10
115
(K10a
94
)
1
Arc Sequences
7 5 1 10 2 9 3 4 6 8
Solving Sequence
2,6
5
3,10
4 9 7 1 8
c
5
c
2
c
4
c
9
c
6
c
1
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
12
4u
11
+ 11u
10
22u
9
+ 33u
8
41u
7
+ 40u
6
33u
5
+ 24u
4
13u
3
+ 8u
2
2u + 1,
2u
11
+ 7u
10
17u
9
+ 29u
8
34u
7
+ 31u
6
16u
5
+ 3u
4
+ u
3
6u
2
+ a + u 4,
u
11
+ 4u
10
10u
9
+ 18u
8
23u
7
+ 23u
6
17u
5
+ 10u
4
8u
3
+ 5u
2
+ b 3u + 1i
I
u
2
= hu
66
3u
65
+ ··· + 77u + 21,
3.70660 × 10
116
u
65
1.01795 × 10
117
u
64
+ ··· + 2.00531 × 10
117
a + 8.92905 × 10
117
,
6.39895 × 10
117
u
65
+ 2.04454 × 10
118
u
64
+ ··· + 2.03396 × 10
118
b 4.22854 × 10
118
i
There are 2 irreducible components with 78 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I.
I
u
1
= hu
12
4u
11
+· · ·−2u+1, 2u
11
+7u
10
+· · ·+a4, u
11
+4u
10
+· · ·+b+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
10
=
2u
11
7u
10
+ ··· u + 4
u
11
4u
10
+ ··· + 3u 1
a
4
=
4u
11
+ 16u
10
+ ··· 16u + 2
u
11
4u
10
+ ··· + 3u + 1
a
9
=
2u
11
7u
10
+ ··· u + 4
u
5
2u
4
+ 4u
3
5u
2
+ 3u 2
a
7
=
4u
11
18u
10
+ ··· + 23u 6
2u
11
+ 7u
10
+ ··· u 2
a
1
=
4u
11
17u
10
+ ··· + 24u 6
u
11
+ 4u
10
+ ··· + 11u
2
6u
a
8
=
4u
11
17u
10
+ ··· + 19u 4
u
11
+ 3u
10
+ ··· + 2u 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 6u
11
+26u
10
70u
9
+137u
8
198u
7
+228u
6
208u
5
+155u
4
112u
3
+67u
2
34u+12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.275611 0.671814I
a = 0.711883 0.702298I
b = 0.26365 2.63645I
4.91597I 11.1152I
u = 0.275611 + 0.671814I
a = 0.711883 + 0.702298I
b = 0.26365 + 2.63645I
4.91597I 11.1152I
u = 0.058582 0.533279I
a = 2.66328 0.11018I
b = 0.009607 0.675486I
3.74262 3.79217I 1.40025 + 6.58435I
u = 0.058582 + 0.533279I
a = 2.66328 + 0.11018I
b = 0.009607 + 0.675486I
3.74262 + 3.79217I 1.40025 6.58435I
u = 0.09726 1.42673I
a = 0.374835 + 0.015507I
b = 0.745540 + 0.810936I
3.74262 + 3.79217I 1.40025 6.58435I
u = 0.09726 + 1.42673I
a = 0.374835 0.015507I
b = 0.745540 0.810936I
3.74262 3.79217I 1.40025 + 6.58435I
u = 0.238381 0.958097I
a = 0.554558 + 1.261575I
b = 0.16536 + 1.71751I
3.76649 + 0.96528I 2.46025 6.19259I
u = 0.238381 + 0.958097I
a = 0.554558 1.261575I
b = 0.16536 1.71751I
3.76649 0.96528I 2.46025 + 6.19259I
u = 0.540477 1.222062I
a = 0.672801 + 0.739823I
b = 0.89278 + 1.69181I
6.92803I 5.92253I
u = 0.540477 + 1.222062I
a = 0.672801 0.739823I
b = 0.89278 1.69181I
6.92803I 5.92253I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.340908 0.230586I
