11a
132
(K11a
132
)
1
Arc Sequences
6 1 8 10 2 4 11 3 5 7 9
Solving Sequence
2,5
6 1
3,10
4 7 9 8 11
c
5
c
1
c
2
c
4
c
6
c
9
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
2
+ u + 1, a
2
b a
3
+ bu au a 1i
I
u
2
= ha 1, b + u 1, u
2
+ u + 1i
I
u
3
= ha
3
+ a
2
+ b + 1, a
2
+ a + u + 1, a
4
+ 2a
3
+ 2a
2
+ a + 1i
I
u
4
= hu
50
4u
49
+ ··· 255u + 62, 4208u
49
16832u
48
+ ··· + 71424a 423590,
158u
49
395u
48
+ ··· + 2304b + 8974i
I
v
1
= ha, v 1, b
3
+ b + 1i
There are 5 irreducible components with 59 representations.
There are 1 irreducible components of dim
C
= 1 for 11a
132
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
2
+ u + 1, a
2
b a
3
+ bu au a 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
3
=
0
u
a
10
=
a
b
a
4
=
a
4
b
2
a
5
b a
4
b + a
5
b
2
a
2
a
3
b + a
4
a
2
b + a
3
+ b
2
+ 2a
2
+ 2a u
a
4
b
2
a
5
b a
4
b + a
5
b
2
a
2
a
2
b + a
3
+ b
2
+ a
2
+ a
a
7
=
a
3
b a
4
a
2
+ u
b
2
a
2
a
3
b 2ba + a
2
+ u + 1
a
9
=
a
a
4
b a
5
a
2
b a
2
+ 2b a + u + 1
a
8
=
a
b
a
11
=
a
3
b + a
4
+ a
2
+ a u
b
2
a
2
+ a
3
b + 2ba a
2
+ b u 1
a
11
=
a
3
b + a
4
+ a
2
+ a u
b
2
a
2
+ a
3
b + 2ba a
2
+ b u 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 = ···
a = ···
b = ···
1.64493 + 2.02988I 6.00000 3.46410I
3
II. I
u
2
= ha 1, b + u 1, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
3
=
0
u
a
10
=
1
u + 1
a
4
=
u
u 1
a
7
=
1
u 1
a
9
=
1
2u
a
8
=
1
u + 1
a
11
=
1
u + 1
a
11
=
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 = 1.00000
b = 1.50000 + 0.86603I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 1.00000
b = 1.50000 0.86603I
2.02988I 3.46410I
5
III. I
u
3
= ha
3
+ a
2
+ b + 1, a
2
+ a + u + 1, a
4
+ 2a
3
+ 2a
2
+ a + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
a
2
a 1
a
6
=
a
2
a 1
a
2
a 1
a
1
=
a
2
+ a + 1
a
2
+ a
a
3
=
0
a
2
a 1
a
10
=
a
a
3
a
2
1
a
4
=
a
3
a
2
a 1
a
3
a
2
2a 1
a
7
=
a
a
3
+ a
2
+ 1
a
9
=
a
1
a
8
=
a
a
3
a
2
1
a
11
=
a
a
3
a
2
1
a
11
=
a
a
3
a
2
1
(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 = 0.500000 0.866025I
a = 1.070696 0.758745I
b = 2.19244 + 0.54788I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 1.070696 + 0.758745I
b = 2.19244 0.54788I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.070696 0.758745I
b = 0.307560 0.318148I
2.02988I 3.46410I
u = 0.500000 0.866025I
a = 0.070696 + 0.758745I
b = 0.307560 + 0.318148I
2.02988I 3.46410I
7
IV. I
u
4
= hu
50
4u
49
+ · · · 255u + 62, 4208u
49
16832u
48
+ · · · +
71424a 423590, 158u
49
395u
48
+ · · · + 2304b + 8974i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
10
=
0.0589158u
49
+ 0.235663u
48
+ ··· 18.3254u + 5.93064
0.0685764u
49
+ 0.171441u
48
+ ··· + 6.12630u 3.89497
