12n
0260
(K12n
0260
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 12 11 3 6 5 7 9
Solving Sequence
6,9
10 5
3,11
2 1 4 8 7 12
c
9
c
5
c
10
c
2
c
1
c
4
c
8
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
10
+ u
9
− 26u
8
+ 4u
7
− 82u
6
+ u
5
− 107u
4
− 4u
3
− 35u
2
+ 8b + 10u + 1,
− 21u
10
+ 7u
9
− 186u
8
+ 20u
7
− 594u
6
− 33u
5
− 785u
4
− 48u
3
− 265u
2
+ 32a + 162u + 15,
u
11
+ 9u
9
+ 2u
8
+ 30u
7
+ 11u
6
+ 44u
5
+ 15u
4
+ 21u
3
− 3u
2
− u − 1i
I
u
2
= hb, −u
2
+ 2a − u − 3, u
3
+ 2u − 1i
I
u
3
= h205u
9
− 272u
8
+ 955u
7
− 1446u
6
+ 1567u
5
− 1260u
4
+ 1037u
3
+ 526u
2
+ 951b + 628u + 381,
− 190u
9
− 235u
8
− 514u
7
− 585u
6
+ 728u
5
− 711u
4
− 590u
3
− 3874u
2
+ 2853a − 1765u − 4737,
u
10
− 2u
9
+ 7u
8
− 12u
7
+ 19u
6
− 21u
5
+ 23u
4
− 11u
3
+ 16u
2
+ 9i
I
u
4
= hb, u
3
+ a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
I
u
5
= h−au + 2b − a − 2u, a
2
+ au + a − 2u, u
2
+ 1i
* 5 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−3u
10
+u
9
+· · ·+8b+1, −21u
10
+7u
9
+· · ·+32a+15, u
11
+9u
9
+· · ·−u−1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
3
=
0.656250u
10
− 0.218750u
9
+ ··· − 5.06250u − 0.468750
3
8
u
10
−
1
8
u
9
+ ··· −
5
4
u −
1
8
a
11
=
u
2
+ 1
−u
4
− 2u
2
a
2
=
0.593750u
10
− 0.0312500u
9
+ ··· − 4.43750u − 0.781250
1
2
u
10
+
9
2
u
8
+ ··· + 5u
2
− 2u
a
1
=
1
4
u
9
+ 2u
7
+ ··· −
1
4
u −
5
4
1
4
u
9
+ 2u
7
+ ··· −
1
4
u −
1
4
a
4
=
0.281250u
10
− 0.0937500u
9
+ ··· − 4.81250u + 0.156250
−
3
8
u
10
+
3
8
u
9
+ ··· −
1
2
u −
1
8
a
8
=
u
3
+ 2u
−
1
4
u
10
− 2u
8
+ ··· +
1
4
u
2
+
5
4
u
a
7
=
u
−
1
4
u
10
− 2u
8
+ ··· +
1
4
u
2
+
5
4
u
a
12
=
−1
1
4
u
9
+ 2u
7
+ ··· −
1
4
u −
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
39
64
u
10
+
27
64
u
9
+
175
32
u
8
+
85
16
u
7
+
619
32
u
6
+
1419
64
u
5
+
2147
64
u
4
+
151
4
u
3
+
1571
64
u
2
+
633
32
u +
163
64
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
+ 18u
10
+ ··· + 1201u + 16
c
2
, c
4
u
11
− 4u
10
+ ··· + 37u − 4
c
3
, c
8
u
11
+ 3u
10
+ ··· + 104u − 32
c
5
, c
6
, c
7
c
9
, c
10
, c
11
u
11
+ 9u
9
+ 2u
8
+ 30u
7
+ 11u
6
+ 44u
5
+ 15u
4
+ 21u
3
− 3u
2
− u − 1
c
12
u
11
+ 13u
9
+ 2u
8
+ 50u
7
+ 25u
6
+ 61u
5
+ 79u
4
+ 71u
3
− 7u
2
− 4u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 14y
10
+ ··· + 1302785y −256
c
2
, c
4
y
11
− 18y
10
+ ··· + 1201y −16
c
3
, c
8
y
11
+ 21y
10
+ ··· + 6464y −1024
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y
11
+ 18y
10
+ ··· − 5y −1
c
12
y
11
+ 26y
10
+ ··· − 40y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.405677 + 0.805557I
a = 0.760429 + 0.630050I
b = 0.11145 + 1.86194I
−11.46380 − 1.35185I 1.54505 + 5.14075I
u = −0.405677 − 0.805557I
a = 0.760429 − 0.630050I
b = 0.11145 − 1.86194I
−11.46380 + 1.35185I 1.54505 − 5.14075I
u = −0.23897 + 1.55675I
a = 0.230096 − 0.384371I
b = 1.033710 + 0.020727I
−10.53020 − 4.19214I 1.36233 + 0.44368I
