12n
0285
(K12n
0285
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 12 2 5 12 7 6 9 10
Solving Sequence
2,6
3 7
4,10
11 1 9 12 5 8
c
2
c
6
c
3
c
10
c
1
c
9
c
12
c
5
c
7
c
4
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
16
+ 4u
15
+ ··· + b 1, u
16
3u
15
+ ··· + 2a 3, u
17
+ 5u
16
+ ··· 3u 2i
I
u
2
= h−u
9
+ 2u
8
4u
7
+ 4u
6
6u
5
+ 4u
4
5u
3
+ 2u
2
+ b 2u + 1,
2u
9
+ 3u
8
6u
7
+ 4u
6
8u
5
+ 4u
4
7u
3
+ u
2
+ a 3u + 2,
u
10
2u
9
+ 4u
8
4u
7
+ 6u
6
5u
5
+ 6u
4
3u
3
+ 3u
2
2u + 1i
I
u
3
= h−u
5
+ u
4
+ u
2
a au u
2
+ b u + 1, u
5
a + 2u
5
+ u
4
u
3
+ a
2
+ 3au + u
2
+ 4u + 4,
u
6
u
5
+ u
4
+ 2u
2
u + 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
16
+4u
15
+· · ·+b1, u
16
3u
15
+· · ·+2a3, u
17
+5u
16
+· · ·3u2i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
a
4
=
u
4
+ u
2
+ 1
u
4
a
10
=
1
2
u
16
+
3
2
u
15
+ ···
1
2
u +
3
2
u
16
4u
15
+ ··· u + 1
a
11
=
1
2
u
16
+
3
2
u
15
+ ···
1
2
u +
3
2
u
16
+ 3u
15
+ ··· 3u 1
a
1
=
u
2
+ 1
u
4
a
9
=
3
2
u
16
+
15
2
u
15
+ ···
7
2
u
5
2
2u
15
+ 7u
14
+ ··· 4u 3
a
12
=
3
2
u
16
+
15
2
u
15
+ ···
7
2
u
5
2
2u
16
+ 9u
15
+ ··· 4u 3
a
5
=
1
2
u
16
+
3
2
u
15
+ ···
1
2
u +
1
2
u
16
6u
15
+ ··· + 3u + 3
a
8
=
5
2
u
16
27
2
u
15
+ ··· +
13
2
u +
13
2
2u
16
12u
15
+ ··· + 6u + 7
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
16
6u
15
20u
14
43u
13
71u
12
91u
11
94u
10
58u
9
+
66u
7
+ 87u
6
+ 92u
5
+ 60u
4
+ 46u
3
+ 17u
2
+ 17u 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 5u
16
+ ··· 35u 4
c
2
, c
6
u
17
5u
16
+ ··· 3u + 2
c
3
u
17
+ 5u
16
+ ··· + 201u + 74
c
4
, c
10
u
17
+ 13u
15
+ ··· + 4u + 1
c
5
, c
7
u
17
u
16
+ ··· + 2u + 1
c
8
, c
11
u
17
10u
16
+ ··· + 27u 4
c
9
, c
12
u
17
+ 3u
16
+ ··· 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 17y
16
+ ··· + 49y 16
c
2
, c
6
y
17
+ 5y
16
+ ··· 35y 4
c
3
y
17
+ 29y
16
+ ··· 126395y 5476
c
4
, c
10
y
17
+ 26y
16
+ ··· 10y 1
c
5
, c
7
y
17
25y
16
+ ··· + 4y 1
c
8
, c
11
y
17
+ 2y
16
+ ··· 143y 16
c
9
, c
12
y
17
15y
16
+ ··· + 16y 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.999402
a = 0.387652
b = 0.774608
3.45333 1.57930
u = 0.225183 + 0.839513I
a = 0.457066 0.305622I
b = 0.260858 0.479622I
0.515123 1.230730I 4.57965 + 6.11157I
u = 0.225183 0.839513I
a = 0.457066 + 0.305622I
b = 0.260858 + 0.479622I
0.515123 + 1.230730I 4.57965 6.11157I
u = 0.902975 + 0.779498I
a = 0.37268 + 1.47715I
b = 1.18710 + 0.79283I
