12a
0800
(K12a
0800
)
A knot diagram
1
Linearized knot diagam
3 8 12 1 10 11 9 2 7 6 5 4
Solving Sequence
5,10
6 11 7
1,12
4 3 9 8 2
c
5
c
10
c
6
c
11
c
4
c
3
c
9
c
7
c
2
c
1
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u,
u
16
− u
15
− 6u
14
+ 5u
13
+ 14u
12
− 8u
11
− 12u
10
− u
9
− 6u
8
+ 14u
7
+ 16u
6
− 6u
5
− 4u
4
− 8u
3
− 4u
2
+ a + 2u,
u
17
− u
16
+ ··· + 4u + 1i
I
u
2
= hu
29
− 10u
27
+ ··· + b + 1, −u
28
+ 9u
26
+ ··· + a + 1, u
30
− u
29
+ ··· + 2u − 1i
I
u
3
= hb + 1, a, u − 1i
* 3 irreducible components of dim
C
= 0, with total 48 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
= hb − u, u
16
− u
15
+ · · · + a + 2u, u
17
− u
16
+ · · · + 4u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
−u
16
+ u
15
+ ··· + 4u
2
− 2u
u
a
12
=
u
3
− 2u
−u
3
+ u
a
4
=
u
15
− u
14
+ ··· − 7u
2
− 4u
−u
2
a
3
=
u
15
− u
14
+ ··· − 5u
2
− 4u
u
4
− 2u
2
a
9
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
−u
8
+ 3u
6
− 3u
4
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
2
=
−u
16
+ u
15
+ ··· + 4u
2
− 3u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
16
+ 6u
15
+ 12u
14
− 38u
13
− 32u
12
+ 92u
11
+ 48u
10
− 78u
9
−
40u
8
− 56u
7
+ 132u
5
+ 48u
4
− 24u
3
− 40u
2
− 50u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
c
11
u
17
+ 3u
16
+ ··· − 20u − 4
c
2
, c
8
u
17
− 3u
16
+ ··· + 4u − 2
c
3
, c
4
, c
5
c
6
, c
10
, c
12
u
17
− u
16
+ ··· + 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
c
11
y
17
+ 19y
16
+ ··· + 8y −16
c
2
, c
8
y
17
+ 3y
16
+ ··· − 20y −4
c
3
, c
4
, c
5
c
6
, c
10
, c
12
y
17
− 15y
16
+ ··· − 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.015819 + 0.919296I
a = 0.01726 − 1.91592I
b = −0.015819 + 0.919296I
12.59050 + 3.33698I −1.74242 − 2.42496I
u = −0.015819 − 0.919296I
a = 0.01726 + 1.91592I
b = −0.015819 − 0.919296I
12.59050 − 3.33698I −1.74242 + 2.42496I
u = −1.25414
a = −3.12432
b = −1.25414
−6.31627 −14.3970
u = −1.245520 + 0.229336I
a = −1.37037 − 2.30613I
b = −1.245520 + 0.229336I
−3.83323 + 4.04550I −10.81210 − 4.36543I
u = −1.245520 − 0.229336I
a = −1.37037 + 2.30613I
b = −1.245520 − 0.229336I
−3.83323 − 4.04550I −10.81210 + 4.36543I
u = −0.080998 + 0.665320I
a = 0.12402 − 1.59512I
b = −0.080998 + 0.665320I
3.24651 + 2.33383I −1.26781 − 4.48047I
u = −0.080998 − 0.665320I
a = 0.12402 + 1.59512I
b = −0.080998 − 0.665320I
3.24651 − 2.33383I −1.26781 + 4.48047I
u = 1.346580 + 0.091150I
a = 1.77757 − 0.81741I
b = 1.346580 + 0.091150I
−10.21630 − 3.30364I −18.3839 + 4.0252I
u = 1.346580 − 0.091150I
a = 1.77757 + 0.81741I
b = 1.346580 − 0.091150I
−10.21630 + 3.30364I −18.3839 − 4.0252I
u = 1.324670 + 0.275245I
a = 0.89163 − 1.84476I
b = 1.324670 + 0.275245I
−5.62027 − 9.18761I −13.0093 + 8.4138I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.324670 − 0.275245I
a = 0.89163 + 1.84476I
b = 1.324670 − 0.275245I
−5.62027 + 9.18761I −13.0093 − 8.4138I
u = −1.289570 + 0.434389I
a = −0.19449 − 2.10821I
b = −1.289570 + 0.434389I
4.65712 + 6.34473I −8.38113 − 3.64612I
u = −1.289570 − 0.434389I
a = −0.19449 + 2.10821I
b = −1.289570 − 0.434389I
4.65712 − 6.34473I −8.38113 + 3.64612I
u = 1.316590 + 0.436364I
a = 0.18956 − 2.01627I
b = 1.316590 + 0.436364I
4.26790 − 13.04860I −8.96446 + 7.94392I
u = 1.316590 − 0.436364I
a = 0.18956 + 2.01627I
b = 1.316590 − 0.436364I
4.26790 + 13.04860I −8.96446 − 7.94392I
u = −0.228864 + 0.240486I
a = 0.626966 − 0.762434I
b = −0.228864 + 0.240486I
−0.289179 + 0.793664I −7.24014 − 8.54497I
u = −0.228864 − 0.240486I
a = 0.626966 + 0.762434I
b = −0.228864 − 0.240486I
−0.289179 − 0.793664I −7.24014 + 8.54497I
6
II.