u = −0.23897 − 1.55675I
a = 0.230096 + 0.384371I
b = 1.033710 − 0.020727I
−10.53020 + 4.19214I 1.36233 − 0.44368I
u = 0.375177
a = −0.576120
b = 0.341658
0.611064 16.3080
u = −0.168597 + 0.298863I
a = −0.19001 − 2.05542I
b = −0.229927 − 0.652177I
−1.60266 − 0.72420I −1.00231 + 3.71560I
u = −0.168597 − 0.298863I
a = −0.19001 + 2.05542I
b = −0.229927 + 0.652177I
−1.60266 + 0.72420I −1.00231 − 3.71560I
u = 0.51296 + 1.70104I
a = 0.76568 − 1.38765I
b = −1.21043 − 2.42503I
11.0221 + 10.7546I −1.36525 − 3.85022I
u = 0.51296 − 1.70104I
a = 0.76568 + 1.38765I
b = −1.21043 + 2.42503I
11.0221 − 10.7546I −1.36525 + 3.85022I
u = 0.11269 + 1.88177I
a = −0.528135 + 1.174060I
b = −1.37564 + 2.29704I
−17.3399 + 2.6033I −1.56897 − 1.12618I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.11269 − 1.88177I
a = −0.528135 − 1.174060I
b = −1.37564 − 2.29704I
−17.3399 − 2.6033I −1.56897 + 1.12618I
6
II. I
u
2
= hb, −u
2
+ 2a − u − 3, u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
−u
−u + 1
a
3
=
1
2
u
2
+
1
2
u +
3
2
0
a
11
=
u
2
+ 1
−u
a
2
=
1
2
u
2
+
3
2
u +
3
2
u − 1
a
1
=
u
u − 1
a
4
=
1
2
u
2
+
1
2
u +
3
2
0
a
8
=
1
0
a
7
=
u
u
2
a
12
=
1
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
7
4
u
2
+
21
4
u +
9
4
7
(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
u
3
+ 2u + 1
c
9
, c
10
, c
11
u
3
+ 2u − 1
c
12
u
3
− 3u
2
+ 5u − 2
8
(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
y
3
+ 4y
2
+ 4y −1
c
12
y
3
+ y
2
+ 13y −4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 0.335258 + 0.401127I
b = 0
−11.08570 − 5.13794I −2.62004 + 6.54094I
u = −0.22670 − 1.46771I
a = 0.335258 − 0.401127I
b = 0
−11.08570 + 5.13794I −2.62004 − 6.54094I
u = 0.453398
a = 1.82948
b = 0
−0.857735 4.99010
10
III. I
u
3
= h205u
9
− 272u
8
+ · · · + 951b + 381, −190u
9
− 235u
8
+ · · · +
2853a − 4737, u
10
− 2u
9
+ · · · + 16u
2
+ 9i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
3
=
0.0665966u
9
+ 0.0823694u
8
+ ··· + 0.618647u + 1.66036
−0.215563u
9
+ 0.286015u
8
+ ··· − 0.660358u − 0.400631
a
11
=
u
2
+ 1
−u
4
− 2u
2
a
2
=
0.0476691u
9
+ 0.322117u
8
+ ··· + 0.653347u + 2.95689
−0.376446u
9
+ 0.323870u
8
+ ··· − 0.865405u + 0.119874
a
1
=
−0.174553u
9
+ 0.433228u
8
+ ··· − 1.23554u + 1.62355
−0.126183u
9
+ 0.264984u
8
+ ··· − 0.435331u − 0.356467
a
4
=
−0.198738u
9
+ 0.850683u
8
+ ··· + 0.197687u + 3.44690
−0.164038u
9
+ 0.744479u
8
+ ··· − 2.36593u − 0.763407
a
8
=
0.123729u
9
− 0.0487206u
8
+ ··· + 1.42131u + 1.13565
0.0357518u
9
− 0.00841220u
8
+ ··· − 0.509989u + 0.217666
a
7
=
0.182615u
9
− 0.239047u
8
+ ··· + 2.75780u + 0.435331
−0.0588854u
9
+ 0.190326u
8
+ ··· + 0.663512u + 0.700315
a
12
=
−0.0483701u
9
+ 0.168244u
8
+ ··· − 0.800210u + 1.98002
−0.126183u
9
+ 0.264984u
8
+ ··· − 0.435331u − 0.356467
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
39
317
u
9
+
140
317
u
8
−
58
317
u
7
+
362
317
u
6
−
373
317
u
5
−