a = 0.292010 0.664299I
b = 0.264083 0.377825I
3.76649 0.96528I 2.46025 + 6.19259I
u = 1.340908 + 0.230586I
a = 0.292010 + 0.664299I
b = 0.264083 + 0.377825I
3.76649 + 0.96528I 2.46025 6.19259I
4
II. I
u
2
=
hu
66
3u
65
+· · ·+77u+21, 3.71×10
116
u
65
1.02×10
117
u
64
+· · ·+2.01×10
117
a+
8.93×10
117
, 6.40×10
117
u
65
+2.04×10
118
u
64
+· · ·+ 2.03×10
118
b4.23×10
118
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
10
=
0.184839u
65
+ 0.507626u
64
+ ··· 17.4166u 4.45269
0.314605u
65
1.00520u
64
+ ··· + 20.3943u + 2.07897
a
4
=
0.308403u
65
+ 0.949029u
64
+ ··· 17.9221u 1.73509
0.481633u
65
1.52717u
64
+ ··· + 24.6346u 0.797674
a
9
=
0.184839u
65
+ 0.507626u
64
+ ··· 17.4166u 4.45269
0.400189u
65
1.21192u
64
+ ··· + 27.8865u + 3.06369
a
7
=
0.188623u
65
0.659710u
64
+ ··· + 18.7072u + 3.61763
0.173642u
65
+ 0.604165u
64
+ ··· + 4.04969u + 3.85319
a
1
=
0.146618u
65
+ 0.462669u
64
+ ··· 32.6222u 8.43124
0.0485928u
65
+ 0.355584u
64
+ ··· + 30.8350u + 9.06616
a
8
=
0.212665u
65
0.744957u
64
+ ··· + 14.8759u + 2.51546
0.257271u
65
+ 0.795851u
64
+ ··· 1.75953u + 2.88552
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.703263u
65
+ 1.38008u
64
+ ··· 130.121u 35.8005
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.195213 0.323364I
a = 0.638585 + 0.816935I
b = 0.242388 + 0.116276I
3.08497 + 10.03664I 2.48774 7.33059I
u = 1.195213 + 0.323364I
a = 0.638585 0.816935I
b = 0.242388 0.116276I
3.08497 10.03664I 2.48774 + 7.33059I
u = 0.734419 0.105022I
a = 0.959921 0.280270I
b = 0.275200 0.957750I
5.41602I 4.57520I
u = 0.734419 + 0.105022I
a = 0.959921 + 0.280270I
b = 0.275200 + 0.957750I
5.41602I 4.57520I
u = 0.687231 0.358682I
a = 1.20075 1.10102I
b = 0.163175 0.430869I
1.38542 + 3.12807I 0.18739 5.83461I
u = 0.687231 + 0.358682I
a = 1.20075 + 1.10102I
b = 0.163175 + 0.430869I
1.38542 3.12807I 0.18739 + 5.83461I
u = 0.68120 1.28817I
a = 0.562935 0.826501I
b = 1.06170 1.94740I
16.6380I 9.37696I
u = 0.68120 + 1.28817I
a = 0.562935 + 0.826501I
b = 1.06170 + 1.94740I
16.6380I 9.37696I
u = 0.643705 0.913849I
a = 0.205125 + 0.869703I
b = 0.02417 + 1.63012I
2.10904 4.58826I 5.96198 + 6.80019I
u = 0.643705 + 0.913849I
a = 0.205125 0.869703I
b = 0.02417 1.63012I
2.10904 + 4.58826I 5.96198 6.80019I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.542734 1.105869I
a = 0.730042 + 0.981803I
b = 0.84096 + 1.94152I
0.80844 7.87674I 4.76079 + 12.34614I
u = 0.542734 + 1.105869I
a = 0.730042 0.981803I
