12n
0073
(K12n
0073
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 4 5 12 6 9 8 11
Solving Sequence
5,10 3,6
2 1 4 7 8 9 11 12
c
5
c
2
c
1
c
4
c
6
c
7
c
9
c
10
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.25963 × 10
16
u
35
+ 4.95553 × 10
16
u
34
+ ··· + 3.15381 × 10
17
b + 2.76334 × 10
17
,
− 2.34713 × 10
17
u
35
+ 2.08730 × 10
17
u
34
+ ··· + 3.15381 × 10
17
a − 4.82090 × 10
17
, u
36
− 2u
35
+ ··· + u − 1i
I
u
2
= hb + 1, −u
8
+ 2u
7
− 3u
6
+ 3u
5
− 4u
4
+ 4u
3
− 3u
2
+ a + 2u − 1, u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 45 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−3.26×10
16
u
35
+4.96×10
16
u
34
+· · ·+3.15×10
17
b+2.76×10
17
, −2.35×
10
17
u
35
+2.09×10
17
u
34
+· · ·+3.15×10
17
a−4.82×10
17
, u
36
−2u
35
+· · ·+u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
0.744221u
35
− 0.661834u
34
+ ··· − 3.44897u + 1.52860
0.103355u
35
− 0.157128u
34
+ ··· + 0.0286142u − 0.876190
a
6
=
1
−u
2
a
2
=
0.847576u
35
− 0.818962u
34
+ ··· − 3.42035u + 0.652406
0.103355u
35
− 0.157128u
34
+ ··· + 0.0286142u − 0.876190
a
1
=
0.376173u
35
− 0.492061u
34
+ ··· + 0.281199u − 0.411800
0.0410701u
35
− 0.0783542u
34
+ ··· − 0.108796u − 0.0990764
a
4
=
0.700268u
35
− 0.600854u
34
+ ··· − 3.50274u + 1.47901
0.106016u
35
− 0.0966188u
34
+ ··· + 0.0456405u − 0.849264
a
7
=
0.376173u
35
− 0.492061u
34
+ ··· + 0.281199u − 0.411800
−0.170748u
35
+ 0.219929u
34
+ ··· − 0.00709182u − 0.161209
a
8
=
0.546921u
35
− 0.711990u
34
+ ··· + 0.288291u − 0.250591
−0.170748u
35
+ 0.219929u
34
+ ··· − 0.00709182u − 0.161209
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
12
=
0.267209u
35
− 0.299525u
34
+ ··· + 0.309131u − 0.445002
0.211046u
35
− 0.287825u
34
+ ··· + 0.274209u + 0.206608
(ii) Obstruction class = −1
(iii) Cusp Shapes =
110199305673793977
315381009766300841
u
35
+
250535878173710291
315381009766300841
u
34
+···−
683728502499797155
28671000887845531
u+
3186236650537876488
315381009766300841
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
36
+ 2u
35
+ ··· + 151u + 1
c
2
, c
4
u
36
− 10u
35
+ ··· + 19u − 1
c
3
, c
6
u
36
+ 3u
35
+ ··· + 512u + 512
c
5
, c
9
u
36
− 2u
35
+ ··· + u − 1
c
7
u
36
− 6u
35
+ ··· + 790797u − 444601
c
8
, c
11
u
36
+ 2u
35
+ ··· + 5u + 1
c
10
u
36
+ 6u
35
+ ··· + u + 1
c
12
u
36
− 22u
35
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
36
+ 74y
35
+ ··· − 20743y + 1
c
2
, c
4
y
36
− 2y
35
+ ··· − 151y + 1
c
3
, c
6
y
36
− 57y
35
+ ··· − 8126464y + 262144
c
5
, c
9
y
36
+ 6y
35
+ ··· + y + 1
c
7
y
36
+ 134y
35
+ ··· − 15572451598723y + 197670049201
c
8
, c
11
y
36
− 22y
35
+ ··· + y + 1
c
10
y
36
+ 50y
35
+ ··· − 35y + 1
c
12
y
36
− 14y
35
+ ··· + 61y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.615843 + 0.841646I
a = 1.43664 − 0.80379I
b = 0.301969 + 0.746984I
4.77425 − 0.10637I 7.48146 + 1.59247I
u = −0.615843 − 0.841646I
a = 1.43664 + 0.80379I
b = 0.301969 − 0.746984I
4.77425 + 0.10637I 7.48146 − 1.59247I
u = −0.732748 + 0.745235I
a = −0.092665 + 1.018320I
b = 0.079470 − 1.108210I
5.18897 − 4.94800I 7.17998 + 5.99105I
u = −0.732748 − 0.745235I
a = −0.092665 − 1.018320I
b = 0.079470 + 1.108210I
5.18897 + 4.94800I 7.17998 − 5.99105I
u = 0.739358 + 0.601439I
a = 0.104781 − 0.811350I
b = 0.083215 + 0.736081I
1.62325 + 1.32416I 4.12502 − 2.60316I
u = 0.739358 − 0.601439I
a = 0.104781 + 0.811350I
b = 0.083215 − 0.736081I
1.62325 − 1.32416I 4.12502 + 2.60316I
u = 0.125468 + 1.044610I