116
317
u
4
+
1289
317
u
3
−
653
317
u
2
+
1355
317
u +
1291
317
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
+ 11u
4
+ 37u
3
+ 30u
2
− 12u + 1)
2
c
2
, c
4
(u
5
− 3u
4
− u
3
+ 6u
2
+ 1)
2
c
3
, c
8
(u
5
− u
4
+ 8u
3
− u
2
− 4u − 4)
2
c
5
, c
6
, c
7
c
9
, c
10
, c
11
u
10
− 2u
9
+ 7u
8
− 12u
7
+ 19u
6
− 21u
5
+ 23u
4
− 11u
3
+ 16u
2
+ 9
c
12
(u
5
+ 6u
3
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− 47y
4
+ 685y
3
− 1810y
2
+ 84y −1)
2
c
2
, c
4
(y
5
− 11y
4
+ 37y
3
− 30y
2
− 12y −1)
2
c
3
, c
8
(y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y −16)
2
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y
10
+ 10y
9
+ ··· + 288y + 81
c
12
(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.334233 + 1.155480I
a = −1.03102 + 1.25338I
b = 1.04912
−5.84264 −6 − 0.349607 + 0.10I
u = 0.334233 − 1.155480I
a = −1.03102 − 1.25338I
b = 1.04912
−5.84264 −6 − 0.349607 + 0.10I
u = −0.447614 + 0.607198I
a = 1.181370 − 0.198963I
b = −0.465884 + 0.485496I
−3.23236 − 1.37362I 4.45374 + 4.59823I
u = −0.447614 − 0.607198I
a = 1.181370 + 0.198963I
b = −0.465884 − 0.485496I
−3.23236 + 1.37362I 4.45374 − 4.59823I
u = 0.011167 + 1.262230I
a = −0.223398 − 0.807514I
b = −0.465884 − 0.485496I
−3.23236 + 1.37362I 4.45374 − 4.59823I
u = 0.011167 − 1.262230I
a = −0.223398 + 0.807514I
b = −0.465884 + 0.485496I
−3.23236 − 1.37362I 4.45374 + 4.59823I
u = 1.28009 + 0.69443I
a = −0.932756 − 0.175792I
b = 0.44133 + 2.86818I
18.4907 + 4.0569I −0.27894 − 1.95729I
u = 1.28009 − 0.69443I
a = −0.932756 + 0.175792I
b = 0.44133 − 2.86818I
18.4907 − 4.0569I −0.27894 + 1.95729I
u = −0.17787 + 1.78975I
a = −0.32754 − 1.78671I
b = 0.44133 − 2.86818I
18.4907 − 4.0569I −0.27894 + 1.95729I
u = −0.17787 − 1.78975I
a = −0.32754 + 1.78671I
b = 0.44133 + 2.86818I
18.4907 + 4.0569I −0.27894 − 1.95729I
14
IV. I
u
4
= hb, u
3
+ a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
3
=
−u
3
− u − 1
0
a
11
=
u
2
+ 1
u
3
+ 2u + 1
a
2
=
−u
3
− 1
−u
3
− u
a
1
=
u
−u
3
− u
a
4
=
−u
3
− u − 1
0
a
8
=
1
0
a
7
=
2u
3
+ u
2
+ 3u + 3
−u
3
− u
2
− u − 2
a
12
=
u
3
+ 2u
−u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 4u + 3
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
8
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
7
u
4
− u
3
+ 2u
2
− 2u + 1
c
9
, c
10
, c
11
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
12
(u
2
+ u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y
4
+ 3y
3
+ 2y
2
+ 1
c
12
(y
2
+ y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −0.500000 − 0.866025I
b = 0
−4.93480 − 2.02988I 1.0000 + 3.46410I
u = −0.621744 − 0.440597I
a = −0.500000 + 0.866025I
b = 0
−4.93480 + 2.02988I 1.0000 − 3.46410I
u = 0.121744 + 1.306620I
a = −0.500000 + 0.866025I
b = 0
−4.93480 + 2.02988I 1.00000 − 3.46410I
u = 0.121744 − 1.306620I
a = −0.500000 − 0.866025I
b = 0
−4.93480 − 2.02988I 1.00000 + 3.46410I
18
V. I
u
5
