11n
109
(K11n
109
)
1
Arc Sequences
6 1 7 9 2 10 1 4 5 7 4
Solving Sequence
6,10
7
2,11
1 3 5 9 4 8
c
6
c
10
c
1
c
2
c
5
c
9
c
4
c
8
c
3
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
10
u
9
+ 4u
8
3u
7
+ 7u
6
4u
5
+ 7u
4
4u
3
+ 4u
2
u + 1,
u
9
3u
7
u
6
4u
5
3u
4
3u
3
3u
2
+ a 2,
2u
9
2u
8
+ 7u
7
5u
6
+ 10u
5
5u
4
+ 8u
3
4u
2
+ b + 3u + 1i
I
u
2
= hu
38
+ 2u
37
+ ··· + 3u + 1, 2.63625 × 10
25
u
37
+ 2.28783 × 10
25
u
36
+ ··· + 3.48267 × 10
25
a + 2.14600 × 10
25
,
2.20003 × 10
25
u
37
4.95447 × 10
25
u
36
+ ··· + 3.48267 × 10
25
b 2.88601 × 10
25
i
There are 2 irreducible components with 48 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
10
u
9
+ · · · u + 1, u
9
3u
7
+ · · · + a 2, 2u
9
2u
8
+ · · · + b + 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
u
9
+ 3u
7
+ u
6
+ 4u
5
+ 3u
4
+ 3u
3
+ 3u
2
+ 2
2u
9
+ 2u
8
7u
7
+ 5u
6
10u
5
+ 5u
4
8u
3
+ 4u
2
3u 1
a
7
=
u
9
2u
8
+ 4u
7
6u
6
+ 6u
5
8u
4
+ 4u
3
7u
2
+ 2u 1
u
9
+ 2u
8
5u
7
+ 6u
6
9u
5
+ 8u
4
8u
3
+ 7u
2
4u + 2
a
2
=
1
0
a
11
=
u
9
+ 2u
7
+ u
6
+ u
5
+ 2u
4
u
3
+ u
2
3u + 2
3u
9
+ 2u
8
10u
7
+ 4u
6
14u
5
+ 3u
4
11u
3
+ 4u
2
3u 2
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
5
=
u
u
a
9
=
u
8
u
7
+ 4u
6
2u
5
+ 7u
4
2u
3
+ 7u
2
2u + 3
u
9
+ u
8
3u
7
+ 2u
6
4u
5
+ u
4
3u
3
u 2
a
4
=
u
7
+ 3u
5
+ 4u
3
+ u
2
+ 3u
2u
9
u
8
+ 6u
7
2u
6
+ 8u
5
u
4
+ 6u
3
2u
2
+ u + 1
a
8
=
u
9
u
8
+ 3u
7
3u
6
+ 4u
5
4u
4
+ 2u
3
4u
2
u
9
+ u
8
4u
7
+ 3u
6
7u
5
+ 4u
4
6u
3
+ 4u
2
3u + 1
a
8
=
u
9
u
8
+ 3u
7
3u
6
+ 4u
5
4u
4
+ 2u
3
4u
2
u
9
+ u
8
4u
7
+ 3u
6
7u
5
+ 4u
4
6u
3
+ 4u
2
3u + 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 = 0.591573 0.895458I
a = 0.659378 + 0.798595I
b = 0.857907 0.485807I
1.88316 + 2.32533I 12.32535 3.44072I
u = 0.591573 + 0.895458I
a = 0.659378 0.798595I
b = 0.857907 + 0.485807I
1.88316 2.32533I 12.32535 + 3.44072I
u = 0.079307 0.642927I
a = 1.31154 + 0.57908I
b = 2.19685 + 0.82100I
2.77192 + 1.74853I 1.51113 2.06464I
u = 0.079307 + 0.642927I
a = 1.31154 0.57908I
b = 2.19685 0.82100I
2.77192 1.74853I 1.51113 + 2.06464I
u = 0.059179 1.329344I
a = 0.437472 + 0.050184I
b = 0.252670 0.807564I
5.58838 1.13850I 4.94587 0.33361I
u = 0.059179 + 1.329344I
a = 0.437472 0.050184I
b = 0.252670 + 0.807564I
5.58838 + 1.13850I 4.94587 + 0.33361I
u = 0.587969 0.580983I
a = 1.01749 1.46349I
b = 0.258029 + 0.056971I
8.26505 0.63915I 14.5970 + 5.3987I
