12n
0234
(K12n
0234
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 12 3 1 6 7 8
Solving Sequence
7,11
12
3,8
4 1 6 10 5 2 9
c
11
c
7
c
3
c
12
c
6
c
10
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
23
+ 16u
21
+ ··· + b − 1, −u
22
+ 15u
20
+ ··· + a + 1, u
24
− 2u
23
+ ··· + u − 1i
I
u
2
= hu
2
+ b − 1, a + 1, u
3
+ u
2
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 27 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
23
+16u
21
+· · ·+b−1, −u
22
+15u
20
+· · ·+a+1, u
24
−2u
23
+· · ·+u−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
3
=
u
22
− 15u
20
+ ··· − 6u − 1
u
23
− 16u
21
+ ··· + u + 1
a
8
=
−u
−u
3
+ u
a
4
=
−2u
23
+ 2u
22
+ ··· − 6u − 2
−3u
23
+ u
22
+ ··· + 2u − 1
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
u
a
10
=
−u
2
+ 1
−u
2
a
5
=
−u
3
+ 2u
−u
3
+ u
a
2
=
−u
23
+ u
22
+ ··· − 5u − 1
−u
23
+ 15u
21
+ ··· + 8u
2
+ 2u
a
9
=
u
8
− 5u
6
+ 7u
4
− 4u
2
+ 1
u
10
− 6u
8
+ 11u
6
− 6u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
23
− 4u
22
− 16u
21
+ 62u
20
+ 110u
19
− 405u
18
− 420u
17
+
1453u
16
+ 940u
15
− 3134u
14
− 1118u
13
+ 4197u
12
+ 260u
11
− 3495u
10
+ 1034u
9
+
1741u
8
− 1269u
7
− 400u
6
+ 550u
5
− 66u
4
− 92u
3
+ 43u
2
+ 22u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 8u
23
+ ··· + 34u + 1
c
2
, c
4
u
24
− 4u
23
+ ··· + 6u − 1
c
3
, c
8
u
24
+ u
23
+ ··· + 36u + 8
c
5
, c
6
, c
7
c
10
, c
11
, c
12
u
24
− 2u
23
+ ··· + u − 1
c
9
u
24
+ 2u
23
+ ··· + 7u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 20y
23
+ ··· − 534y + 1
c
2
, c
4
y
24
− 8y
23
+ ··· − 34y + 1
c
3
, c
8
y
24
− 21y
23
+ ··· − 1040y + 64
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
24
− 34y
23
+ ··· − 21y + 1
c
9
y
24
+ 26y
23
+ ··· − 21y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.065440 + 0.105091I
a = −0.101056 − 1.017940I
b = 0.427572 + 0.787716I
−4.78298 − 2.15986I −14.8245 + 3.7042I
u = 1.065440 − 0.105091I
a = −0.101056 + 1.017940I
b = 0.427572 − 0.787716I
−4.78298 + 2.15986I −14.8245 − 3.7042I
u = −1.044560 + 0.293303I
a = 0.060787 − 0.485482I
b = 0.951447 + 0.437799I
0.73559 + 1.57187I −10.92931 − 1.48898I
u = −1.044560 − 0.293303I
a = 0.060787 + 0.485482I
b = 0.951447 − 0.437799I
0.73559 − 1.57187I −10.92931 + 1.48898I
u = −1.08841
a = −1.48671
b = −1.10653
−6.35267 −13.5900
u = −1.158950 + 0.291958I
a = 0.215884 + 1.148480I
b = −0.234965 − 0.552300I
−0.44591 + 7.86280I −12.55352 − 5.99165I
u = −1.158950 − 0.291958I
a = 0.215884 − 1.148480I
b = −0.234965 + 0.552300I
−0.44591 − 7.86280I −12.55352 + 5.99165I
u = 1.29716
a = −0.674636
b = 0.0175724
−7.12889 −9.59420
u = 0.413733 + 0.547171I
a = −0.390896 + 1.056560I
b = −0.969122 + 0.621169I
4.52862 − 4.98340I −8.25902 + 6.13145I
u = 0.413733 − 0.547171I
a = −0.390896 − 1.056560I
b = −0.969122 − 0.621169I
4.52862 + 4.98340I −8.25902 − 6.13145I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.292302 + 0.564711I