= h−au + 2b − a − 2u, a
2
+ au + a − 2u, u
2
+ 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
1
a
5
=
−u
0
a
3
=
a
1
2
au +
1
2
a + u
a
11
=
0
1
a
2
=
−
1
2
au +
1
2
a − u
1
2
au +
1
2
a + u
a
1
=
−
1
2
au −
1
2
a − 2u − 1
−
1
2
au −
1
2
a − 2u
a
4
=
1
2
au +
1
2
a − 1
−
1
2
au −
1
2
a − 2u
a
8
=
u
1
2
au −
1
2
a − 2
a
7
=
u
1
2
au −
1
2
a + u − 2
a
12
=
−1
−
1
2
au −
1
2
a − 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− 3u + 1)
2
c
2
(u
2
+ u − 1)
2
c
3
, c
8
u
4
+ 3u
2
+ 1
c
4
(u
2
− u − 1)
2
c
5
, c
6
, c
7
c
9
, c
10
, c
11
(u
2
+ 1)
2
c
12
u
4
+ 7u
2
+ 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 7y + 1)
2
c
2
, c
4
(y
2
− 3y + 1)
2
c
3
, c
8
(y
2
+ 3y + 1)
2
c
5
, c
6
, c
7
c
9
, c
10
, c
11
(y + 1)
4
c
12
(y
2
+ 7y + 1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.618034 + 0.618034I
b = 1.61803I
−12.1725 −4.00000
u = 1.000000I
a = −1.61803 − 1.61803I
b = − 0.618034I
−4.27683 −4.00000
u = − 1.000000I
a = 0.618034 − 0.618034I
b = − 1.61803I
−12.1725 −4.00000
u = − 1.000000I
a = −1.61803 + 1.61803I
b = 0.618034I
−4.27683 −4.00000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
7
(u
2
− 3u + 1)
2
(u
5
+ 11u
4
+ 37u
3
+ 30u
2
− 12u + 1)
2
· (u
11
+ 18u
10
+ ··· + 1201u + 16)
c
2
(u − 1)
7
(u
2
+ u − 1)
2
(u
5
− 3u
4
− u
3
+ 6u
2
+ 1)
2
· (u
11
− 4u
10
+ ··· + 37u − 4)
c
3
, c
8
u
7
(u
4
+ 3u
2
+ 1)(u
5
− u
4
+ 8u
3
− u
2
− 4u − 4)
2
· (u
11
+ 3u
10
+ ··· + 104u − 32)
c
4
(u + 1)
7
(u
2
− u − 1)
2
(u
5
− 3u
4
− u
3
+ 6u
2
+ 1)
2
· (u
11
− 4u
10
+ ··· + 37u − 4)
c
5
, c
6
, c
7
(u
2
+ 1)
2
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
10
− 2u
9
+ 7u
8
− 12u
7
+ 19u
6
− 21u
5
+ 23u
4
− 11u
3
+ 16u
2
+ 9)
· (u
11
+ 9u
9
+ 2u
8
+ 30u
7
+ 11u
6
+ 44u
5
+ 15u
4
+ 21u
3
− 3u
2
− u − 1)
c
9
, c
10
, c
11
(u
2
+ 1)
2
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
10
− 2u
9
+ 7u
8
− 12u
7
+ 19u
6
− 21u
5
+ 23u
4
− 11u
3
+ 16u
2
+ 9)
· (u
11
+ 9u
9
+ 2u
8
+ 30u
7
+ 11u
6
+ 44u
5
+ 15u
4
+ 21u
3
− 3u
2
− u − 1)
c
12
(u
2
+ u + 1)
2
(u
3
− 3u
2
+ 5u − 2)(u
4
+ 7u
2
+ 1)(u
5
+ 6u
3
+ u − 1)
2
· (u
11
+ 13u
9
+ 2u
8
+ 50u
7
+ 25u
6
+ 61u
5
+ 79u
4
+ 71u
3
− 7u
2
− 4u − 4)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
7
(y
2
− 7y + 1)
2
(y
5
− 47y
4
+ 685y
3
− 1810y
2
+ 84y −1)
2
· (y
11
− 14y
10
+ ··· + 1302785y −256)
c
2
, c
4
(y −1)
7
(y
2
− 3y + 1)
2
(y
5
− 11y
4
+ 37y
3
− 30y
2
− 12y −1)
2
· (y
11
− 18y
10
+ ··· + 1201y −16)
c
3
, c
8
y
7
(y
2
+ 3y + 1)
2
(y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y −16)
2
· (y
11
+ 21y
10
+ ··· + 6464y −1024)
c
5
, c
6
, c
7
c
9
, c
10
, c
11
(y + 1)
4
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
10
+ 10y
9
+ ··· + 288y + 81)(y
11
+ 18y
10
+ ··· − 5y −1)
c
12
(y
2
+ y + 1)
2
(y
2
+ 7y + 1)
2
(y
3
+ y
2
+ 13y −4)
· ((y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1)
2
)(y
11
+ 26y
10
+ ··· − 40y −16)
24
