 5u
4
− 6u
3
− 8u
2
+ a − u + 2,
u
9
+ 3u
8
− 3u
7
− 15u
6
− 3u
5
+ 22u
4
+ 15u
3
− 5u
2
− 5u + 1i
I
u
2
= h−u
4
+ u
3
+ 3u
2
+ b − u − 2, u
4
− u
3
− 3u
2
+ a + u + 3, u
6
− 2u
5
− 3u
4
+ 5u
3
+ 4u
2
− 3u − 1i
* 2 irreducible components of dim
C
= 0, with total 15 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
7
+ u
6
− 5u
5
− 5u
4
+ 6u
3
+ 8u
2
+ b + u − 1, −u
7
− u
6
+ 5u
5
+
5u
4
− 6u
3
− 8u
2
+ a − u + 2, u
9
+ 3u
8
+ · · · − 5u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
u
7
+ u
6
− 5u
5
− 5u
4
+ 6u
3
+ 8u
2
+ u − 2
−u
7
− u
6
+ 5u
5
+ 5u
4
− 6u
3
− 8u
2
− u + 1
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−1
−u
7
− u
6
+ 5u
5
+ 5u
4
− 6u
3
− 8u
2
− u + 1
a
7
=
−u
7
− u
6
+ 5u
5
+ 5u
4
− 6u
3
− 8u
2
− u + 2
u
8
− 6u
6
+ 12u
4
+ 3u
3
− 9u
2
− 5u + 1
a
6
=
−u
7
− u
6
+ 5u
5
+ 5u
4
− 6u
3
− 8u
2
− u + 2
−u
8
− 2u
7
+ 4u
6
+ 9u
5
− 2u
4
− 11u
3
− 6u
2
a
2
=
u
8
+ u
7
− 5u
6
− 5u
5
+ 7u
4
+ 8u
3
− u
2
− 2u
−3u
8
− 3u
7
+ 16u
6
+ 15u
5
− 25u
4
− 26u
3
+ 8u
2
+ 12u − 2
a
5
=
7u
8
+ 7u
7
− 37u
6
− 33u
5
+ 56u
4
+ 54u
3
− 17u
2
− 22u + 6
−2u
8
− 3u
7
+ 11u
6
+ 13u
5
− 17u
4
− 19u
3
+ 5u
2
+ 8u − 2
a
11
=
u
8
+ u
7
− 5u
6
− 5u
5
+ 7u
4
+ 8u
3
− 2u
2
− 3u + 1
−3u
8
− 3u
7
+ 16u
6
+ 15u
5
− 24u
4
− 25u
3
+ 6u
2
+ 11u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
8
+ 11u
7
− 6u
6
− 54u
5
− 23u
4
+ 73u
3
+ 62u
2
− 4u − 17
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
u
9
− u
8
+ 18u
7
+ 15u
6
− 15u
5
+ 3u
4
− 13u
3
− 4u
2
− 2u − 1
c
2
, c
5
u
9
+ 15u
7
− 31u
6
+ 42u
5
− 72u
4
+ 30u
3
+ 10u
2
− 7u + 1
c
3
, c
7
u
9
− 9u
8
+ 38u
7
− 94u
6
+ 144u
5
− 132u
4
+ 57u
3
+ 8u
2
− 20u + 8
c
6
, c
11
u
9
+ 2u
8
+ 12u
7
+ 4u
6
+ 39u
5
+ 14u
4
+ 8u
3
+ 11u
2
− u − 1
c
8
, c
9
, c
12
u
9
− 3u
8
− 3u
7
+ 15u
6
− 3u
5
− 22u
4
+ 15u
3
+ 5u
2
− 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
y
9
+ 35y
8
+ ··· − 4y −1
c
2
, c
5
y
9
+ 30y
8
+ ··· + 29y −1
c
3
, c
7
y
9
− 5y
8
+ ··· + 272y −64
c
6
, c
11
y
9
+ 20y
8
+ ··· + 23y −1
c
8
, c
9
, c
12
y
9
− 15y
8
+ ··· + 35y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.803718 + 0.480044I
a = −0.545358 + 0.558768I
b = −0.454642 − 0.558768I
1.43676 − 1.39156I −1.67156 + 5.14855I
u = −0.803718 − 0.480044I
