12n
0304
(K12n
0304
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 2 5 11 12 5 1 8 9
Solving Sequence
2,5
6 3
7,9
10 1 11 4 12 8
c
5
c
2
c
6
c
9
c
1
c
10
c
4
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
40
− 5u
39
+ ··· + b − 2, 2u
39
+ 5u
38
+ ··· + 2a − 7, u
41
+ 3u
40
+ ··· − 4u + 1i
I
u
2
= hb, a
2
− au − u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 47 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−2u
40
−5u
39
+· · ·+b−2, 2u
39
+5u
38
+· · ·+2a−7, u
41
+3u
40
+· · ·−4u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
9
=
−u
39
−
5
2
u
38
+ ··· + 4u +
7
2
2u
40
+ 5u
39
+ ··· − 12u + 2
a
10
=
2u
40
+ 4u
39
+ ··· − 8u +
11
2
2u
40
+ 5u
39
+ ··· − 12u + 2
a
1
=
u
3
u
5
− u
3
+ u
a
11
=
u
40
+
3
2
u
39
+ ··· −
1
2
u + 4
3
2
u
40
+
9
2
u
39
+ ··· −
21
2
u +
3
2
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
12
=
−
1
2
u
38
− u
37
+ ··· − 6u −
1
2
−u
12
+ 2u
10
− 4u
8
+ 4u
6
+ 2u
5
− 3u
4
− 2u
3
+ 2u
2
+ 2u
a
8
=
1
2
u
39
+ u
38
+ ··· +
5
2
u + 2
−
1
2
u
40
−
3
2
u
39
+ ··· +
3
2
u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
19
2
u
40
− 20u
39
+ ··· + 75u −
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
41
+ 15u
40
+ ··· + 54u + 1
c
2
, c
5
u
41
+ 3u
40
+ ··· − 4u + 1
c
3
u
41
− 3u
40
+ ··· + 2u + 1
c
4
, c
9
u
41
− u
40
+ ··· + 32u + 64
c
7
, c
8
, c
11
c
12
u
41
− 4u
40
+ ··· − u − 1
c
10
u
41
+ 6u
40
+ ··· − 25u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
41
+ 25y
40
+ ··· + 1814y −1
c
2
, c
5
y
41
− 15y
40
+ ··· + 54y −1
c
3
y
41
− 35y
40
+ ··· + 54y −1
c
4
, c
9
y
41
+ 35y
40
+ ··· − 23552y −4096
c
7
, c
8
, c
11
c
12
y
41
− 46y
40
+ ··· − y −1
c
10
y
41
+ 38y
40
+ ··· + 419y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.554966 + 0.821093I
a = 0.497003 − 1.229360I
b = −0.36616 + 1.39788I
−2.41826 − 3.80802I 3.60583 + 3.47707I
u = −0.554966 − 0.821093I
a = 0.497003 + 1.229360I
b = −0.36616 − 1.39788I
−2.41826 + 3.80802I 3.60583 − 3.47707I
u = −0.827605 + 0.515025I
a = −0.328705 + 0.303123I
b = −0.241607 + 0.924234I
1.70931 + 2.07723I 4.47890 − 3.71404I
u = −0.827605 − 0.515025I
a = −0.328705 − 0.303123I
b = −0.241607 − 0.924234I
1.70931 − 2.07723I 4.47890 + 3.71404I
u = 0.776023 + 0.578782I
a = 1.72542 + 0.58271I
b = −0.884835 + 0.226791I
1.60480 − 0.80076I 5.09218 − 0.58482I
u = 0.776023 − 0.578782I
a = 1.72542 − 0.58271I
b = −0.884835 − 0.226791I
1.60480 + 0.80076I 5.09218 + 0.58482I
u = −1.04869
a = 0.0572806
b = −1.17662
3.30786 2.08440
u = −0.613160 + 0.856728I
a = −0.94977 + 1.28570I
b = 0.59049 − 1.37882I
4.75528 − 6.89936I 6.92603 + 2.94505I
u = −0.613160 − 0.856728I
a = −0.94977 − 1.28570I
b = 0.59049 + 1.37882I
4.75528 + 6.89936I 6.92603 − 2.94505I
u = −0.851706 + 0.644533I
a = 0.208645 − 1.142410I