b = 0.84096 1.94152I
0.80844 + 7.87674I 4.76079 12.34614I
u = 0.499077 1.182907I
a = 0.593936 0.759817I
b = 0.02839 1.69149I
3.08497 10.03664I 2.48774 + 7.33059I
u = 0.499077 + 1.182907I
a = 0.593936 + 0.759817I
b = 0.02839 + 1.69149I
3.08497 + 10.03664I 2.48774 7.33059I
u = 0.458890 0.841270I
a = 0.920320 0.973850I
b = 1.33518 1.22881I
3.75920 + 1.29912I 3.74536 + 0.15014I
u = 0.458890 + 0.841270I
a = 0.920320 + 0.973850I
b = 1.33518 + 1.22881I
3.75920 1.29912I 3.74536 0.15014I
u = 0.396945 1.221128I
a = 0.512607 0.542422I
b = 0.12573 1.45523I
3.75920 + 1.29912I 3.74536 + 0.15014I
u = 0.396945 + 1.221128I
a = 0.512607 + 0.542422I
b = 0.12573 + 1.45523I
3.75920 1.29912I 3.74536 0.15014I
u = 0.360202 1.059137I
a = 0.876995 + 0.622085I
b = 1.60783 + 2.05818I
2.39843 6.38163I 4.17951 + 8.03965I
u = 0.360202 + 1.059137I
a = 0.876995 0.622085I
b = 1.60783 2.05818I
2.39843 + 6.38163I 4.17951 8.03965I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.359059 0.750855I
a = 1.76508 + 0.73462I
b = 0.445942 + 0.062419I
4.14504 4.87522I 2.26179 + 9.07875I
u = 0.359059 + 0.750855I
a = 1.76508 0.73462I
b = 0.445942 0.062419I
4.14504 + 4.87522I 2.26179 9.07875I
u = 0.355245 1.033541I
a = 0.811664 + 0.456344I
b = 0.532448 + 0.602433I
2.30167 + 0.78056I 4.09013 4.30344I
u = 0.355245 + 1.033541I
a = 0.811664 0.456344I
b = 0.532448 0.602433I
2.30167 0.78056I 4.09013 + 4.30344I
u = 0.290517 0.266226I
a = 1.60596 + 0.54000I
b = 0.235922 0.223315I
1.36445 + 0.78938I 4.43693 1.27648I
u = 0.290517 + 0.266226I
a = 1.60596 0.54000I
b = 0.235922 + 0.223315I
1.36445 0.78938I 4.43693 + 1.27648I
u = 0.287386 0.983998I
a = 0.652202 + 1.134297I
b = 0.48213 + 1.72142I
4.24208 0.93364I 15.9161 + 3.0658I
u = 0.287386 + 0.983998I
a = 0.652202 1.134297I
b = 0.48213 1.72142I
4.24208 + 0.93364I 15.9161 3.0658I
u = 0.280686 0.541514I
a = 2.16077 0.75386I
b = 0.525952 0.325099I
4.11853 + 3.35398I 7.32610 + 2.92910I
u = 0.280686 + 0.541514I
a = 2.16077 + 0.75386I
b = 0.525952 + 0.325099I
4.11853 3.35398I 7.32610 2.92910I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.08218 1.58909I
a = 0.482900 0.200981I
b = 0.717726 0.694699I
4.14504 + 4.87522I 2.26179 9.07875I
u = 0.08218 + 1.58909I
a = 0.482900 + 0.200981I
b = 0.717726 + 0.694699I
4.14504 4.87522I 2.26179 + 9.07875I
u = 0.009834 0.802079I
a = 1.31271 + 0.72770I
b = 0.10267 + 1.70261I
0.88395 2.01054I 3.45858 + 2.97810I
u = 0.009834 + 0.802079I
a = 1.31271 0.72770I