u = 0.587969 + 0.580983I
a = 1.01749 + 1.46349I
b = 0.258029 0.056971I
8.26505 + 0.63915I 14.5970 5.3987I
u = 0.642090 1.139228I
a = 0.697194 0.500245I
b = 1.34430 + 1.39077I
6.43677 4.34705I 13.62063 + 3.59101I
u = 0.642090 + 1.139228I
a = 0.697194 + 0.500245I
b = 1.34430 1.39077I
6.43677 + 4.34705I 13.62063 3.59101I
3
II. I
u
2
=
hu
38
+2u
37
+· · · +3u +1, 2.64× 10
25
u
37
+2.29 ×10
25
u
36
+· · · +3.48 ×10
25
a +
2.15 × 10
25
, 2.20× 10
25
u
37
4.95 × 10
25
u
36
+ · · · + 3.48 × 10
25
b 2.89 × 10
25
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.756963u
37
0.656918u
36
+ ··· + 0.582663u 0.616194
0.631707u
37
+ 1.42261u
36
+ ··· + 2.60943u + 0.828678
a
7
=
1.81988u
37
3.09009u
36
+ ··· 4.84948u 1.27598
1.84428u
37
+ 2.49676u
36
+ ··· + 5.29511u + 0.631271
a
2
=
1
0
a
11
=
0.101427u
37
+ 0.518797u
36
+ ··· + 0.997777u + 0.694773
0.719948u
37
0.842438u
36
+ ··· 0.294641u + 0.0155744
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
5
=
u
u
a
9
=
0.0856053u
37
+ 0.615474u
36
+ ··· + 3.50601u + 0.400009
0.0396514u
37
+ 0.150216u
36
+ ··· 0.313917u 0.187524
a
4
=
0.442400u
37
+ 0.747467u
36
+ ··· + 2.30248u + 1.92979
0.664711u
37
1.67661u
36
+ ··· 3.97167u 1.27334
a
8
=
0.613714u
37
0.654017u
36
+ ··· + 0.274755u 0.0950486
1.06980u
37
+ 1.25852u
36
+ ··· + 4.01771u + 0.607524
a
8
=
0.613714u
37
0.654017u
36
+ ··· + 0.274755u 0.0950486
1.06980u
37
+ 1.25852u
36
+ ··· + 4.01771u + 0.607524
(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 = 1.081328 0.386328I
a = 0.867653 + 0.857446I
b = 0.206832 0.161047I
1.95602 6.72677I 10.60803 + 3.77299I
u = 1.081328 + 0.386328I
a = 0.867653 0.857446I
b = 0.206832 + 0.161047I
1.95602 + 6.72677I 10.60803 3.77299I
u = 0.698047 1.223377I
a = 1.030832 + 0.525008I
b = 1.83321 1.27479I
0.64182 + 13.08695I 8.09856 7.03946I
u = 0.698047 + 1.223377I
a = 1.030832 0.525008I
b = 1.83321 + 1.27479I
0.64182 13.08695I 8.09856 + 7.03946I
u = 0.693371
a = 2.22543
b = 0.225023
7.32047 11.7759
u = 0.668607 0.872893I
a = 0.572440 0.409656I
b = 0.912959 + 0.050023I
1.01360 + 2.58424I 2.68887 3.99949I
u = 0.668607 + 0.872893I
a = 0.572440 + 0.409656I
b = 0.912959 0.050023I
1.01360 2.58424I 2.68887 + 3.99949I
u = 0.534109 1.201299I
a = 0.673458 0.887023I
b = 1.21349 + 1.64286I
4.11680 + 4.64818I 8.30001 4.11714I
u = 0.534109 + 1.201299I
a = 0.673458 + 0.887023I
b = 1.21349 1.64286I
4.11680 4.64818I 8.30001 + 4.11714I