b = 0.123688 − 0.806905I
10.80640 + 2.51167I 3.89555 − 1.99801I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.851706 − 0.644533I
a = 0.208645 + 1.142410I
b = 0.123688 + 0.806905I
10.80640 − 2.51167I 3.89555 + 1.99801I
u = 0.607006 + 0.678640I
a = −2.19270 − 0.14272I
b = 1.058510 − 0.256762I
8.39222 + 0.77239I 9.02070 + 0.54313I
u = 0.607006 − 0.678640I
a = −2.19270 + 0.14272I
b = 1.058510 + 0.256762I
8.39222 − 0.77239I 9.02070 − 0.54313I
u = −0.468903 + 0.772766I
a = 0.050516 + 1.153690I
b = 0.094850 − 1.373440I
−2.97142 + 0.55275I 2.30390 − 2.65158I
u = −0.468903 − 0.772766I
a = 0.050516 − 1.153690I
b = 0.094850 + 1.373440I
−2.97142 − 0.55275I 2.30390 + 2.65158I
u = 0.917493 + 0.603542I
a = −1.30840 − 0.95232I
b = 0.987613 − 0.046245I
1.14349 − 3.91148I 2.83783 + 7.19272I
u = 0.917493 − 0.603542I
a = −1.30840 + 0.95232I
b = 0.987613 + 0.046245I
1.14349 + 3.91148I 2.83783 − 7.19272I
u = 1.145870 + 0.098490I
a = 0.557107 + 1.022870I
b = −0.43743 + 1.53434I
−1.90305 − 5.90674I 0.65317 + 3.76442I
u = 1.145870 − 0.098490I
a = 0.557107 − 1.022870I
b = −0.43743 − 1.53434I
−1.90305 + 5.90674I 0.65317 − 3.76442I
u = 1.150510 + 0.030891I
a = −0.178270 − 1.012240I
b = 0.13939 − 1.59211I
−8.52969 − 2.39687I −2.76728 + 3.02467I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.150510 − 0.030891I
a = −0.178270 + 1.012240I
b = 0.13939 + 1.59211I
−8.52969 + 2.39687I −2.76728 − 3.02467I
u = 0.879522 + 0.766671I
a = 0.349468 − 0.635389I
b = −0.011712 + 0.469579I
3.60760 − 2.89390I −5.91956 + 4.41976I
u = 0.879522 − 0.766671I
a = 0.349468 + 0.635389I
b = −0.011712 − 0.469579I
3.60760 + 2.89390I −5.91956 − 4.41976I
u = −0.318016 + 0.760424I
a = −0.76042 − 1.22775I
b = 0.286457 + 1.371580I
3.09276 + 3.48744I 6.36068 − 3.00060I
u = −0.318016 − 0.760424I
a = −0.76042 + 1.22775I
b = 0.286457 − 1.371580I
3.09276 − 3.48744I 6.36068 + 3.00060I
u = −0.803311 + 0.150840I
a = 0.146401 − 0.020062I
b = 0.412782 − 0.447090I
−1.348790 + 0.350630I −5.34995 − 0.74571I
u = −0.803311 − 0.150840I
a = 0.146401 + 0.020062I
b = 0.412782 + 0.447090I
−1.348790 − 0.350630I −5.34995 + 0.74571I
u = −1.057040 + 0.554919I
a = −1.211900 − 0.182135I
b = −0.13020 + 1.48466I
0.94110 + 1.27702I 2.95191 − 1.84094I
u = −1.057040 − 0.554919I
a = −1.211900 + 0.182135I
b = −0.13020 − 1.48466I
0.94110 − 1.27702I 2.95191 + 1.84094I
u = 1.004470 + 0.646364I
a = 1.16107 + 1.33962I
b = −1.216110 − 0.239593I
7.24069 − 5.93073I 6.53956 + 5.01754I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.004470 − 0.646364I
a = 1.16107 − 1.33962I
b = −1.216110 + 0.239593I
7.24069 + 5.93073I 6.53956 − 5.01754I
u = 0.891105 + 0.826186I
a = −0.68311 + 1.33083I
b = 0.038155 − 1.007650I
10.04850 − 3.07313I 5.86480 + 2.85684I
u = 0.891105 − 0.826186I
a = −0.68311 − 1.33083I