I
u
2
= hu
29
−10u
27
+· · ·+b+1, −u
28
+9u
26
+· · ·+a+1, u
30
−u
29
+· · ·+2u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
28
− 9u
26
+ ··· − 5u − 1
−u
29
+ 10u
27
+ ··· − u − 1
a
12
=
u
3
− 2u
−u
3
+ u
a
4
=
u
27
− 10u
25
+ ··· − 16u
2
− 6u
−u
29
+ 9u
27
+ ··· + u − 2
a
3
=
−u
25
+ 8u
23
+ ··· − 5u − 1
−2u
29
+ 19u
27
+ ··· + u − 2
a
9
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
−u
8
+ 3u
6
− 3u
4
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
2
=
u
22
− 7u
20
+ ··· − 4u − 1
−2u
29
+ 20u
27
+ ··· + u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
29
−40u
27
−4u
26
+ 176u
25
+ 36u
24
−416u
23
−140u
22
+ 476u
21
+ 280u
20
+ 56u
19
−
228u
18
− 892u
17
− 176u
16
+ 920u
15
+ 540u
14
+ 112u
13
− 300u
12
− 784u
11
− 236u
10
+
296u
9
+ 284u
8
+ 240u
7
+ 20u
6
− 112u
5
− 76u
4
− 48u
3
− 12u
2
− 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
c
11
(u
15
+ 3u
14
+ ··· + 8u
2
− 1)
2
c
2
, c
8
(u
15
+ u
14
+ ··· + 2u + 1)
2
c
3
, c
4
, c
5
c
6
, c
10
, c
12
u
30
− u
29
+ ··· + 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
c
11
(y
15
+ 19y
14
+ ··· + 16y −1)
2
c
2
, c
8
(y
15
+ 3y
14
+ ··· + 8y
2
− 1)
2
c
3
, c
4
, c
5
c
6
, c
10
, c
12
y
30
− 21y
29
+ ··· − 16y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.003710 + 0.352470I
a = −0.310106 + 0.106023I
b = 1.197040 + 0.205439I
−3.26563 − 1.73642I −11.57231 + 4.08118I
u = −1.003710 − 0.352470I
a = −0.310106 − 0.106023I
b = 1.197040 − 0.205439I
−3.26563 + 1.73642I −11.57231 − 4.08118I
u = −0.039142 + 0.923066I
a = −1.25369 + 1.52176I
b = 1.299550 − 0.440363I
8.49724 + 8.19235I −5.30498 − 5.35870I
u = −0.039142 − 0.923066I
a = −1.25369 − 1.52176I
b = 1.299550 + 0.440363I
8.49724 − 8.19235I −5.30498 + 5.35870I
u = 0.006457 + 0.907657I
a = 1.26734 + 1.55206I
b = −1.275180 − 0.450373I
8.68612 − 1.54935I −4.90398 + 0.66420I
u = 0.006457 − 0.907657I
a = 1.26734 − 1.55206I
b = −1.275180 + 0.450373I
8.68612 + 1.54935I −4.90398 − 0.66420I
u = −1.09543
a = 0.681427
b = 0.231455
−2.03422 −3.51620
u = −1.144780 + 0.271378I
a = 0.776168 + 0.778536I
b = 0.050886 − 0.582477I
0.109911 + 1.108490I −4.48602 − 0.68443I
u = −1.144780 − 0.271378I
a = 0.776168 − 0.778536I
b = 0.050886 + 0.582477I
0.109911 − 1.108490I −4.48602 + 0.68443I
u = 1.197040 + 0.205439I
a = 0.192214 − 0.213199I
b = −1.003710 + 0.352470I
−3.26563 − 1.73642I −11.57231 + 4.08118I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.197040 − 0.205439I
a = 0.192214 + 0.213199I
b = −1.003710 − 0.352470I
−3.26563 + 1.73642I −11.57231 − 4.08118I
u = 1.245200 + 0.056118I
a = −0.303735 + 0.483850I
b = −0.497721 − 0.447731I
−4.53214 − 1.75942I −14.8508 + 5.0146I
u = 1.245200 − 0.056118I
a = −0.303735 − 0.483850I
b = −0.497721 + 0.447731I
−4.53214 + 1.75942I −14.8508 − 5.0146I
u = −0.191672 + 0.711539I
a = −0.90899 + 1.57838I
b = 1.261970 − 0.268055I
−0.87635 + 5.68434I −7.79510 − 7.47679I