a
4
=
0.00634241u
49
0.0149530u
48
+ ··· 3.42792u + 2.03763
0.0312500u
49
0.00520833u
48
+ ··· 13.7031u + 4.67188
a
7
=
0.0568016u
49
+ 0.242832u
48
+ ··· 12.1412u + 2.05516
0.0759549u
49
+ 0.319444u
48
+ ··· 11.9714u + 2.53299
a
9
=
0.0589158u
49
+ 0.235663u
48
+ ··· 18.3254u + 5.93064
0.0182292u
49
+ 0.0368924u
48
+ ··· + 9.77908u 3.89497
a
8
=
0.0276658u
49
+ 0.0481631u
48
+ ··· 4.88399u + 2.25095
0.0329861u
49
0.293403u
48
+ ··· + 36.0286u 10.9184
a
11
=
0.00243616u
49
+ 0.0722446u
48
+ ··· 12.2518u + 4.46237
0.0625000u
49
0.192708u
48
+ ··· + 14.2604u 4.51042
a
11
=
0.00243616u
49
+ 0.0722446u
48
+ ··· 12.2518u + 4.46237
0.0625000u
49
0.192708u
48
+ ··· + 14.2604u 4.51042
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.019766 0.461213I
a = 0.071558 + 1.030492I
b = 0.191463 0.629136I
3.76854 4.51039I 0.38340 + 5.29534I
u = 1.019766 + 0.461213I
a = 0.071558 1.030492I
b = 0.191463 + 0.629136I
3.76854 + 4.51039I 0.38340 5.29534I
u = 0.941593 0.502895I
a = 0.168999 1.339255I
b = 0.080265 + 0.551571I
0.83094 10.80584I 3.02453 + 5.61837I
u = 0.941593 + 0.502895I
a = 0.168999 + 1.339255I
b = 0.080265 0.551571I
0.83094 + 10.80584I 3.02453 5.61837I
u = 0.89791 1.14735I
a = 0.655518 0.432065I
b = 1.66605 0.00348I
3.45411 + 3.73566I 11.2594 8.9167I
u = 0.89791 + 1.14735I
a = 0.655518 + 0.432065I
b = 1.66605 + 0.00348I
3.45411 3.73566I 11.2594 + 8.9167I
u = 0.736653 0.765266I
a = 0.789004 0.601404I
b = 0.249158 0.443278I
3.34886 + 2.77656I 9.46596 3.37700I
u = 0.736653 + 0.765266I
a = 0.789004 + 0.601404I
b = 0.249158 + 0.443278I
3.34886 2.77656I 9.46596 + 3.37700I
u = 0.718487 0.241043I
a = 0.520846 0.286396I
b = 0.077804 + 0.646051I
1.86829 + 2.54813I 2.68166 3.38783I
u = 0.718487 + 0.241043I
a = 0.520846 + 0.286396I
b = 0.077804 0.646051I
1.86829 2.54813I 2.68166 + 3.38783I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.701751 1.159991I
a = 0.952790 + 0.181167I
b = 2.56729 + 0.06253I
1.61095 + 10.69364I 2.44370 7.82760I
u = 0.701751 + 1.159991I
a = 0.952790 0.181167I
b = 2.56729 0.06253I
1.61095 10.69364I 2.44370 + 7.82760I
u = 0.691337 1.130817I
a = 1.131584 0.210462I
b = 2.91187 0.28170I
2.7651 + 16.7788I 5.20232 9.33540I
u = 0.691337 + 1.130817I
a = 1.131584 + 0.210462I
b = 2.91187 + 0.28170I
2.7651 16.7788I 5.20232 + 9.33540I
u = 0.447528 0.991757I
a = 0.180442 + 0.191884I
b = 0.744838 + 0.700057I
0.55196 + 1.45362I 0.849896 + 0.307578I
u = 0.447528 + 0.991757I
a = 0.180442 0.191884I
b = 0.744838 0.700057I
0.55196 1.45362I 0.849896 0.307578I
u = 0.209377 0.627431I
a = 0.901724 0.148839I
b = 0.012762 + 0.432089I
0.051801 + 1.349276I 0.36388 5.58397I
u = 0.209377 + 0.627431I