a = 0.966320 + 0.109826I
b = 0.454746 − 0.102565I
−2.34146 + 2.27465I 2.44627 − 4.29475I
u = 0.125468 − 1.044610I
a = 0.966320 − 0.109826I
b = 0.454746 + 0.102565I
−2.34146 − 2.27465I 2.44627 + 4.29475I
u = −0.912979 + 0.603282I
a = −0.094847 + 0.581208I
b = 0.428044 − 0.685537I
4.35891 + 2.74036I 7.57970 − 3.16606I
u = −0.912979 − 0.603282I
a = −0.094847 − 0.581208I
b = 0.428044 + 0.685537I
4.35891 − 2.74036I 7.57970 + 3.16606I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.552810 + 1.007290I
a = 1.032000 + 0.667957I
b = 0.536480 − 0.548160I
0.20181 + 3.58839I 2.59766 − 4.47078I
u = 0.552810 − 1.007290I
a = 1.032000 − 0.667957I
b = 0.536480 + 0.548160I
0.20181 − 3.58839I 2.59766 + 4.47078I
u = 0.403187 + 0.692172I
a = 0.222101 − 1.316930I
b = −0.848909 + 0.718608I
−0.01273 + 3.75640I 1.96178 − 8.67374I
u = 0.403187 − 0.692172I
a = 0.222101 + 1.316930I
b = −0.848909 − 0.718608I
−0.01273 − 3.75640I 1.96178 + 8.67374I
u = −0.656810 + 1.069490I
a = 0.849616 − 0.843755I
b = 0.705777 + 0.623024I
2.76036 − 8.54206I 5.00446 + 8.46696I
u = −0.656810 − 1.069490I
a = 0.849616 + 0.843755I
b = 0.705777 − 0.623024I
2.76036 + 8.54206I 5.00446 − 8.46696I
u = 0.959801 + 0.917573I
a = −0.795562 − 0.722155I
b = 1.04335 + 1.28065I
15.4107 + 4.2831I 6.51475 − 3.16359I
u = 0.959801 − 0.917573I
a = −0.795562 + 0.722155I
b = 1.04335 − 1.28065I
15.4107 − 4.2831I 6.51475 + 3.16359I
u = −0.110883 + 0.661998I
a = −0.567675 + 0.716209I
b = −1.295050 − 0.200690I
−2.17080 − 1.28901I −2.65057 + 3.66135I
u = −0.110883 − 0.661998I
a = −0.567675 − 0.716209I
b = −1.295050 + 0.200690I
−2.17080 + 1.28901I −2.65057 − 3.66135I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.981174 + 0.904102I
a = −0.759912 + 0.643457I
b = 1.05191 − 1.17838I
11.26030 + 0.66616I 3.25844 + 0.20862I
u = −0.981174 − 0.904102I
a = −0.759912 − 0.643457I
b = 1.05191 + 1.17838I
11.26030 − 0.66616I 3.25844 − 0.20862I
u = 0.914043 + 0.991327I
a = 0.58196 + 1.72130I
b = 1.10681 − 1.16652I
15.1611 + 2.6010I 6.17050 − 1.49093I
u = 0.914043 − 0.991327I
a = 0.58196 − 1.72130I
b = 1.10681 + 1.16652I
15.1611 − 2.6010I 6.17050 + 1.49093I
u = 1.000680 + 0.916853I
a = −0.805879 − 0.579891I
b = 1.13554 + 1.14001I
15.0538 − 5.8256I 6.10110 + 2.81591I
u = 1.000680 − 0.916853I
a = −0.805879 + 0.579891I
b = 1.13554 − 1.14001I
15.0538 + 5.8256I 6.10110 − 2.81591I
u = −0.914948 + 1.013510I
a = 0.49309 − 1.64704I
b = 1.15250 + 1.09178I
10.89350 − 7.61965I 2.72159 + 4.20211I
u = −0.914948 − 1.013510I
a = 0.49309 + 1.64704I
b = 1.15250 − 1.09178I
10.89350 + 7.61965I 2.72159 − 4.20211I
u = 0.931682 + 1.021610I
a = 0.39982 + 1.67662I
b = 1.22688 − 1.08995I
14.6962 + 12.8981I 5.54056 − 6.97685I
u = 0.931682 − 1.021610I
a = 0.39982 − 1.67662I
b = 1.22688 + 1.08995I
14.6962 − 12.8981I 5.54056 + 6.97685I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.473309 + 0.394862I
a = 2.44876 − 1.56604I
b = −0.697476 − 0.163056I
0.829785 − 0.751885I 3.89282 − 2.36905I
u = 0.473309 − 0.394862I
a = 2.44876 + 1.56604I
b = −0.697476 + 0.163056I
0.829785 + 0.751885I 3.89282 + 2.36905I
u = −0.261492 + 0.555195I
a = −0.24853 + 1.82453I
b = −0.984945 − 0.277413I
−1.87419 − 0.91390I −3.92701 + 0.44517I
u = −0.261492 − 0.555195I
a = −0.24853 − 1.82453I
b = −0.984945 + 0.277413I
−1.87419 + 0.91390I −3.92701 − 0.44517I
u = 0.560238
a = 0.651078
b = 0.0833533
1.12215 9.27350
u = −0.387160
a = 9.00889
b = −1.04399
−0.292584 54.7300
8
II.
