11n
62
(K11n
62
)
A knot diagram
1
Linearized knot diagam
4 1 9 2 9 11 10 3 1 7 6
Solving Sequence
1,9 4,10
3 2 5 6 8 7 11
c
9
c
3
c
2
c
4
c
5
c
8
c
7
c
11
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2464615243943u
19
+ 4244123981661u
18
+ ··· + 9728979932592b + 197070286033,
− 5509715115239u
19
− 10822359944445u
18
+ ··· + 9728979932592a + 3267475985807,
u
20
+ 2u
19
+ ··· + 5u
2
+ 1i
I
u
2
= hb, −u
3
− u
2
+ a − u, u
4
+ u
3
+ u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 24 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
= h2.46 × 10
12
u
19
+ 4.24 × 10
12
u
18
+ · · · + 9.73 × 10
12
b + 1.97 ×
10
11
, −5.51 × 10
12
u
19
− 1.08 × 10
13
u
18
+ · · · + 9.73 × 10
12
a + 3.27 ×
10
12
, u
20
+ 2u
19
+ · · · + 5u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
9
=
1
0
a
4
=
0.566320u
19
+ 1.11238u
18
+ ··· + 2.54326u − 0.335850
−0.253327u
19
− 0.436235u
18
+ ··· + 1.56632u − 0.0202560
a
10
=
1
u
2
a
3
=
0.819647u
19
+ 1.54862u
18
+ ··· + 0.976937u − 0.315594
−0.253327u
19
− 0.436235u
18
+ ··· + 1.56632u − 0.0202560
a
2
=
0.819647u
19
+ 1.54862u
18
+ ··· + 0.976937u − 0.315594
−0.396913u
19
− 0.728464u
18
+ ··· + 2.38597u − 0.110931
a
5
=
−0.650241u
19
− 1.16470u
18
+ ··· + 2.95229u − 0.131187
0.105052u
19
+ 0.296152u
18
+ ··· − 1.03621u + 0.246713
a
6
=
−0.755292u
19
− 1.46085u
18
+ ··· + 3.98849u − 0.377900
0.105052u
19
+ 0.296152u
18
+ ··· − 1.03621u + 0.246713
a
8
=
−0.246713u
19
− 0.388373u
18
+ ··· − 0.520256u − 1.03621
0.135781u
19
+ 0.563424u
18
+ ··· − 0.131187u + 0.650241
a
7
=
−0.296445u
19
− 0.863365u
18
+ ··· − 0.142357u − 1.79150
0.253735u
19
+ 0.918450u
18
+ ··· − 0.0814545u + 1.02577
a
11
=
−0.408108u
19
− 0.952751u
18
+ ··· + 1.33641u − 1.23004
−0.300860u
19
− 0.485772u
18
+ ··· − 0.821932u + 0.544643
a
11
=
−0.408108u
19
− 0.952751u
18
+ ··· + 1.33641u − 1.23004
−0.300860u
19
− 0.485772u
18
+ ··· − 0.821932u + 0.544643
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
815395233485
405374163858
u
19
+
427506532937
135124721286
u
18
+ ··· −
421979351069
405374163858
u −
2038924430399
405374163858
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
20
− 5u
19
+ ··· − 4u + 1
c
2
u
20
+ 3u
19
+ ··· − 4u + 1
c
3
, c
8
u
20
− u
19
+ ··· − 8u + 16
c
5
u
20
+ 2u
19
+ ··· + 154u + 445
c
6
, c
7
, c
10
c
11
u
20
+ 2u
19
+ ··· + 2u + 1
c
9
u
20
− 2u
19
+ ··· + 5u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
20
− 3y
19
+ ··· + 4y + 1
c
2
y
20
+ 33y
19
+ ··· + 4y + 1
c
3
, c
8
y
20
− 27y
19
+ ··· − 1344y + 256
c
5
y
20
+ 38y
19
+ ··· + 4809874y + 198025
c
6
, c
7
, c
10
c
11
y
20
+ 22y
19
+ ··· + 10y + 1
c
9
y
20
+ 26y
19
+ ··· + 10y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.673071 + 0.753931I
a = 0.535459 − 0.003526I
b = 0.731278 + 0.210088I
0.02154 − 2.08472I 2.36846 + 5.36236I
u = 0.673071 − 0.753931I
a = 0.535459 + 0.003526I
b = 0.731278 − 0.210088I
0.02154 + 2.08472I 2.36846 − 5.36236I
u = −0.094946 + 0.739352I
a = 1.098280 + 0.336321I
b = 0.448296 + 1.074360I
−4.77753 + 2.99094I −0.69176 − 3.46155I
u = −0.094946 − 0.739352I
a = 1.098280 − 0.336321I
b = 0.448296 − 1.074360I
−4.77753 − 2.99094I −0.69176 + 3.46155I
u = −0.177522 + 0.687359I
a = −0.716990 + 0.247004I
b = −0.553957 + 0.621299I
0.995000 − 0.993446I 5.17867 + 4.04800I
u = −0.177522 − 0.687359I
a = −0.716990 − 0.247004I
b = −0.553957 − 0.621299I
0.995000 + 0.993446I 5.17867 − 4.04800I
u = −1.12972 + 0.93010I
a = −0.456176 − 0.088007I
b = −0.757198 + 0.007629I