a = −0.545358 − 0.558768I
b = −0.454642 + 0.558768I
1.43676 + 1.39156I −1.67156 − 5.14855I
u = −1.39574
a = 0.458311
b = −1.45831
−1.87529 −5.03640
u = 0.479009
a = 0.602594
b = −1.60259
−8.12479 0.759950
u = 1.56290 + 0.23534I
a = 0.115329 − 1.184520I
b = −1.11533 + 1.18452I
9.69068 + 4.28297I −4.64998 − 2.95733I
u = 1.56290 − 0.23534I
a = 0.115329 + 1.184520I
b = −1.11533 − 1.18452I
9.69068 − 4.28297I −4.64998 + 2.95733I
u = 0.189912
a = −1.48814
b = 0.488141
−0.677543 −15.0670
u = −1.89577 + 0.05938I
a = 0.64365 + 1.55034I
b = −1.64365 − 1.55034I
−16.4807 − 5.9861I −4.50677 + 1.85792I
u = −1.89577 − 0.05938I
a = 0.64365 − 1.55034I
b = −1.64365 + 1.55034I
−16.4807 + 5.9861I −4.50677 − 1.85792I
5
II. I
u
2
= h−u
4
+ u
3
+ 3u
2
+ b − u − 2, u
4
− u
3
− 3u
2
+ a + u + 3, u
6
− 2u
5
−
3u
4
+ 5u
3
+ 4u
2
− 3u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−u
4
+ u
3
+ 3u
2
− u − 3
u
4
− u
3
− 3u
2
+ u + 2
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−1
u
4
− u
3
− 3u
2
+ u + 2
a
7
=
u
4
− u
3
− 3u
2
+ u + 3
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u − 2
a
6
=
u
4
− u
3
− 3u
2
+ u + 3
−u
4
+ u
3
+ 3u
2
− u − 3
a
2
=
−u
5
+ 2u
4
+ 3u
3
− 5u
2
− 3u + 2
u
5
− u
4
− 4u
3
+ 2u
2
+ 4u − 1
a
5
=
u
5
− u
4
− 3u
3
+ u
2
+ 3u
−2u
5
+ u
4
+ 6u
3
− 4u − 1
a
11
=
u
5
− 2u
4
− 3u
3
+ 4u
2
+ 4u − 1
−u
5
+ 2u
4
+ 3u
3
− 4u
2
− 3u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
− 11u
4
− 9u
3
+ 30u
2
+ 10u − 24
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
6
+ 2u
4
− 4u
3
− 3u
2
+ 4u − 1
c
2
u
6
− u
5
+ 2u
4
− 5u
2
+ 5u − 1
c
3
u
6
− u
5
− 2u
4
+ 4u
3
− 2u + 1
c
5
u
6
+ u
5
+ 2u
4
− 5u
2
− 5u − 1
c
6
u
6
+ u
5
+ u
4
− 2u
3
− 4u
2
− 3u − 1
c
7
u
6
+ u
5
− 2u
4
− 4u
3
+ 2u + 1
c
8
, c
9
u
6
− 2u
5
− 3u
4
+ 5u
3
+ 4u
2
− 3u − 1
c
10
u
6
+ 2u
4
+ 4u
3
− 3u
2
− 4u − 1
c
11
u
6
− u
5
+ u
4
+ 2u
3
− 4u
2
+ 3u − 1
c
12
u
6
+ 2u
5
− 3u
4
− 5u
3
+ 4u
2
+ 3u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
y
6
+ 4y
5
− 2y
4
− 30y
3
+ 37y
2
− 10y + 1
c
2
, c
5
y
6
+ 3y
5
− 6y
4
− 12y
3
+ 21y
2
− 15y + 1
c
3
, c
7
y
6
− 5y
5
+ 12y
4
− 18y
3
+ 12y
2
− 4y + 1
c
6
, c
11
y
6
+ y
5
− 3y
4
− 8y
3
+ 2y
2
− y + 1
c
8
, c
9
, c
12
y
6
− 10y
5
+ 37y
4
− 63y
3
+ 52y
2