u = 0.496586 1.000335I
a = 0.91537 + 1.15790I
b = 1.72217 0.16503I
3.05076 + 4.70281I 7.22690 4.71362I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.496586 + 1.000335I
a = 0.91537 1.15790I
b = 1.72217 + 0.16503I
3.05076 4.70281I 7.22690 + 4.71362I
u = 0.447680 0.663750I
a = 1.43961 0.77935I
b = 2.07802 0.46283I
1.90577 0.72497I 8.59505 1.33995I
u = 0.447680 + 0.663750I
a = 1.43961 + 0.77935I
b = 2.07802 + 0.46283I
1.90577 + 0.72497I 8.59505 + 1.33995I
u = 0.437061 1.002291I
a = 0.812401 + 0.746887I
b = 1.011531 + 0.515601I
3.43318 + 1.20443I 6.92259 2.66519I
u = 0.437061 + 1.002291I
a = 0.812401 0.746887I
b = 1.011531 0.515601I
3.43318 1.20443I 6.92259 + 2.66519I
u = 0.326007 0.583311I
a = 0.117692 + 0.462876I
b = 1.34426 1.15183I
2.08165 + 2.22554I 9.62442 6.36612I
u = 0.326007 + 0.583311I
a = 0.117692 0.462876I
b = 1.34426 + 1.15183I
2.08165 2.22554I 9.62442 + 6.36612I
u = 0.284870
a = 1.40010
b = 0.198545
0.644934 15.5787
u = 0.19948 1.53203I
a = 0.152978 + 0.524099I
b = 0.205003 1.335942I
4.80130 2.15880I 9.55926 + 4.13726I
u = 0.19948 + 1.53203I
a = 0.152978 0.524099I
b = 0.205003 + 1.335942I
4.80130 + 2.15880I 9.55926 4.13726I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.198690 0.927706I
a = 0.639094 + 0.144480I
b = 1.268122 0.556517I
1.65932 + 1.72508I 3.99615 5.00557I
u = 0.198690 + 0.927706I
a = 0.639094 0.144480I
b = 1.268122 + 0.556517I
1.65932 1.72508I 3.99615 + 5.00557I
u = 0.297248 1.257894I
a = 0.663933 0.642369I
b = 1.199794 + 0.637937I
8.06788 1.04561I 1.73854 + 0.76531I
u = 0.297248 + 1.257894I
a = 0.663933 + 0.642369I
b = 1.199794 0.637937I
8.06788 + 1.04561I 1.73854 0.76531I
u = 0.337441 0.249883I
a = 2.42316 2.21260I
b = 0.635847 0.037727I
7.79085 0.03851I 8.05729 1.80582I
u = 0.337441 + 0.249883I
a = 2.42316 + 2.21260I
b = 0.635847 + 0.037727I
7.79085 + 0.03851I 8.05729 + 1.80582I
u = 0.379743 0.859856I
a = 0.773786 + 1.078396I
b = 1.190231 0.562474I
1.30138 1.64549I 4.54049 1.93386I
u = 0.379743 + 0.859856I
a = 0.773786 1.078396I
b = 1.190231 + 0.562474I
1.30138 + 1.64549I 4.54049 + 1.93386I
u = 0.526939 1.137219I
a = 0.696229 0.307990I
b = 1.33219 + 1.36680I
5.23331 4.18634I 5.68349 + 3.29449I
u = 0.526939 + 1.137219I
a = 0.696229 + 0.307990I
b = 1.33219 1.36680I
5.23331 + 4.18634I 5.68349 3.29449I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.561660 1.148651I