a = 0.90790 − 1.36582I
b = 0.567413 + 0.008562I
4.89005 + 1.34187I −6.87930 + 0.47018I
u = 0.292302 − 0.564711I
a = 0.90790 + 1.36582I
b = 0.567413 − 0.008562I
4.89005 − 1.34187I −6.87930 − 0.47018I
u = −0.544664
a = 0.533142
b = 0.444791
−0.910827 −10.5330
u = −0.260496 + 0.277785I
a = 1.25655 − 0.85026I
b = −0.092184 − 0.632306I
−0.635329 + 0.918549I −9.38568 − 7.31949I
u = −0.260496 − 0.277785I
a = 1.25655 + 0.85026I
b = −0.092184 + 0.632306I
−0.635329 − 0.918549I −9.38568 + 7.31949I
u = 0.266177
a = −2.54350
b = 0.862360
−2.02344 2.35380
u = 1.73483 + 0.06615I
a = −1.05402 + 1.43450I
b = −1.58632 + 2.38877I
−9.17857 − 2.99479I −11.67464 + 0.80624I
u = 1.73483 − 0.06615I
a = −1.05402 − 1.43450I
b = −1.58632 − 2.38877I
−9.17857 + 2.99479I −11.67464 − 0.80624I
u = −1.74821 + 0.02394I
a = −0.42595 + 2.65836I
b = −0.53152 + 4.79437I
−14.9721 + 2.6782I −14.7087 − 2.4859I
u = −1.74821 − 0.02394I
a = −0.42595 − 2.65836I
b = −0.53152 − 4.79437I
−14.9721 − 2.6782I −14.7087 + 2.4859I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.75360
a = −0.586507
b = −1.87176
−16.6700 −14.3680
u = 1.76989 + 0.07670I
a = 1.04117 − 2.39816I
b = 2.07334 − 4.41830I
−11.0119 − 9.4629I −13.6249 + 4.9785I
u = 1.76989 − 0.07670I
a = 1.04117 + 2.39816I
b = 2.07334 + 4.41830I
−11.0119 + 9.4629I −13.6249 − 4.9785I
u = −1.81183
a = −1.26255
b = −2.55776
−18.6696 −7.58970
7
II. I
u
2
= hu
2
+ b − 1, a + 1, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
3
=
−1
−u
2
+ 1
a
8
=
−u
u
2
− u − 1
a
4
=
−1
−u
2
+ 1
a
1
=
−u
2
+ 1
−u
2
+ u + 1
a
6
=
u
u
a
10
=
−u
2
+ 1
−u
2
a
5
=
u
2
− 1
u
2
− u − 1
a
2
=
−u
2
−2u
2
+ u + 2
a
9
=
−u
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ u − 23
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
c
9
u
3
− u
2
− 2u + 1
c
10
, c
11
, c
12
u
3
+ u
2
− 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y − 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −1.00000
b = −0.554958
−7.98968 −20.1980
u = −0.445042
a = −1.00000
b = 0.801938
−2.34991 −23.2470
u = −1.80194
a = −1.00000
b = −2.24698
−19.2692 −21.5550
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
24
+ 8u
23
+ ··· + 34u + 1)
c
2
((u − 1)
3
)(u
24
− 4u
23
+ ··· + 6u − 1)
c
3
, c
8
u
3
(u
24
+ u
23
+ ··· + 36u + 8)
c
4
((u + 1)
3
)(u
24
− 4u
23
+ ··· + 6u − 1)
c
5
, c
6
, c
7
(u
3
− u
2
− 2u + 1)(u
24
− 2u
23
+ ··· + u − 1)
c
9
(u
3
− u
2
− 2u + 1)(u
24
+ 2u
23
+ ··· + 7u + 1)
c
10
, c
11
, c
12
(u
3
+ u
2
− 2u − 1)(u
24
− 2u
23
+ ··· + u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
3
)(y
24
+ 20y
23
+ ··· − 534y + 1)
c
2
, c
4
((y − 1)
3
)(y
24
− 8y
23
+ ··· − 34y + 1)
c
3
, c
8
y
3
(y
24
− 21y
23
+ ··· − 1040y + 64)
c
5
, c
6
, c
7
c
10
, c
11
, c
12
(y
3
− 5y
2
+ 6y − 1)(y
24
− 34y
23
+ ··· − 21y + 1)
c
9
(y
3
− 5y
2
+ 6y − 1)(y
24
+ 26y
23
+ ··· − 21y + 1)
13