5.52032 2.59660I 1.98291 + 2.57071I
u = 0.902975 0.779498I
a = 0.37268 1.47715I
b = 1.18710 0.79283I
5.52032 + 2.59660I 1.98291 2.57071I
u = 1.013990 + 0.825716I
a = 1.04491 1.15054I
b = 1.67420 + 0.11880I
2.66130 + 5.14311I 3.31224 2.06906I
u = 1.013990 0.825716I
a = 1.04491 + 1.15054I
b = 1.67420 0.11880I
2.66130 5.14311I 3.31224 + 2.06906I
u = 0.792289 + 1.041050I
a = 1.139370 + 0.603283I
b = 1.78949 + 0.05972I
4.68149 3.71646I 2.79518 + 2.82261I
u = 0.792289 1.041050I
a = 1.139370 0.603283I
b = 1.78949 0.05972I
4.68149 + 3.71646I 2.79518 2.82261I
u = 0.065367 + 0.651578I
a = 0.870762 0.440011I
b = 0.015136 0.797710I
0.691744 1.119700I 6.04337 + 5.63794I
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.065367 0.651578I
a = 0.870762 + 0.440011I
b = 0.015136 + 0.797710I
0.691744 + 1.119700I 6.04337 5.63794I
u = 0.354914 + 0.549103I
a = 0.916291 + 0.819903I
b = 0.934133 + 0.713219I
0.02980 + 3.01264I 7.09672 + 0.35044I
u = 0.354914 0.549103I
a = 0.916291 0.819903I
b = 0.934133 0.713219I
0.02980 3.01264I 7.09672 0.35044I
u = 0.398162 + 1.288650I
a = 0.300696 0.167666I
b = 0.720067 + 0.510972I
7.74200 + 4.95608I 4.59429 5.14858I
u = 0.398162 1.288650I
a = 0.300696 + 0.167666I
b = 0.720067 0.510972I
7.74200 4.95608I 4.59429 + 5.14858I
u = 0.883711 + 1.061310I
a = 0.91247 1.36079I
b = 2.22990 0.92943I
1.89495 12.06910I 4.30600 + 6.19242I
u = 0.883711 1.061310I
a = 0.91247 + 1.36079I
b = 2.22990 + 0.92943I
1.89495 + 12.06910I 4.30600 6.19242I
6
II.
I
u
2
= h−u
9
+2u
8
+· · ·+b +1, 2u
9
+3u
8
+· · ·+a +2, u
10
2u
9
+· · ·2u +1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
a
4
=
u
4
+ u
2
+ 1
u
4
a
10
=
2u
9
3u
8
+ 6u
7
4u
6
+ 8u
5
4u
4
+ 7u
3
u
2
+ 3u 2
u
9
2u
8
+ 4u
7
4u
6
+ 6u
5
4u
4
+ 5u
3
2u
2
+ 2u 1
a
11
=
2u
9
3u
8
+ 6u
7
4u
6
+ 8u
5
4u
4
+ 7u
3
u
2
+ 3u 2
u
9
2u
8
+ 4u
7
4u
6
+ 6u
5
5u
4
+ 5u
3
3u
2
+ 2u 2
a
1
=
u
2
+ 1
u
4
a
9
=
2u
9
3u
8
+ 6u
7
5u
6
+ 9u
5
6u
4
+ 8u
3
3u
2
+ 4u 3
u
9
2u
8
+ 4u
7
5u
6
+ 7u
5
6u
4
+ 6u
3
4u
2
+ 3u 2
a
12
=
2u
9
+ 3u
8
6u
7
+ 5u
6
9u
5
+ 6u
4
8u
3
+ 3u
2
4u + 3
u
9
+ u
8
2u
7
+ u
6
3u
5
+ u
4
3u
3
+ u
2
2u + 1
a
5
=
3u
9
+ 5u
8
10u
7
+ 8u
6
14u
5
+ 10u
4
13u
3
+ 4u
2
6u + 5
u
9
+ 2u
8
4u
7
+ 3u
6
5u
5
+ 4u
4
5u
3
+ u
2
2u + 2
a
8
=
6u
9
9u
8
+ 18u
7
14u
6
+ 26u
5
16u
4
+ 23u
3
7u
2
+ 11u 8
3u
9
4u
8
+ 8u
7
6u
6
+ 12u
5
6u
4
+ 10u
3
3u
2
+ 5u 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
9
+ 4u
8
5u
7
+ 5u
6
2u
5
+ u
4