− 17y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.123140 + 0.280028I
a = −0.484226 + 0.358962I
b = −0.515774 − 0.358962I
1.070880 − 0.298492I −3.25325 − 1.22821I
u = −1.123140 − 0.280028I
a = −0.484226 − 0.358962I
b = −0.515774 + 0.358962I
1.070880 + 0.298492I −3.25325 + 1.22821I
u = 0.779219
a = −1.85322
b = 0.853215
−5.05469 −5.15680
u = −0.272443
a = −2.53061
b = 1.53061
−8.45292 −24.3820
u = 1.86975 + 0.14034I
a = 0.176141 − 0.745556I
b = −1.176140 + 0.745556I
12.26270 + 2.92755I −2.47722 − 2.29256I
u = 1.86975 − 0.14034I
a = 0.176141 + 0.745556I
b = −1.176140 − 0.745556I
12.26270 − 2.92755I −2.47722 + 2.29256I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
6
+ 2u
4
− 4u
3
− 3u
2
+ 4u − 1)
· (u
9
− u
8
+ 18u
7
+ 15u
6
− 15u
5
+ 3u
4
− 13u
3
− 4u
2
− 2u − 1)
c
2
(u
6
− u
5
+ 2u
4
− 5u
2
+ 5u − 1)
· (u
9
+ 15u
7
− 31u
6
+ 42u
5
− 72u
4
+ 30u
3
+ 10u
2
− 7u + 1)
c
3
(u
6
− u
5
− 2u
4
+ 4u
3
− 2u + 1)
· (u
9
− 9u
8
+ 38u
7
− 94u
6
+ 144u
5
− 132u
4
+ 57u
3
+ 8u
2
− 20u + 8)
c
5
(u
6
+ u
5
+ 2u
4
− 5u
2
− 5u − 1)
· (u
9
+ 15u
7
− 31u
6
+ 42u
5
− 72u
4
+ 30u
3
+ 10u
2
− 7u + 1)
c
6
(u
6
+ u
5
+ u
4
− 2u
3
− 4u
2
− 3u − 1)
· (u
9
+ 2u
8
+ 12u
7
+ 4u
6
+ 39u
5
+ 14u
4
+ 8u
3
+ 11u
2
− u − 1)
c
7
(u
6
+ u
5
− 2u
4
− 4u
3
+ 2u + 1)
· (u
9
− 9u
8
+ 38u
7
− 94u
6
+ 144u
5
− 132u
4
+ 57u
3
+ 8u
2
− 20u + 8)
c
8
, c
9
(u
6
− 2u
5
− 3u
4
+ 5u
3
+ 4u
2
− 3u − 1)
· (u
9
− 3u
8
− 3u
7
+ 15u
6
− 3u
5
− 22u
4
+ 15u
3
+ 5u
2
− 5u − 1)
c
10
(u
6
+ 2u
4
+ 4u
3
− 3u
2
− 4u − 1)
· (u
9
− u
8
+ 18u
7
+ 15u
6
− 15u
5
+ 3u
4
− 13u
3
− 4u
2
− 2u − 1)
c
11
(u
6
− u
5
+ u
4
+ 2u
3
− 4u
2
+ 3u − 1)
· (u
9
+ 2u
8
+ 12u
7
+ 4u
6
+ 39u
5
+ 14u
4
+ 8u
3
+ 11u
2
− u − 1)
c
12
(u
6
+ 2u
5
− 3u
4
− 5u
3
+ 4u
2
+ 3u − 1)
· (u
9
− 3u
8
− 3u
7
+ 15u
6
− 3u
5
− 22u
4
+ 15u
3
+ 5u
2
− 5u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
(y
6
+ 4y
5
+ ··· − 10y + 1)(y
9
+ 35y
8
+ ··· − 4y −1)
c
2
, c
5
(y
6
+ 3y
5
+ ··· − 15y + 1)(y
9
+ 30y
8
+ ··· + 29y −1)
c
3
, c
7
(y
6
− 5y
5
+ ··· − 4y + 1)(y
9
− 5y
8
+ ··· + 272y −64)
c
6
, c
11
(y
6
+ y
5
− 3y
4
− 8y
3
+ 2y
2
− y + 1)(y
9
+ 20y
8
+ ··· + 23y −1)
c
8
, c
9
, c
12
(y
6
− 10y
5
+ 37y
4
− 63y
3
+ 52y
2
− 17y + 1)
· (y
9
− 15y
8
+ ··· + 35y −1)
11