I
u
2
= hb +1, −u
8
+2u
7
+· · · +a−1, u
9
−u
8
+2u
7
−u
6
+3u
5
−u
4
+2u
3
+u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
u
8
− 2u
7
+ 3u
6
− 3u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1
−1
a
6
=
1
−u
2
a
2
=
u
8
− 2u
7
+ 3u
6
− 3u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u
−1
a
1
=
−1
0
a
4
=
u
8
− 2u
7
+ 3u
6
− 3u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1
−1
a
7
=
1
−u
2
a
8
=
u
2
+ 1
−u
2
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
12
=
u
8
+ u
6
+ u
4
− 1
−u
8
+ u
7
− u
6
+ 2u
5
− u
4
+ 2u
3
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
8
− 4u
6
+ 3u
5
− 10u
4
+ u
3
− 7u
2
+ 6u − 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
6
u
9
c
4
(u + 1)
9
c
5
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
7
, c
10
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
8
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
9
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
11
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
12
u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
6
y
9
c
5
, c
9
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
7
, c
10
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
8
, c
11
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
12
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.140343 + 0.966856I
a = −1.004430 + 0.297869I
b = −1.00000
−3.42837 − 2.09337I −6.19892 + 4.26451I
u = −0.140343 − 0.966856I
a = −1.004430 − 0.297869I
b = −1.00000
−3.42837 + 2.09337I −6.19892 − 4.26451I
u = −0.628449 + 0.875112I
a = −0.275254 + 0.816341I
b = −1.00000
−1.02799 − 2.45442I −0.00914 + 2.54651I
u = −0.628449 − 0.875112I
a = −0.275254 − 0.816341I
b = −1.00000
−1.02799 + 2.45442I −0.00914 − 2.54651I
u = 0.796005 + 0.733148I
a = 0.070080 − 0.850995I
b = −1.00000
2.72642 − 1.33617I 5.35644 + 0.59665I
u = 0.796005 − 0.733148I
a = 0.070080 + 0.850995I
b = −1.00000
2.72642 + 1.33617I 5.35644 − 0.59665I
u = 0.728966 + 0.986295I
a = −0.195086 − 0.635552I
b = −1.00000
1.95319 + 7.08493I 3.81555 − 4.89194I
u = 0.728966 − 0.986295I
a = −0.195086 + 0.635552I
b = −1.00000
1.95319 − 7.08493I 3.81555 + 4.89194I
u = −0.512358
a = 3.80937
b = −1.00000
−0.446489 −9.92790
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
36
+ 2u
35
+ ··· + 151u + 1)
c
2
((u − 1)
9
)(u
36
− 10u
35
+ ··· + 19u − 1)
c
3
, c
6
u
9
(u
36
+ 3u
35
+ ··· + 512u + 512)
c
4
((u + 1)
9
)(u
36
− 10u
35
+ ··· + 19u − 1)
c
5
(u
9
− u
8
+ ··· + u + 1)(u
36
− 2u
35
+ ··· + u − 1)
c
7
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
36
− 6u
35
+ ··· + 790797u − 444601)
c
8
(u
9
− u
8
+ ··· − u + 1)(u
36
+ 2u
35
+ ··· + 5u + 1)
c
9
(u
9
+ u
8
+ ··· + u − 1)(u
36
− 2u
35
+ ··· + u − 1)
c
10
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
36
+ 6u
35
+ ··· + u + 1)
c
11
(u
9
+ u
8
+ ··· − u − 1)(u
36
+ 2u
35
+ ··· + 5u + 1)
c
12
(u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1)
· (u
36
− 22u
35
+ ··· + u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
36
+ 74y
35
+ ··· − 20743y + 1)
c
2
, c
4
((y −1)
9
)(y
36
− 2y
35
+ ··· − 151y + 1)
c
3
, c
6
y
9
(y
36
− 57y
35
+ ··· − 8126464y + 262144)
c
5
, c
9
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
36
+ 6y
35
+ ··· + y + 1)
c
7
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
36
+ 134y
35
+ ··· − 15572451598723y + 197670049201)
c
8
, c
11
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
36
− 22y
35
+ ··· + y + 1)
c
10
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
36
+ 50y
35
+ ··· − 35y + 1)
c
12
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
36
− 14y
35
+ ··· + 61y + 1)
14
