10
101
(K10a
45
)
A knot diagram
1
Linearized knot diagam
6 9 7 10 3 1 5 2 8 4
Solving Sequence
3,9 2,6
1 5 8 10 4 7
c
2
c
1
c
5
c
8
c
9
c
4
c
7
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
16
− 32u
15
+ ··· + 2b + 12, 9u
16
− 51u
15
+ ··· + 2a + 43, u
17
− 6u
16
+ ··· + 26u − 4i
I
u
2
= h170u
7
a
3
− 173u
7
a
2
+ ··· − 479a + 513, 2u
7
a
3
− 11u
7
a
2
+ ··· − 38a + 27,
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
I
u
3
= hu
6
− 2u
4
+ 2u
2
+ b + u − 1, u
6
− u
5
− u
4
+ 3u
2
+ a − 1, u
7
− u
6
− u
5
+ u
4
+ 2u
3
− u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 56 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
=
h7u
16
−32u
15
+· · ·+2b+12, 9u
16
−51u
15
+· · ·+2a+43, u
17
−6u
16
+· · ·+26u−4i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
−u
2
a
6
=
−
9
2
u
16
+
51
2
u
15
+ ··· + 127u −
43
2
−
7
2
u
16
+ 16u
15
+ ··· +
87
2
u − 6
a
1
=
−
9
4
u
16
+ 10u
15
+ ··· +
95
4
u − 3
3
2
u
16
− 8u
15
+ ··· −
79
2
u + 7
a
5
=
−8u
16
+
83
2
u
15
+ ··· +
341
2
u −
55
2
−
7
2
u
16
+ 16u
15
+ ··· +
87
2
u − 6
a
8
=
u
−u
3
+ u
a
10
=
−u
3
u
5
− u
3
+ u
a
4
=
−u
16
+
9
2
u
15
+ ··· +
53
2
u −
7
2
1
2
u
16
− 2u
15
+ ··· +
7
2
u
2
+
1
2
u
a
7
=
−
11
4
u
16
+ 15u
15
+ ··· +
293
4
u − 12
−
3
2
u
16
+ 8u
15
+ ··· +
65
2
u − 5
(ii) Obstruction class = −1
(iii) Cusp Shapes = −15u
16
+ 81u
15
− 169u
14
+ 62u
13
+ 429u
12
− 933u
11
+ 562u
10
+
878u
9
− 2024u
8
+ 1325u
7
+ 679u
6
− 1802u
5
+ 1076u
4
+ 221u
3
− 663u
2
+ 360u − 78
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
17
+ 7u
15
+ ··· + u + 1
c
2
, c
8
u
17
+ 6u
16
+ ··· + 26u + 4
c
3
u
17
+ 18u
16
+ ··· + 2816u + 256
c
5
, c
7
u
17
+ u
16
+ ··· + 6u + 1
c
9
u
17
+ 6u
16
+ ··· + 188u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
17
+ 14y
16
+ ··· + 7y −1
c
2
, c
8
y
17
− 6y
16
+ ··· + 188y −16
c
3
y
17
+ 2y
16
+ ··· + 524288y −65536
c
5
, c
7
y
17
+ 3y
16
+ ··· + 6y −1
c
9
y
17
+ 10y
16
+ ··· + 14704y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.902416 + 0.208075I
a = −1.79366 − 0.59487I
b = 1.199630 + 0.242688I
−3.24705 + 0.67841I −12.7998 − 8.2767I
u = −0.902416 − 0.208075I
a = −1.79366 + 0.59487I
b = 1.199630 − 0.242688I
−3.24705 − 0.67841I −12.7998 + 8.2767I
u = 0.938877 + 0.582285I
a = −1.134880 + 0.826949I
b = 0.715526 + 0.898293I
−0.99442 − 4.22945I −12.33800 + 5.21456I
u = 0.938877 − 0.582285I
a = −1.134880 − 0.826949I
b = 0.715526 − 0.898293I
−0.99442 + 4.22945I −12.33800 − 5.21456I
u = 0.739806 + 0.493958I
a = 0.794662 − 0.257822I
b = 0.240261 − 0.634801I
