11a
97
(K11a
97
)
1
Arc Sequences
6 1 10 9 2 3 11 5 4 7 8
Solving Sequence
2,6
1 3
5,8
9 4 11 7 10
c
1
c
2
c
5
c
8
c
4
c
11
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
4
+ 2a
3
+ a
2
6a + 3, a
3
4a
2
+ 3b 3a + 3, 2a
3
5a
2
6a + 3u + 6i
I
u
2
= hu
2
+ u + 1, b + u + 1, a u 1i
I
u
3
= hu
41
2u
40
+ ··· + 3u 3,
2228818024373u
40
+ 5180258455780u
39
+ ··· + 12914375539278a 3530185376379,
1708198037735u
40
+ 4948231923855u
39
+ ··· + 73181461389242b 39698161867904i
There are 3 irreducible components with 47 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
= ha
4
+2a
3
+a
2
6a +3, a
3
4a
2
+3b 3a +3, 2a
3
5a
2
6a +3u +6i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
2
3
a
3
+
5
3
a
2
+ 2a 2
a
1
=
1
2
3
a
3
+
5
3
a
2
+ 2a 3
a
3
=
2
3
a
3
+
5
3
a
2
+ 2a 2
2
3
a
3
5
3
a
2
2a + 2
a
5
=
2
3
a
3
5
3
a
2
2a + 2
2
3
a
3
+
5
3
a
2
+ 2a 2
a
8
=
a
1
3
a
3
+
4
3
a
2
+ a 1
a
9
=
2
3
a
3
+
5
3
a
2
+ 2a 3
1
3
a
3
1
3
a
2
+ 2
a
4
=
1
3
a
3
1
3
a
2
+ 1
a
3
3a
2
4a + 5
a
11
=
a + 1
1
3
a
3
+
1
3
a
2
+ a 2
a
7
=
1
2
3
a
3
+
5
3
a
2
+ 2a 3
a
10
=
a
1
3
a
3
4
3
a
2
a + 1
a
10
=
a
1
3
a
3
4
3
a
2
a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.72474 1.57313I
b = 0.50000 + 2.28024I
6.57974 2.02988I 6.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 1.72474 + 1.57313I
b = 0.50000 2.28024I
6.57974 + 2.02988I 6.00000 3.46410I
u = 0.500000 0.866025I
a = 0.724745 0.158919I
b = 0.500000 0.548188I
6.57974 2.02988I 6.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.724745 + 0.158919I
b = 0.500000 + 0.548188I
6.57974 + 2.02988I 6.00000 3.46410I
3
II. I
u
2
= hu
2
+ u + 1, b + u + 1, a u 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u 1
a
3
=
u
u
a
5
=
u
u
a
8
=
u + 1
u 1
a
9
=
u + 1
u 1
a
4
=
u
u
a
11
=
u + 2
2u 2
a
7
=
1
u + 1
a
10
=
u + 1
u 1
a
10
=
u + 1
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 3.46410I
5
III. I
u
3
=
hu
41
2u
40
+· · ·+3u3, 2.23×10
12
u
40
+5.18×10
12
u
39
+· · ·+1.29×10
13
a
3.53 × 10
12
, 1.71 × 10
12
u
40
+ 4.95 × 10
12
u
39
+ · · · + 7.32 × 10
13
b 3.97 × 10
13
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
5
=
u
u
a
8
=
0.172584u
40
0.401123u
39
+ ··· + 0.246650u + 0.273353
0.0233420u
40
0.0676159u
39
+ ··· 1.80205u + 0.542462
a
9
=
0.130291u
40
0.140124u
39
+ ··· + 1.20514u 0.237411
0.0189515u
40
0.328615u
39
+ ··· 2.76054u + 1.05323
a
4
=
0.0472070u
40
0.641152u
39
+ ··· 2.24095u + 0.880067
0.196698u
40
+ 0.0788085u
39
+ ··· + 1.04339u 1.78184
a
11
=
0.690959u
40
1.15909u
39
+ ··· 0.00321786u 0.758520
0.717875u
40
+ 1.35352u
39
+ ··· + 0.812265u + 0.184990
a
7
=
u
5
2u
3
u
u
7
u
5
+ u
a
10
=
0.230618u
40
0.252059u
39
+ ··· 0.153593u 2.09818
0.0887266u
40
0.162139u
39
+ ··· 0.563961u + 1.01840
a
10
=
0.230618u
40
0.252059u
39
+ ··· 0.153593u 2.09818
0.0887266u
40