b = 0.10267 1.70261I
0.88395 + 2.01054I 3.45858 2.97810I
u = 0.043435 0.637728I
a = 0.990765 + 0.135589I
b = 0.20429 + 2.57613I
4.25960I 0.329447I
u = 0.043435 + 0.637728I
a = 0.990765 0.135589I
b = 0.20429 2.57613I
4.25960I 0.329447I
u = 0.149058 1.034506I
a = 0.536031 + 1.213218I
b = 0.01609 + 1.77821I
4.29717 + 0.35366I 9.83486 + 1.20455I
u = 0.149058 + 1.034506I
a = 0.536031 1.213218I
b = 0.01609 1.77821I
4.29717 0.35366I 9.83486 1.20455I
u = 0.198274 1.381686I
a = 0.412578 0.143942I
b = 1.21850 0.82029I
4.11853 + 3.35398I 7.32610 + 2.92910I
u = 0.198274 + 1.381686I
a = 0.412578 + 0.143942I
b = 1.21850 + 0.82029I
4.11853 3.35398I 7.32610 2.92910I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.271930 1.178514I
a = 0.056295 + 0.677501I
b = 0.784129 + 1.017889I
3.01082 + 3.10826I 5.64725 5.61918I
u = 0.271930 + 1.178514I
a = 0.056295 0.677501I
b = 0.784129 1.017889I
3.01082 3.10826I 5.64725 + 5.61918I
u = 0.342978 1.152934I
a = 0.758574 0.538085I
b = 0.26317 2.11021I
2.39843 + 6.38163I 4.17951 8.03965I
u = 0.342978 + 1.152934I
a = 0.758574 + 0.538085I
b = 0.26317 + 2.11021I
2.39843 6.38163I 4.17951 + 8.03965I
u = 0.430277 1.187344I
a = 0.452422 0.414846I
b = 0.032996 1.401181I
1.38542 + 3.12807I 0.18739 5.83461I
u = 0.430277 + 1.187344I
a = 0.452422 + 0.414846I
b = 0.032996 + 1.401181I
1.38542 3.12807I 0.18739 + 5.83461I
u = 0.448635 1.259821I
a = 0.723067 + 0.828539I
b = 1.19204 + 1.54947I
1.26966 + 7.40298I 6.35867 10.60309I
u = 0.448635 + 1.259821I
a = 0.723067 0.828539I
b = 1.19204 1.54947I
1.26966 7.40298I 6.35867 + 10.60309I
u = 0.596579 1.045744I
a = 0.582714 0.323025I
b = 0.93558 1.21382I
0.88395 + 2.01054I 3.45858 2.97810I
u = 0.596579 + 1.045744I
a = 0.582714 + 0.323025I
b = 0.93558 + 1.21382I
0.88395 2.01054I 3.45858 + 2.97810I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.610321 0.270672I
a = 0.559429 + 0.188105I
b = 0.341171 0.464433I
1.36445 + 0.78938I 4.43693 1.27648I
u = 0.610321 + 0.270672I
a = 0.559429 0.188105I
b = 0.341171 + 0.464433I
1.36445 0.78938I 4.43693 + 1.27648I
u = 0.662737 0.747286I
a = 0.256901 1.089226I
b = 1.08534 1.50551I
2.10904 + 4.58826I 5.96198 6.80019I
u = 0.662737 + 0.747286I
a = 0.256901 + 1.089226I
b = 1.08534 + 1.50551I
2.10904 4.58826I 5.96198 + 6.80019I
u = 0.68953 1.34019I
a = 0.487703 0.655892I
b = 1.01534 1.90636I
0.80844 + 7.87674I 4.76079 12.34614I
u = 0.68953 + 1.34019I
a = 0.487703 + 0.655892I
b = 1.01534 + 1.90636I