+ 3u
3
4u
2
+ 5u 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
4u
9
+ 12u
8
24u
7
+ 38u
6
41u
5
+ 34u
4
19u
3
+ 9u
2
2u + 1
c
2
u
10
2u
9
+ 4u
8
4u
7
+ 6u
6
5u
5
+ 6u
4
3u
3
+ 3u
2
2u + 1
c
3
u
10
+ 2u
9
+ 8u
8
+ 4u
7
+ 10u
5
+ u
4
6u
3
+ 16u
2
10u + 5
c
4
, c
10
u
10
+ u
8
+ 4u
7
11u
6
5u
5
+ 18u
4
6u
3
2u + 1
c
5
, c
7
u
10
+ u
9
4u
8
5u
7
3u
6
+ u
5
+ 15u
4
+ 15u
3
+ 11u
2
+ 4u + 1
c
6
u
10
+ 2u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 5u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 2u + 1
c
8
u
10
7u
9
+ 19u
8
24u
7
+ 9u
6
+ 16u
5
27u
4
+ 13u
3
+ 6u
2
10u + 5
c
9
, c
12
u
10
3u
9
+ u
8
+ 4u
7
5u
6
+ 6u
4
u
3
u
2
+ 2u + 1
c
11
u
10
+ 7u
9
+ 19u
8
+ 24u
7
+ 9u
6
16u
5
27u
4
13u
3
+ 6u
2
+ 10u + 5
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 8y
9
+ ··· + 14y + 1
c
2
, c
6
y
10
+ 4y
9
+ 12y
8
+ 24y
7
+ 38y
6
+ 41y
5
+ 34y
4
+ 19y
3
+ 9y
2
+ 2y + 1
c
3
y
10
+ 12y
9
+ ··· + 60y + 25
c
4
, c
10
y
10
+ 2y
9
+ ··· 4y + 1
c
5
, c
7
y
10
9y
9
+ ··· + 6y + 1
c
8
, c
11
y
10
11y
9
+ ··· 40y + 25
c
9
, c
12
y
10
7y
9
+ ··· 6y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.378370 + 0.962478I
a = 0.467882 + 0.091620I
b = 0.034500 + 0.828197I
0.560424 0.074153I 4.22332 0.05050I
u = 0.378370 0.962478I
a = 0.467882 0.091620I
b = 0.034500 0.828197I
0.560424 + 0.074153I 4.22332 + 0.05050I
u = 0.549385 + 0.711068I
a = 0.483620 0.417086I
b = 0.789580 0.577598I
0.35639 3.68459I 1.74176 + 8.84832I
u = 0.549385 0.711068I
a = 0.483620 + 0.417086I
b = 0.789580 + 0.577598I
0.35639 + 3.68459I 1.74176 8.84832I
u = 0.485122 + 1.143680I
a = 0.522993 0.643782I
b = 0.836616 0.985073I
8.64742 + 3.94137I 8.44290 2.10467I
u = 0.485122 1.143680I
a = 0.522993 + 0.643782I
b = 0.836616 + 0.985073I
8.64742 3.94137I 8.44290 + 2.10467I
u = 0.946362 + 0.955964I
a = 0.97616 1.25675I
b = 2.01415 0.37924I
10.15450 + 3.46808I 1.22554 2.41931I
u = 0.946362 0.955964I
a = 0.97616 + 1.25675I
b = 2.01415 + 0.37924I
10.15450 3.46808I 1.22554 + 2.41931I
u = 0.496271 + 0.410325I
a = 1.53110 + 1.96143I
b = 0.646542 + 0.815887I
6.23787 + 0.27295I 4.31756 + 1.10366I
u = 0.496271 0.410325I
a = 1.53110 1.96143I
b = 0.646542 0.815887I
6.23787 0.27295I 4.31756 1.10366I
10
III. I
u
3
= h−u
5
+ u
4
+ u
2
a au u
2
+ b u + 1, u
5
a + 2u
5
+ u
4
u
3
+ a
2
+
3au + u
2
+ 4u + 4, u
6
u
5
+ u
4
+ 2u
2
u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
a
4
=
u
4
+ u
2
+ 1
u
4
a
10