−0.323057 − 0.236182I −11.08521 − 0.74956I
u = 0.739806 − 0.493958I
a = 0.794662 + 0.257822I
b = 0.240261 + 0.634801I
−0.323057 + 0.236182I −11.08521 + 0.74956I
u = 0.602874 + 0.959066I
a = −0.051648 − 0.335588I
b = −0.85046 + 1.32525I
9.54876 + 8.47221I −3.97806 − 4.13044I
u = 0.602874 − 0.959066I
a = −0.051648 + 0.335588I
b = −0.85046 − 1.32525I
9.54876 − 8.47221I −3.97806 + 4.13044I
u = 0.465319 + 1.172900I
a = −0.076760 + 0.308019I
b = 0.134443 − 0.808764I
7.94985 − 2.22960I 2.70903 + 2.09494I
u = 0.465319 − 1.172900I
a = −0.076760 − 0.308019I
b = 0.134443 + 0.808764I
7.94985 + 2.22960I 2.70903 − 2.09494I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.101610 + 0.741547I
a = 1.82798 − 0.24157I
b = −1.09788 − 1.34726I
7.9938 − 14.6875I −6.10908 + 8.19550I
u = 1.101610 − 0.741547I
a = 1.82798 + 0.24157I
b = −1.09788 + 1.34726I
7.9938 + 14.6875I −6.10908 − 8.19550I
u = −1.311020 + 0.221936I
a = 0.907067 − 0.771746I
b = −0.650500 + 0.629679I
1.56788 + 6.73537I −9.26043 − 8.18250I
u = −1.311020 − 0.221936I
a = 0.907067 + 0.771746I
b = −0.650500 − 0.629679I
1.56788 − 6.73537I −9.26043 + 8.18250I
u = 1.18518 + 0.83889I
a = −0.962583 + 0.017371I
b = 0.662446 + 0.685312I
5.79881 − 4.87487I −3.72990 + 6.85875I
u = 1.18518 − 0.83889I
a = −0.962583 − 0.017371I
b = 0.662446 − 0.685312I
5.79881 + 4.87487I −3.72990 − 6.85875I
u = 0.359530
a = 0.979665
b = 0.293070
−0.661408 −14.8170
6
II. I
u
2
= h170u
7
a
3
− 173u
7
a
2
+ · · · − 479a + 513, 2u
7
a
3
− 11u
7
a
2
+ · · · −
38a + 27, u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
−u
2
a
6
=
a
−3.95349a
3
u
7
+ 4.02326a
2
u
7
+ ··· + 11.1395a − 11.9302
a
1
=
1.97674a
3
u
7
− 3.02326a
2
u
7
+ ··· − 13.0698a + 16.9302
1.02326a
2
u
7
− 0.511628u
7
+ ··· − 0.139535a
2
+ 1.06977
a
5
=
−3.95349a
3
u
7
+ 4.02326a
2
u
7
+ ··· + 12.1395a − 11.9302
−3.95349a
3
u
7
+ 4.02326a
2
u
7
+ ··· + 11.1395a − 11.9302
a
8
=
u
−u
3
+ u
a
10
=
−u
3
u
5
− u
3
+ u
a
4
=
−3u
7
a
3
+ 3u
7
a
2
+ ··· + 12a − 13
−2.04651a
3
u
7
+ 1.97674a
2
u
7
+ ··· + 6.86047a − 8.06977
a
7
=
3.06977a
3
u
7
− 2.95349a
2
u
7
+ ··· − 4.79070a + 4.13953
2.04651a
3
u
7
− 1.97674a
2
u
7
+ ··· − 6.86047a + 7.06977
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
352
43
u
7
a
3
+
340
43
u
7
a
2
+ ··· +
1180
43
a −
1818
43
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
32
+ u
31
+ ··· − 10u + 1
c
2
, c
8
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
4
c
3
(u
2
− u + 1)
16
c
5
, c
7
u
32
− 9u
31
+ ··· − 602u + 73
c
9
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