0.162139u
39
+ ··· 0.563961u + 1.01840
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.822005 0.179047I
a = 0.775116 0.651519I
b = 1.148091 0.477252I
6.01336 4.06181I 3.66785 + 3.84288I
u = 0.822005 + 0.179047I
a = 0.775116 + 0.651519I
b = 1.148091 + 0 .477252I
6.01336 + 4.06181I 3.66785 3.84288I
u = 0.808241 0.837807I
a = 1.31887 0.91833I
b = 0.29744 + 1.87603I
10.76739 + 2.97655I 6.11781 2.74414I
u = 0.808241 + 0.837807I
a = 1.31887 + 0 .91833I
b = 0.29744 1.87603I
10.76739 2.97655I 6.11781 + 2.74414I
u = 0.553626 0.237053I
a = 0.479923 + 1 .216925I
b = 0.537241 0.109773I
0.16445 1.48990I 0.43975 + 5.27239I
u = 0.553626 + 0.237053I
a = 0.479923 1.216925I
b = 0.537241 + 0.109773I
0.16445 + 1.48990I 0.43975 5.27239I
u = 0.549965 0.794582I
a = 1.26894 + 0.70417I
b = 0.01617 1.53684I
4.91839 + 2.21626I 0.59639 3.86290I
u = 0.549965 + 0.794582I
a = 1.26894 0.70417I
b = 0.01617 + 1 .53684I
4.91839 2.21626I 0.59639 + 3.86290I
u = 0.528320 1.193695I
a = 0.52219 + 1 .77170I
b = 1.31673 1.60526I
9.03060 + 9.03490I 6.35139 7.01689I
u = 0.528320 + 1.193695I
a = 0.52219 1.77170I
b = 1.31673 + 1.60526I
9.03060 9.03490I 6.35139 + 7.01689I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.477572 1.096484I
a = 0.662533 1.133144I
b = 1.37683 + 1 .11032I
2.54750 + 5.63680I 3.76761 8.15870I
u = 0.477572 + 1.096484I
a = 0.662533 + 1.133144I
b = 1.37683 1.11032I
2.54750 5.63680I 3.76761 + 8.15870I
u = 0.387666 1.042530I
a = 0.286727 + 0 .046522I
b = 0.937246 0.308680I
3.17977 + 1.35362I 6.58078 0.48937I
u = 0.387666 + 1.042530I
a = 0.286727 0.046522I
b = 0.937246 + 0.308680I
3.17977 1.35362I 6.58078 + 0.48937I
u = 0.353024 1.209190I
a = 0.886737 + 0 .641631I
b = 0.212121 0.039074I
10.25757 0.21541I 8.38655 + 0.44300I
u = 0.353024 + 1.209190I
a = 0.886737 0.641631I
b = 0.212121 + 0.039074I
10.25757 + 0.21541I 8.38655 0.44300I
u = 0.274226 0.759914I
a = 0.07085 1.54503I
b = 0.388660 + 1.032105I
2.03507 + 1.22420I 5.30574 + 2.47978I
u = 0.274226 + 0.759914I
a = 0.07085 + 1 .54503I
b = 0.388660 1.032105I
2.03507 1.22420I 5.30574 2.47978I
u = 0.077478 0.878687I
a = 1.06600 1.67133I
b = 0.54877 + 1 .67327I
7.87303 + 0.41748I 9.30153 + 0.50767I
u = 0.077478 + 0.878687I
a = 1.06600 + 1 .67133I
b = 0.54877 1.67327I
7.87303 0.41748I 9.30153 0.50767I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.278438 1.307289I
a = 0.383720 + 0.224681I
b = 0.853784 + 0.125711I
19.5976 + 2.5619I 9.48958 0.43711I
u = 0.278438 + 1 .307289I
a = 0.383720 0.224681I
b = 0.853784 0.125711I
19.5976 2.5619I 9.48958 + 0.43711I
u = 0.394567 1.198377I
a = 0.563836 + 0.339653I
b = 1.80562 0.74664I
11.70312 1.34196I 7.30625 + 0.70220I
u = 0.394567 + 1 .198377I
a = 0.563836 0.339653I
b = 1.80562 + 0 .74664I
11.70312 + 1.34196I 7.30625 0.70220I
u = 0.419594 0.580623I