u = −0.191672 − 0.711539I
a = −0.90899 − 1.57838I
b = 1.261970 + 0.268055I
−0.87635 − 5.68434I −7.79510 + 7.47679I
u = 1.261970 + 0.268055I
a = −0.566534 + 0.872590I
b = −0.191672 − 0.711539I
−0.87635 − 5.68434I −7.79510 + 7.47679I
u = 1.261970 − 0.268055I
a = −0.566534 − 0.872590I
b = −0.191672 + 0.711539I
−0.87635 + 5.68434I −7.79510 − 7.47679I
u = −0.497721 + 0.447731I
a = −0.134683 + 1.055100I
b = 1.245200 − 0.056118I
−4.53214 + 1.75942I −14.8508 − 5.0146I
u = −0.497721 − 0.447731I
a = −0.134683 − 1.055100I
b = 1.245200 + 0.056118I
−4.53214 − 1.75942I −14.8508 + 5.0146I
u = −1.256250 + 0.462320I
a = −0.578274 − 0.179714I
b = 1.279350 + 0.437720I
4.73497 − 3.25615I −8.32867 + 2.40088I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.256250 − 0.462320I
a = −0.578274 + 0.179714I
b = 1.279350 − 0.437720I
4.73497 + 3.25615I −8.32867 − 2.40088I
u = 1.279350 + 0.437720I
a = 0.556509 − 0.222909I
b = −1.256250 + 0.462320I
4.73497 − 3.25615I −8.32867 + 2.40088I
u = 1.279350 − 0.437720I
a = 0.556509 + 0.222909I
b = −1.256250 − 0.462320I
4.73497 + 3.25615I −8.32867 − 2.40088I
u = −1.275180 + 0.450373I
a = 0.690774 + 1.153910I
b = 0.006457 − 0.907657I
8.68612 + 1.54935I −4.90398 − 0.66420I
u = −1.275180 − 0.450373I
a = 0.690774 − 1.153910I
b = 0.006457 + 0.907657I
8.68612 − 1.54935I −4.90398 + 0.66420I
u = 1.299550 + 0.440363I
a = −0.651100 + 1.156960I
b = −0.039142 − 0.923066I
8.49724 − 8.19235I −5.30498 + 5.35870I
u = 1.299550 − 0.440363I
a = −0.651100 − 1.156960I
b = −0.039142 + 0.923066I
8.49724 + 8.19235I −5.30498 − 5.35870I
u = 0.050886 + 0.582477I
a = 0.99593 + 1.97518I
b = −1.144780 − 0.271378I
0.109911 − 1.108490I −4.48602 + 0.68443I
u = 0.050886 − 0.582477I
a = 0.99593 − 1.97518I
b = −1.144780 + 0.271378I
0.109911 + 1.108490I −4.48602 − 0.68443I
u = 0.231455
a = −3.22507
b = −1.09543
−2.03422 −3.51620
12
III. I
u
3
= hb + 1, a, u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
1
a
6
=
1
1
a
11
=
−1
0
a
7
=
0
1
a
1
=
0
−1
a
12
=
−1
0
a
4
=
1
−1
a
3
=
0
−1
a
9
=
0
1
a
8
=
0
1
a
2
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
9
, c
11
u
c
3
, c
4
, c
10
u + 1
c
5
, c
6
, c
12
u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
9
, c
11
y
c
3
, c
4
, c
5
c
6
, c
10
, c
12
y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
c
11
u(u
15
+ 3u
14
+ ··· + 8u
2
− 1)
2
(u
17
+ 3u
16
+ ··· − 20u − 4)
c
2
, c
8
u(u
15
+ u
14
+ ··· + 2u + 1)
2
(u
17
− 3u
16
+ ··· + 4u − 2)
c
3
, c
4
, c
10
(u + 1)(u
17
− u
16
+ ··· + 4u + 1)(u
30
− u
29
+ ··· + 2u − 1)
c
5
, c
6
, c
12
(u − 1)(u
17
− u
16
+ ··· + 4u + 1)(u
30
− u
29
+ ··· + 2u − 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
c
11
y(y
15
+ 19y
14
+ ··· + 16y −1)
2
(y
17
+ 19y
16
+ ··· + 8y −16)
c
2
, c
8
y(y
15
+ 3y
14
+ ··· + 8y
2
− 1)
2
(y
17
+ 3y
16
+ ··· − 20y −4)
c
3
, c
4
, c
5
c
6
, c
10
, c
12
(y −1)(y
17
− 15y
16
+ ··· − 2y −1)(y
30
− 21y
29
+ ··· − 16y + 1)
18