−7.51526 + 3.82239I 0.11541 − 4.60594I
u = −1.12972 − 0.93010I
a = −0.456176 + 0.088007I
b = −0.757198 − 0.007629I
−7.51526 − 3.82239I 0.11541 + 4.60594I
u = −0.382707 + 0.237846I
a = 2.61545 + 0.82402I
b = −0.767833 + 0.639917I
−7.81656 + 1.26535I −3.51291 − 0.02866I
u = −0.382707 − 0.237846I
a = 2.61545 − 0.82402I
b = −0.767833 − 0.639917I
−7.81656 − 1.26535I −3.51291 + 0.02866I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.18268 + 1.66062I
a = 0.734667 − 0.455842I
b = 1.90756 − 0.02696I
3.14125 + 1.95377I −0.018441 − 0.726692I
u = 0.18268 − 1.66062I
a = 0.734667 + 0.455842I
b = 1.90756 + 0.02696I
3.14125 − 1.95377I −0.018441 + 0.726692I
u = 0.177052 + 0.214813I
a = −2.67119 + 2.37138I
b = 0.317909 + 0.453091I
−1.76070 − 0.62769I −5.52555 − 1.68478I
u = 0.177052 − 0.214813I
a = −2.67119 − 2.37138I
b = 0.317909 − 0.453091I
−1.76070 + 0.62769I −5.52555 + 1.68478I
u = −0.21357 + 1.74539I
a = −0.684239 − 0.489378I
b = −1.90198 − 0.23325I
9.48767 + 1.51858I 3.11046 + 0.47571I
u = −0.21357 − 1.74539I
a = −0.684239 + 0.489378I
b = −1.90198 + 0.23325I
9.48767 − 1.51858I 3.11046 − 0.47571I
u = 0.24322 + 1.81257I
a = 0.640240 − 0.509585I
b = 1.85951 − 0.39598I
9.21922 − 6.23574I 2.42777 + 5.05678I
u = 0.24322 − 1.81257I
a = 0.640240 + 0.509585I
b = 1.85951 + 0.39598I
9.21922 + 6.23574I 2.42777 − 5.05678I
u = −0.27754 + 1.87563I
a = −0.595500 − 0.521593I
b = −1.78359 − 0.53915I
2.29524 + 9.61446I −0.95211 − 4.92599I
u = −0.27754 − 1.87563I
a = −0.595500 + 0.521593I
b = −1.78359 + 0.53915I
2.29524 − 9.61446I −0.95211 + 4.92599I
6
II. I
u
2
= hb, −u
3
− u
2
+ a − u, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
9
=
1
0
a
4
=
u
3
+ u
2
+ u
0
a
10
=
1
u
2
a
3
=
u
3
+ u
2
+ u
0
a
2
=
u
3
+ u
2
+ u
u
a
5
=
0
−u
a
6
=
u
−u
a
8
=
1
0
a
7
=
u
2
+ 1
−u
3
− u
2
− 1
a
11
=
−u
3
u
3
+ u
a
11
=
−u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 5u − 1
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
8
u
4
c
5
, c
9
u
4
+ u
3
+ u
2
+ 1
c
6
, c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
10
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
8
(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
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
6
, c
7
, c
10
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = −0.547424 + 1.120870I
b = 0
−1.43393 − 1.41510I −0.82145 + 5.62908I
u = 0.351808 − 0.720342I
a = −0.547424 − 1.120870I
b = 0
−1.43393 + 1.41510I −0.82145 − 5.62908I
u = −0.851808 + 0.911292I
a = 0.547424 + 0.585652I
b = 0
−8.43568 + 3.16396I −5.67855 − 1.65351I
u = −0.851808 − 0.911292I
a = 0.547424 − 0.585652I
b = 0
−8.43568 − 3.16396I −5.67855 + 1.65351I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
20
− 5u
19
+ ··· − 4u + 1)
c
2
((u + 1)
4
)(u
20
+ 3u
19
+ ··· − 4u + 1)
c
3
, c
8
u
4
(u
20
− u
19
+ ··· − 8u + 16)
c
4
((u + 1)
4
)(u
20
− 5u
19
+ ··· − 4u + 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
20
+ 2u
19
+ ··· + 154u + 445)
c
6
, c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
c
9
(u
4
+ u
3
+ u
2
+ 1)(u
20
− 2u
19
+ ··· + 5u
2
+ 1)
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
20
− 3y
19
+ ··· + 4y + 1)
c
2
((y −1)
4
)(y
20
+ 33y
19
+ ··· + 4y + 1)
c
3
, c
8
y
4
(y
20
− 27y
19
+ ··· − 1344y + 256)
c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 38y
19
+ ··· + 4809874y + 198025)
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
20
+ 22y
19
+ ··· + 10y + 1)
c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 26y
19
+ ··· + 10y + 1)
12