a = 0.226420 + 0 .874449I
b = 0.435854 0.678681I
0.667792 1.033691I 4.88372 + 5.04725I
u = 0.419594 + 0 .580623I
a = 0.226420 0.874449I
b = 0.435854 + 0 .678681I
0.667792 + 1.033691I 4.88372 5.04725I
u = 0.452830 1.177818I
a = 0.92923 + 1.38200I
b = 1.086792 0.743720I
6.74821 4.23995I 4.67532 + 3.10579I
u = 0.452830 + 1 .177818I
a = 0.92923 1.38200I
b = 1.086792 + 0 .743720I
6.74821 + 4.23995I 4.67532 3.10579I
u = 0.458387 0.977753I
a = 0.600173 0.717159I
b = 0.704854 + 0.299478I
0.52713 2.73009I 3.23724 + 4.11462I
u = 0.458387 + 0 .977753I
a = 0.600173 + 0 .717159I
b = 0.704854 0.299478I
0.52713 + 2.73009I 3.23724 4.11462I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.503223 1.190273I
a = 0.62585 1.34135I
b = 1.82748 + 1.76713I
10.92988 7.37639I 5.80773 + 5.55165I
u = 0.503223 + 1 .190273I
a = 0.62585 + 1 .34135I
b = 1.82748 1.76713I
10.92988 + 7.37639I 5.80773 5.55165I
u = 0.588276 1.220760I
a = 0.14502 + 1.89433I
b = 1.31646 2.30729I
17.6792 12.1700I 7.60138 + 6.11091I
u = 0.588276 + 1 .220760I
a = 0.14502 1.89433I
b = 1.31646 + 2 .30729I
17.6792 + 12.1700I 7.60138 6.11091I
u = 0.659432 0.806619I
a = 0.979250 0.969514I
b = 0.427164 + 1 .293522I
2.68106 2.54195I 5.78064 + 4.47516I
u = 0.659432 + 0 .806619I
a = 0.979250 + 0.969514I
b = 0.427164 1.293522I
2.68106 + 2.54195I 5.78064 4.47516I
u = 0.711439
a = 0.256315
b = 1.25148
3.43392 0.468188
u = 0.780672 0.127078I
a = 0.71324 + 1.67556I
b = 0.459261 + 0 .338203I
7.81669 + 2.63616I 2.57791 2.58719I
u = 0.780672 + 0 .127078I
a = 0.71324 1.67556I
b = 0.459261 0.338203I
7.81669 2.63616I 2.57791 + 2.58719I
u = 0.940985 0.247429I
a = 1.20898 0.77385I
b = 0.919721 0.921654I
14.7091 + 6.6164I 5.20491 2.71827I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.940985 + 0 .247429I
a = 1.20898 + 0 .77385I
b = 0.919721 + 0.921654I
14.7091 6.6164I 5.20491 + 2.71827I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)(u
41
+ 2u
40
+ ··· + 3 u + 3)
c
2
(u
2
+ u + 1)
3
(u
41
+ 22u
40
+ ··· 33u 9)
c
3
, c
4
, c
8
u
2
(u
2
+ 2)
2
(u
41
+ u
40
+ ··· + 8u 4)
c
5
(u
2
u + 1)(u
2
+ u + 1)
2
(u
41
+ 2u
40
+ ··· + 3 u + 3)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
41
+ 2u
40
+ ··· + 327 u 87)
c
7
(u 1)
2
(u + 1)
4
(u
41
+ 3u
40
+ ··· 16u + 3)
c
9
u
2
(u
2
+ 2)
2
(u
41
+ u
40
+ ··· + 8u 4)
c
10
(u 1)
4
(u + 1)
2
(u
41
+ 3u
40
+ ··· 16u + 3)
c
11
(u 1)
4
(u + 1)
2
(u
41
+ 3u
40
+ ··· 16u + 3)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
41
+ 22y
40
+ ··· 33y 9)
c
2
(y
2
+ y + 1)
3
(y
41
2y
40
+ ··· + 423 y 81)
c
3
, c
8
y
2
(y + 2)
4
(y
41
+ 51y
40
+ ··· 128y 16)
c
4
, c
9
y
2
(y + 2)
4
(y
41
+ 51y
40
+ ··· 128y 16)
c
6
(y
2
+ y + 1)
3
(y
41
26y
40
+ ··· 166425y 7569)
c
7
, c
10
(y 1)
6
(y
41
43y
40
+ ··· + 4y 9)
c
11
(y 1)
6
(y
41
43y
40
+ ··· + 4y 9)
12