=
a
u
5
u
4
u
2
a + au + u
2
+ u 1
a
11
=
a
u
5
u
4
+ au + u
2
+ u 1
a
1
=
u
2
+ 1
u
4
a
9
=
u
4
a + u
5
u
3
a u
4
+ u
2
a + u
3
+ a + u
u
4
a + 2u
5
u
3
a 2u
4
+ u
3
+ au + u
2
+ 2u 1
a
12
=
u
4
a u
5
+ u
3
a + u
4
u
2
a u
3
a 2u
u
4
a 2u
5
+ u
3
a + 2u
4
u
2
a 2u
3
a 4u + 1
a
5
=
3u
5
+ 3u
4
u
2
a 2u
3
+ au + u
2
a 7u + 3
u
4
a 4u
5
+ u
3
a + 4u
4
2u
2
a 3u
3
+ au + u
2
2a 8u + 3
a
8
=
2u
4
a 3u
5
+ 2u
3
a + 2u
4
2u
2
a 2u
3
3a 5u
2u
4
a 6u
5
+ 2u
3
a + 5u
4
u
2
a 3u
3
au u
2
2a 10u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
4u
2
8u 10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1)
2
c
2
, c
6
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
c
3
(u
6
u
5
+ 9u
4
+ 20u
2
u + 5)
2
c
4
, c
10
u
12
+ u
11
+ ··· 30u + 187
c
5
, c
7
u
12
+ u
11
+ ··· 12u + 1
c
8
, c
11
(u
6
+ 3u
5
u
4
8u
3
2u
2
+ 5u + 3)
2
c
9
, c
12
u
12
+ 3u
11
+ ··· + 184u + 41
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1)
2
c
2
, c
6
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
c
3
(y
6
+ 17y
5
+ 121y
4
+ 368y
3
+ 490y
2
+ 199y + 25)
2
c
4
, c
10
y
12
+ 11y
11
+ ··· + 67916y + 34969
c
5
, c
7
y
12
9y
11
+ ··· + 48y + 1
c
8
, c
11
(y
6
11y
5
+ 45y
4
84y
3
+ 78y
2
37y + 9)
2
c
9
, c
12
y
12
9y
11
+ ··· + 92y + 1681
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.716019 + 0.809696I
a = 0.235532 0.660644I
b = 1.31075 1.14824I
1.93691 2.65597I 6.41885 + 3.39809I
u = 0.716019 + 0.809696I
a = 1.232850 0.459046I
b = 0.342200 + 0.700158I
1.93691 2.65597I 6.41885 + 3.39809I
u = 0.716019 0.809696I
a = 0.235532 + 0.660644I
b = 1.31075 + 1.14824I
1.93691 + 2.65597I 6.41885 3.39809I
u = 0.716019 0.809696I
a = 1.232850 + 0.459046I
b = 0.342200 0.700158I
1.93691 + 2.65597I 6.41885 3.39809I
u = 0.283231 + 0.633899I
a = 0.83956 + 1.55687I
b = 1.80934 + 1.85329I
6.83783 + 1.10871I 11.53615 6.18117I
u = 0.283231 + 0.633899I
a = 0.14932 3.37698I
b = 0.035938 0.941207I
6.83783 + 1.10871I 11.53615 6.18117I
u = 0.283231 0.633899I
a = 0.83956 1.55687I
b = 1.80934 1.85329I
6.83783 1.10871I 11.53615 + 6.18117I
u = 0.283231 0.633899I
a = 0.14932 + 3.37698I
b = 0.035938 + 0.941207I
6.83783 1.10871I 11.53615 + 6.18117I
u = 0.932789 + 0.951611I
a = 0.94860 1.12119I
b = 1.61310 0.56686I
8.77474 + 3.42721I 6.04500 2.25224I
u = 0.932789 + 0.951611I
a = 0.96910 + 1.38183I
b = 2.30544 + 0.27711I
8.77474 + 3.42721I 6.04500 2.25224I
14
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.932789 0.951611I