32
+ 27y
31
+ ··· + 72y + 1
c
2
, c
8
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
4
c
3
(y
2
+ y + 1)
16
c
5
, c
7
y
32
+ 11y
31
+ ··· + 23912y + 5329
c
9
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = 0.506748 + 0.672291I
b = −0.652472 + 0.678771I
3.89415 + 0.89865I −5.41522 − 2.95331I
u = −0.570868 + 0.730671I
a = −0.193541 − 0.783831I
b = −0.485561 − 0.705520I
3.89415 − 3.16112I −5.41522 + 3.97489I
u = −0.570868 + 0.730671I
a = 0.503876 − 0.375809I
b = 0.69485 + 1.61093I
3.89415 − 3.16112I −5.41522 + 3.97489I
u = −0.570868 + 0.730671I
a = 0.342363 + 0.176286I
b = −0.236280 − 0.950229I
3.89415 + 0.89865I −5.41522 − 2.95331I
u = −0.570868 − 0.730671I
a = 0.506748 − 0.672291I
b = −0.652472 − 0.678771I
3.89415 − 0.89865I −5.41522 + 2.95331I
u = −0.570868 − 0.730671I
a = −0.193541 + 0.783831I
b = −0.485561 + 0.705520I
3.89415 + 3.16112I −5.41522 − 3.97489I
u = −0.570868 − 0.730671I
a = 0.503876 + 0.375809I
b = 0.69485 − 1.61093I
3.89415 + 3.16112I −5.41522 − 3.97489I
u = −0.570868 − 0.730671I
a = 0.342363 − 0.176286I
b = −0.236280 + 0.950229I
3.89415 − 0.89865I −5.41522 + 2.95331I
u = 0.855237 + 0.665892I
a = −0.524115 − 0.290681I
b = 0.407531 − 1.001570I
7.09422 − 0.54861I −2.27708 + 0.10386I
u = 0.855237 + 0.665892I
a = 0.10958 − 1.41649I
b = −1.60671 + 1.59050I
7.09422 − 4.60838I −2.27708 + 7.03206I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.855237 + 0.665892I
a = −2.13751 + 0.52423I
b = 0.473109 + 0.696143I
7.09422 − 4.60838I −2.27708 + 7.03206I
u = 0.855237 + 0.665892I
a = 2.31080 − 1.01943I
b = −1.82102 − 1.12347I
7.09422 − 0.54861I −2.27708 + 0.10386I
u = 0.855237 − 0.665892I
a = −0.524115 + 0.290681I
b = 0.407531 + 1.001570I
7.09422 + 0.54861I −2.27708 − 0.10386I
u = 0.855237 − 0.665892I
a = 0.10958 + 1.41649I
b = −1.60671 − 1.59050I
7.09422 + 4.60838I −2.27708 − 7.03206I
u = 0.855237 − 0.665892I
a = −2.13751 − 0.52423I
b = 0.473109 − 0.696143I
7.09422 + 4.60838I −2.27708 − 7.03206I
u = 0.855237 − 0.665892I
a = 2.31080 + 1.01943I
b = −1.82102 + 1.12347I
7.09422 + 0.54861I −2.27708 − 0.10386I
u = 1.09818
a = 1.40909 + 0.27112I
b = −0.797129 − 0.510365I
−1.56793 − 2.02988I −11.86404 + 3.46410I
u = 1.09818
a = 1.40909 − 0.27112I
b = −0.797129 + 0.510365I
−1.56793 + 2.02988I −11.86404 − 3.46410I
u = 1.09818
a = −0.79078 + 1.34206I
b = 0.736952 − 0.614594I
−1.56793 + 2.02988I −11.86404 − 3.46410I
u = 1.09818
a = −0.79078 − 1.34206I
b = 0.736952 + 0.614594I
−1.56793 − 2.02988I −11.86404 + 3.46410I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.031810 + 0.655470I