b = 0.038155 + 1.007650I
10.04850 + 3.07313I 5.86480 − 2.85684I
u = −1.066370 + 0.630817I
a = 1.70734 + 0.08552I
b = −0.22070 − 1.52920I
−4.70291 + 4.72137I 0. − 2.49884I
u = −1.066370 − 0.630817I
a = 1.70734 − 0.08552I
b = −0.22070 + 1.52920I
−4.70291 − 4.72137I 0. + 2.49884I
u = −1.067020 + 0.674301I
a = −2.03641 − 0.00933I
b = 0.45680 + 1.50432I
−3.95261 + 9.40997I 2.00000 − 7.76812I
u = −1.067020 − 0.674301I
a = −2.03641 + 0.00933I
b = 0.45680 − 1.50432I
−3.95261 − 9.40997I 2.00000 + 7.76812I
u = −1.061600 + 0.708976I
a = 2.31095 − 0.08552I
b = −0.65644 − 1.43319I
3.38790 + 12.73300I 5.12799 − 7.34194I
u = −1.061600 − 0.708976I
a = 2.31095 + 0.08552I
b = −0.65644 + 1.43319I
3.38790 − 12.73300I 5.12799 + 7.34194I
u = 0.530791
a = −3.58500
b = 0.511824
8.14249 16.8230
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.153263
a = 3.39924
b = −0.382289
0.765123 13.2670
9
II. I
u
2
= hb, a
2
− au − u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
9
=
a
0
a
10
=
a
0
a
1
=
u
2
− 1
−u
2
a
11
=
−au
−u
2
a + au + a
a
4
=
1
0
a
12
=
u
2
− a − u
−u
2
a
8
=
au − u + 1
u
2
a − au − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a − 3u
2
+ a + 8u + 7
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
10
(u
2
− u − 1)
3
c
11
, c
12
(u
2
+ u − 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.198308 + 1.205210I
b = 0
11.90680 − 2.82812I 11.55793 + 3.24268I
u = 0.877439 + 0.744862I
a = 0.075747 − 0.460350I
b = 0
4.01109 − 2.82812I 14.0681 + 1.5771I
u = 0.877439 − 0.744862I
a = −0.198308 − 1.205210I
b = 0
11.90680 + 2.82812I 11.55793 − 3.24268I
u = 0.877439 − 0.744862I
a = 0.075747 + 0.460350I
b = 0
4.01109 + 2.82812I 14.0681 − 1.5771I
u = −0.754878
a = 1.08457
b = 0
−0.126494 0.954070
u = −0.754878
a = −2.83945
b = 0
7.76919 −5.20600
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
2
)(u
41
+ 15u
40
+ ··· + 54u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
41
+ 3u
40
+ ··· − 4u + 1)
c
3
((u
3
− u
2
+ 2u − 1)
2
)(u
41
− 3u
40
+ ··· + 2u + 1)
c
4
, c
9
u
6
(u
41
− u
40
+ ··· + 32u + 64)
c
5
((u
3
− u
2
+ 1)
2
)(u
41
+ 3u
40
+ ··· − 4u + 1)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
41
+ 15u
40
+ ··· + 54u + 1)
c
7
, c
8
((u
2
− u − 1)
3
)(u
41
− 4u
40
+ ··· − u − 1)
c
10
((u
2
− u − 1)
3
)(u
41
+ 6u
40
+ ··· − 25u − 1)
c
11
, c
12
((u
2
+ u − 1)
3
)(u
41
− 4u
40
+ ··· − u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
41
+ 25y
40
+ ··· + 1814y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
41
− 15y
40
+ ··· + 54y −1)
c
3
((y
3
+ 3y
2
+ 2y −1)
2
)(y
41
− 35y
40
+ ··· + 54y −1)
c
4
, c
9
y
6
(y
41
+ 35y
40
+ ··· − 23552y −4096)
c
7
, c
8
, c
11
c
12
((y
2
− 3y + 1)
3
)(y
41
− 46y
40
+ ··· − y −1)
c
10
((y
2
− 3y + 1)
3
)(y
41
+ 38y
40
+ ··· + 419y −1)
15