a = 0.937344 0.576864I
b = 1.79023 + 0.38453I
6.14363 7.57123I 4.89087 + 5.76194I
u = 0.561660 + 1.148651I
a = 0.937344 + 0.576864I
b = 1.79023 0.38453I
6.14363 + 7.57123I 4.89087 5.76194I
u = 0.775393 0.236076I
a = 0.707229 1.077228I
b = 0.640042 + 0.030719I
3.53482 + 2.58667I 7.16756 2.58418I
u = 0.775393 + 0.236076I
a = 0.707229 + 1.077228I
b = 0.640042 0.030719I
3.53482 2.58667I 7.16756 + 2.58418I
u = 0.784887 0.992111I
a = 0.907975 + 0.168508I
b = 1.69223 0.81013I
4.61276 5.70694I 9.04865 + 6.07255I
u = 0.784887 + 0.992111I
a = 0.907975 0.168508I
b = 1.69223 + 0.81013I
4.61276 + 5.70694I 9.04865 6.07255I
u = 0.913408 0.776527I
a = 0.363262 + 0.855243I
b = 0.392732 0.086905I
5.31271 0.53174I 10.57597 + 0.24868I
u = 0.913408 + 0.776527I
a = 0.363262 0.855243I
b = 0.392732 + 0.086905I
5.31271 + 0.53174I 10.57597 0.24868I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
10
u
9
+ 4u
8
3u
7
+ 7u
6
4u
5
+ 7u
4
4u
3
+ 4u
2
u + 1)
(u
38
+ 2u
37
+ ··· + 3u + 1)
c
2
(u
10
+ 7u
9
+ ··· + 7u + 1)(u
38
+ 20u
37
+ ··· 7u + 1)
c
3
(u
10
+ 2u
8
+ u
7
4u
6
2u
5
2u
4
2u
3
+ 8u
2
2u + 1)
(u
38
+ u
37
+ ··· + 130u 29)
c
4
(u
10
6u
8
u
7
+ 13u
6
+ 4u
5
12u
4
5u
3
+ 4u
2
+ 2u + 1)
(u
38
+ u
37
+ ··· 24u 19)
c
5
(u
10
+ u
9
+ 4u
8
+ 3u
7
+ 7u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 4u
2
+ u + 1)
(u
38
+ 2u
37
+ ··· + 3u + 1)
c
6
(u
10
2u
9
u
8
+ 3u
7
+ u
5
2u
4
2u
3
+ 2u
2
+ 1)
(u
38
+ 3u
37
+ ··· 94u 11)
c
7
(u
10
+ 2u
8
2u
7
2u
6
+ u
5
+ 3u
3
u
2
2u + 1)
(u
38
+ u
37
+ ··· 39u 2)
c
8
, c
9
(u
10
6u
8
+ u
7
+ 13u
6
4u
5
12u
4
+ 5u
3
+ 4u
2
2u + 1)
(u
38
+ u
37
+ ··· 24u 19)
c
10
(u
10
+ 2u
9
u
8
3u
7
u
5
2u
4
+ 2u
3
+ 2u
2
+ 1)
(u
38
+ 3u
37
+ ··· 94u 11)
c
11
(u
10
3u
9
+ u
8
+ 5u
7
7u
6
+ 3u
5
+ 4u
4
7u
3
+ 6u
2
3u + 1)
(u
38
+ 2u
37
+ ··· 31u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
10
+ 7y
9
+ ··· + 7y + 1)(y
38
+ 20y
37
+ ··· 7y + 1)
c
2
(y
10
y
9
+ ··· 5y + 1)(y
38
+ 4y
37
+ ··· 95y + 1)
c
3
(y
10
+ 4y
9
+ ··· + 12y + 1)(y
38
+ 41y
37
+ ··· + 1138y + 841)
c
4
, c
8
, c
9
(y
10
12y
9
+ ··· + 4y + 1)(y
38
35y
37
+ ··· 6y + 361)
c
6
, c
10
(y
10
6y
9
+ 13y
8
9y
7
6y
6
+ 9y
5
+ 6y
4
12y
3
+ 4y + 1)
(y
38
17y
37
+ ··· 1686y + 121)
c
7
(y
10
+ 4y
9
12y
7
+ 6y
6
+ 9y
5
6y
4
9y
3
+ 13y
2
6y + 1)
(y
38
+ 37y
37
+ ··· 325y + 4)
c
11
(y
10
7y
9
+ 17y
8
13y
7
3y
6
+ y
5
+ 6y
4
+ 3y
3
+ 2y
2
+ 3y + 1)
(y
38
38y
37
+ ··· 415y + 1)
10