a = 1.42398 + 0.08840I
b = −0.594791 + 0.693288I
2.55512 + 4.41365I −7.42845 − 1.83007I
u = −1.031810 + 0.655470I
a = 1.24001 + 0.82764I
b = −0.595404 + 0.907218I
2.55512 + 8.47342I −7.42845 − 8.75827I
u = −1.031810 + 0.655470I
a = −0.350327 + 0.360263I
b = −0.302783 − 0.841128I
2.55512 + 4.41365I −7.42845 − 1.83007I
u = −1.031810 + 0.655470I
a = −2.16539 − 0.12217I
b = 1.17222 − 1.61062I
2.55512 + 8.47342I −7.42845 − 8.75827I
u = −1.031810 − 0.655470I
a = 1.42398 − 0.08840I
b = −0.594791 − 0.693288I
2.55512 − 4.41365I −7.42845 + 1.83007I
u = −1.031810 − 0.655470I
a = 1.24001 − 0.82764I
b = −0.595404 − 0.907218I
2.55512 − 8.47342I −7.42845 + 8.75827I
u = −1.031810 − 0.655470I
a = −0.350327 − 0.360263I
b = −0.302783 + 0.841128I
2.55512 − 4.41365I −7.42845 + 1.83007I
u = −1.031810 − 0.655470I
a = −2.16539 + 0.12217I
b = 1.17222 + 1.61062I
2.55512 − 8.47342I −7.42845 + 8.75827I
u = −0.603304
a = 1.41265 + 0.18976I
b = −0.287361 + 1.327250I
4.08977 − 2.02988I −9.89446 + 3.46410I
u = −0.603304
a = 1.41265 − 0.18976I
b = −0.287361 − 1.327250I
4.08977 + 2.02988I −9.89446 − 3.46410I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.603304
a = 0.40258 + 3.33383I
b = −0.605158 − 0.218634I
4.08977 + 2.02988I −9.89446 − 3.46410I
u = −0.603304
a = 0.40258 − 3.33383I
b = −0.605158 + 0.218634I
4.08977 − 2.02988I −9.89446 + 3.46410I
13
III. I
u
3
= hu
6
− 2u
4
+ 2u
2
+ b + u − 1, u
6
− u
5
− u
4
+ 3u
2
+ a − 1, u
7
− u
6
−
u
5
+ u
4
+ 2u
3
− u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
−u
2
a
6
=
−u
6
+ u
5
+ u
4
− 3u
2
+ 1
−u
6
+ 2u
4
− 2u
2
− u + 1
a
1
=
u
6
− u
4
− u
3
+ u
2
+ 2u + 1
u
5
− u
3
− u
2
+ u + 1
a
5
=
−2u
6
+ u
5
+ 3u
4
− 5u
2
− u + 2
−u
6
+ 2u
4
− 2u
2
− u + 1
a
8
=
u
−u
3
+ u
a
10
=
−u
3
u
5
− u
3
+ u
a
4
=
−2u
6
+ u
5
+ 2u
4
− 4u
2
− u + 2
−u
a
7
=
u
6
− u
5
− 2u
4
+ u
3
+ 3u
2
− 3
u
6
− u
5
− u
4
+ 2u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
6
+ 4u
5
+ u
4
− u
3
− 2u
2
+ 4u − 4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
7
+ 4u
5
+ u
4
+ 6u
3
+ 2u
2
+ 4u + 1
c
2
u
7
− u
6
− u
5
+ u
4
+ 2u
3
− u
2
− u + 1
c
3
u
7
+ u
6
+ u
5
+ u
4
− u
2
− u − 1
c
5
, c
7
u
7
− u
6
+ u
5
− u
3
+ u
2
− u + 1
c
6
, c
10
u
7
+ 4u
5
− u
4
+ 6u
3
− 2u
2
+ 4u − 1
c
8
u
7
+ u
6
− u
5
− u
4
+ 2u
3
+ u
2
− u − 1
c
9
u
7
+ 3u
6
+ 7u
5
+ 9u
4
+ 10u
3
+ 7u
2
+ 3u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
7
+ 8y
6
+ 28y
5
+ 55y
4
+ 64y
3
+ 42y
2
+ 12y − 1
c
2
, c
8
y
7
− 3y
6
+ 7y
5
− 9y
4
+ 10y