a = 0.94860 + 1.12119I
b = 1.61310 + 0.56686I
8.77474 3.42721I 6.04500 + 2.25224I
u = 0.932789 0.951611I
a = 0.96910 1.38183I
b = 2.30544 0.27711I
8.77474 3.42721I 6.04500 + 2.25224I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1)
2
· (u
10
4u
9
+ 12u
8
24u
7
+ 38u
6
41u
5
+ 34u
4
19u
3
+ 9u
2
2u + 1)
· (u
17
+ 5u
16
+ ··· 35u 4)
c
2
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
· (u
10
2u
9
+ 4u
8
4u
7
+ 6u
6
5u
5
+ 6u
4
3u
3
+ 3u
2
2u + 1)
· (u
17
5u
16
+ ··· 3u + 2)
c
3
(u
6
u
5
+ 9u
4
+ 20u
2
u + 5)
2
· (u
10
+ 2u
9
+ 8u
8
+ 4u
7
+ 10u
5
+ u
4
6u
3
+ 16u
2
10u + 5)
· (u
17
+ 5u
16
+ ··· + 201u + 74)
c
4
, c
10
(u
10
+ u
8
+ 4u
7
11u
6
5u
5
+ 18u
4
6u
3
2u + 1)
· (u
12
+ u
11
+ ··· 30u + 187)(u
17
+ 13u
15
+ ··· + 4u + 1)
c
5
, c
7
(u
10
+ u
9
4u
8
5u
7
3u
6
+ u
5
+ 15u
4
+ 15u
3
+ 11u
2
+ 4u + 1)
· (u
12
+ u
11
+ ··· 12u + 1)(u
17
u
16
+ ··· + 2u + 1)
c
6
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
· (u
10
+ 2u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 5u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 2u + 1)
· (u
17
5u
16
+ ··· 3u + 2)
c
8
(u
6
+ 3u
5
u
4
8u
3
2u
2
+ 5u + 3)
2
· (u
10
7u
9
+ 19u
8
24u
7
+ 9u
6
+ 16u
5
27u
4
+ 13u
3
+ 6u
2
10u + 5)
· (u
17
10u
16
+ ··· + 27u 4)
c
9
, c
12
(u
10
3u
9
+ u
8
+ 4u
7
5u
6
+ 6u
4
u
3
u
2
+ 2u + 1)
· (u
12
+ 3u
11
+ ··· + 184u + 41)(u
17
+ 3u
16
+ ··· 8u + 1)
c
11
(u
6
+ 3u
5
u
4
8u
3
2u
2
+ 5u + 3)
2
· (u
10
+ 7u
9
+ 19u
8
+ 24u
7
+ 9u
6
16u
5
27u
4
13u
3
+ 6u
2
+ 10u + 5)
· (u
17
10u
16
+ ··· + 27u 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
6
+ 9y
5
+ ··· + 3y + 1)
2
)(y
10
+ 8y
9
+ ··· + 14y + 1)
· (y
17
+ 17y
16
+ ··· + 49y 16)
c
2
, c
6
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
· (y
10
+ 4y
9
+ 12y
8
+ 24y
7
+ 38y
6
+ 41y
5
+ 34y
4
+ 19y
3
+ 9y
2
+ 2y + 1)
· (y
17
+ 5y
16
+ ··· 35y 4)
c
3
(y
6
+ 17y
5
+ 121y
4
+ 368y
3
+ 490y
2
+ 199y + 25)
2
· (y
10
+ 12y
9
+ ··· + 60y + 25)(y
17
+ 29y
16
+ ··· 126395y 5476)
c
4
, c
10
(y
10
+ 2y
9
+ ··· 4y + 1)(y
12
+ 11y
11
+ ··· + 67916y + 34969)
· (y
17
+ 26y
16
+ ··· 10y 1)
c
5
, c
7
(y
10
9y
9
+ ··· + 6y + 1)(y
12
9y
11
+ ··· + 48y + 1)
· (y
17
25y
16
+ ··· + 4y 1)
c
8
, c
11
(y
6
11y
5
+ 45y
4
84y
3
+ 78y
2
37y + 9)
2
· (y
10
11y
9
+ ··· 40y + 25)(y
17
+ 2y
16
+ ··· 143y 16)
c
9
, c
12
(y
10
7y
9
+ ··· 6y + 1)(y
12
9y
11
+ ··· + 92y + 1681)
· (y
17
15y
16
+ ··· + 16y 1)
17