0.80844 7.87674I 4.76079 + 12.34614I
u = 0.71942 1.28265I
a = 0.597921 + 0.685138I
b = 0.74662 + 1.21858I
1.26966 + 7.40298I 6.35867 10.60309I
u = 0.71942 + 1.28265I
a = 0.597921 0.685138I
b = 0.74662 1.21858I
1.26966 7.40298I 6.35867 + 10.60309I
u = 0.759989 0.676773I
a = 0.936124 + 0.526319I
b = 0.102958 0.423467I
2.30167 + 0.78056I 4.09013 4.30344I
u = 0.759989 + 0.676773I
a = 0.936124 0.526319I
b = 0.102958 + 0.423467I
2.30167 0.78056I 4.09013 + 4.30344I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.783137 0.250577I
a = 0.12180 + 1.46589I
b = 0.275861 + 0.163091I
3.01082 + 3.10826I 5.64725 5.61918I
u = 0.783137 + 0.250577I
a = 0.12180 1.46589I
b = 0.275861 0.163091I
3.01082 3.10826I 5.64725 + 5.61918I
u = 1.303580 0.315783I
a = 0.380960 + 0.662558I
b = 0.546036 + 0.347944I
4.24208 0.93364I 15.9161 + 3.0658I
u = 1.303580 + 0.315783I
a = 0.380960 0.662558I
b = 0.546036 0.347944I
4.24208 + 0.93364I 15.9161 3.0658I
u = 1.334981 0.373687I
a = 0.304697 0.689631I
b = 0.203678 0.341465I
4.29717 0.35366I 9.83486 1.20455I
u = 1.334981 + 0.373687I
a = 0.304697 + 0.689631I
b = 0.203678 + 0.341465I
4.29717 + 0.35366I 9.83486 + 1.20455I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
12
2u
10
+ u
8
6u
7
+ 12u
5
+ 4u
4
8u
3
+ u
2
+ 3u + 3)
(u
66
+ u
65
+ ··· 1128u + 193)
c
2
, c
9
(u
12
4u
11
+ ··· 2u + 1)(u
66
+ 3u
65
+ ··· 77u + 21)
c
3
(u
12
+ 2u
11
u
9
+ 6u
8
+ 12u
7
+ 2u
6
11u
5
5u
4
+ u
3
+ u
2
+ 1)
(u
66
+ 5u
65
+ ··· + 7u + 3)
c
4
(u
12
2u
10
+ u
8
+ 6u
7
12u
5
+ 4u
4
+ 8u
3
+ u
2
3u + 3)
(u
66
+ u
65
+ ··· 1128u + 193)
c
5
, c
6
(u
12
+ 4u
11
+ ··· + 2u + 1)(u
66
+ 3u
65
+ ··· 77u + 21)
c
7
(u
12
+ 3u
10
+ u
9
+ 3u
8
+ 4u
7
+ 3u
6
+ 4u
5
+ 3u
4
+ u
3
+ 3u
2
+ 1)
(u
66
+ u
65
+ ··· + 31u + 3)
c
8
(u
12
+ 3u
10
u
9
+ 3u
8
4u
7
+ 3u
6
4u
5
+ 3u
4
u
3
+ 3u
2
+ 1)
(u
66
+ u
65
+ ··· + 31u + 3)
c
10
(u
12
2u
11
+ u
9
+ 6u
8
12u
7
+ 2u
6
+ 11u
5
5u
4
u
3
+ u
2
+ 1)
(u
66
+ 5u
65
+ ··· + 7u + 3)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
12
4y
11
+ ··· 3y + 9)(y
66
+ 5y
65
+ ··· + 1235072y + 37249)
c
2
, c
5
, c
6
c
9
(y
12
+ 6y
11
+ ··· + 12y + 1)(y
66
+ 35y
65
+ ··· + 7259y + 441)
c
3
(y
12
4y
11
+ ··· + 2y + 1)(y
66
+ y
65
+ ··· + 149y + 9)
c
4
(y
12
4y
11
+ ··· 3y + 9)(y
66
+ 5y
65
+ ··· + 1235072y + 37249)
c
7
(y
12
+ 6y
11
+ ··· + 6y + 1)(y
66
+ 3y
65
+ ··· 31y + 9)
c
8
(y
12
+ 6y
11
+ ··· + 6y + 1)(y
66
+ 3y
65
+ ··· 31y + 9)
c
10
(y
12
4y
11
+ ··· + 2y + 1)(y
66
+ y
65
+ ··· + 149y + 9)
14