3
− 7y
2
+ 3y − 1
c
3
y
7
+ y
6
− y
5
− y
4
+ 2y
3
+ y
2
− y − 1
c
5
, c
7
y
7
+ y
6
− y
5
− 2y
4
+ y
3
+ y
2
− y − 1
c
9
y
7
+ 5y
6
+ 15y
5
+ 23y
4
+ 10y
3
− 7y
2
− 5y − 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.793128 + 0.750889I
a = 0.905690 + 0.804408I
b = −0.888952 − 0.354053I
6.43224 + 2.89342I −4.05142 − 2.86813I
u = −0.793128 − 0.750889I
a = 0.905690 − 0.804408I
b = −0.888952 + 0.354053I
6.43224 − 2.89342I −4.05142 + 2.86813I
u = −0.879508
a = −1.71136
b = 1.06630
−3.03629 −10.8170
u = 0.610619 + 0.459179I
a = 0.114923 − 1.389810I
b = −0.362477 − 1.085130I
5.06800 + 1.30245I −2.75170 + 0.65887I
u = 0.610619 − 0.459179I
a = 0.114923 + 1.389810I
b = −0.362477 + 1.085130I
5.06800 − 1.30245I −2.75170 − 0.65887I
u = 1.122260 + 0.611121I
a = −1.164940 − 0.276203I
b = 0.218278 + 0.857268I
3.17738 − 5.75449I −4.78830 + 6.98275I
u = 1.122260 − 0.611121I
a = −1.164940 + 0.276203I
b = 0.218278 − 0.857268I
3.17738 + 5.75449I −4.78830 − 6.98275I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
7
+ 4u
5
+ u
4
+ 6u
3
+ 2u
2
+ 4u + 1)(u
17
+ 7u
15
+ ··· + u + 1)
· (u
32
+ u
31
+ ··· − 10u + 1)
c
2
(u
7
− u
6
− u
5
+ u
4
+ 2u
3
− u
2
− u + 1)
· ((u
8
− u
7
+ ··· + 2u − 1)
4
)(u
17
+ 6u
16
+ ··· + 26u + 4)
c
3
(u
2
− u + 1)
16
(u
7
+ u
6
+ u
5
+ u
4
− u
2
− u − 1)
· (u
17
+ 18u
16
+ ··· + 2816u + 256)
c
5
, c
7
(u
7
− u
6
+ u
5
− u
3
+ u
2
− u + 1)(u
17
+ u
16
+ ··· + 6u + 1)
· (u
32
− 9u
31
+ ··· − 602u + 73)
c
6
, c
10
(u
7
+ 4u
5
− u
4
+ 6u
3
− 2u
2
+ 4u − 1)(u
17
+ 7u
15
+ ··· + u + 1)
· (u
32
+ u
31
+ ··· − 10u + 1)
c
8
(u
7
+ u
6
− u
5
− u
4
+ 2u
3
+ u
2
− u − 1)
· ((u
8
− u
7
+ ··· + 2u − 1)
4
)(u
17
+ 6u
16
+ ··· + 26u + 4)
c
9
(u
7
+ 3u
6
+ 7u
5
+ 9u
4
+ 10u
3
+ 7u
2
+ 3u + 1)
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
4
· (u
17
+ 6u
16
+ ··· + 188u + 16)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
(y
7
+ 8y
6
+ 28y
5
+ 55y
4
+ 64y
3
+ 42y
2
+ 12y − 1)
· (y
17
+ 14y
16
+ ··· + 7y − 1)(y
32
+ 27y
31
+ ··· + 72y + 1)
c
2
, c
8
(y
7
− 3y
6
+ 7y
5
− 9y
4
+ 10y
3
− 7y
2
+ 3y − 1)
· (y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
4
· (y
17
− 6y
16
+ ··· + 188y − 16)
c
3
(y
2
+ y + 1)
16
(y
7
+ y
6
− y
5
− y
4
+ 2y
3
+ y
2
− y − 1)
· (y
17
+ 2y
16
+ ··· + 524288y − 65536)
c
5
, c
7
(y
7
+ y
6
+ ··· − y − 1)(y
17
+ 3y
16
+ ··· + 6y − 1)
· (y
32
+ 11y
31
+ ··· + 23912y + 5329)
c
9
(y
7
+ 5y
6
+ 15y
5
+ 23y
4
+ 10y
3
− 7y
2
− 5y − 1)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
4
· (y
17
+ 10y
16
+ ··· + 14704y − 256)
19