a = 0.901724 + 0.148839I
b = 0.012762 0.432089I
0.051801 1.349276I 0.36388 + 5.58397I
u = 0.092353 1.013551I
a = 0.446565 + 0.667133I
b = 0.032575 + 0.725453I
2.36463 + 4.68098I 6.72560 6.42701I
u = 0.092353 + 1.013551I
a = 0.446565 0.667133I
b = 0.032575 0.725453I
2.36463 4.68098I 6.72560 + 6.42701I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.045175 1.293489I
a = 1.062546 + 0.205077I
b = 2.81945 + 0.26090I
7.73894 8.36358I 8.86999 + 6.30335I
u = 0.045175 + 1.293489I
a = 1.062546 0.205077I
b = 2.81945 0.26090I
7.73894 + 8.36358I 8.86999 6.30335I
u = 0.117739 1.161839I
a = 1.211044 0.364258I
b = 2.60623 0.51420I
10.17264 + 2.94051I 12.16356 1.23495I
u = 0.117739 + 1.161839I
a = 1.211044 + 0.364258I
b = 2.60623 + 0.51420I
10.17264 2.94051I 12.16356 + 1.23495I
u = 0.289618 0.801801I
a = 0.698995 + 0.532493I
b = 0.94110 1.43970I
2.47998 1.07634I 10.74745 1.48815I
u = 0.289618 + 0.801801I
a = 0.698995 0.532493I
b = 0.94110 + 1.43970I
2.47998 + 1.07634I 10.74745 + 1.48815I
u = 0.304509 1.329569I
a = 0.839300 + 0.182529I
b = 2.50002 + 0.00892I
4.37509 1.36766I 12.53381 + 2.65466I
u = 0.304509 + 1.329569I
a = 0.839300 0.182529I
b = 2.50002 0.00892I
4.37509 + 1.36766I 12.53381 2.65466I
u = 0.445173 1.219661I
a = 0.838458 + 0.291675I
b = 2.50047 + 0.12588I
4.31677 1.36210I 11.75254 + 0.18127I
u = 0.445173 + 1.219661I
a = 0.838458 0.291675I
b = 2.50047 0.12588I
4.31677 + 1.36210I 11.75254 0.18127I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.563809 1.077477I
a = 1.011794 0.132575I
b = 2.62044 0.38536I
2.46832 6.35887I 5.91510 + 6.52323I
u = 0.563809 + 1.077477I
a = 1.011794 + 0.132575I
b = 2.62044 + 0.38536I
2.46832 + 6.35887I 5.91510 6.52323I
u = 0.599521 0.343884I
a = 0.25189 1.41967I
b = 0.163023 + 0.400734I
0.46677 + 1.70599I 3.14456 3.69461I
u = 0.599521 + 0.343884I
a = 0.25189 + 1.41967I
b = 0.163023 0.400734I
0.46677 1.70599I 3.14456 + 3.69461I
u = 0.601973 1.087874I
a = 1.282710 + 0.144527I
b = 2.78509 + 0.28888I
7.03977 10.39489I 8.44081 + 7.80630I
u = 0.601973 + 1.087874I
a = 1.282710 0.144527I
b = 2.78509 0.28888I
7.03977 + 10.39489I 8.44081 7.80630I
u = 0.660397 1.034533I
a = 0.626656 + 0.778248I
b = 0.984980 + 0.689745I
2.08583 10.64973I 2.52459 + 8.66549I
u = 0.660397 + 1.034533I
a = 0.626656 0.778248I
b = 0.984980 0.689745I
2.08583 + 10.64973I 2.52459 8.66549I
u = 0.672195 0.888770I
a = 0.170058 + 0.070657I
b = 0.499104 1.028336I
0.22553 2.62229I 0.95524 + 3.73688I
u = 0.672195 + 0.888770I
a = 0.170058 0.070657I
b = 0.499104 + 1.028336I
0.22553 + 2.62229I 0.95524 3.73688I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.686258 1.016054I
a = 0.319897 0.653343I
b = 0.318257 0.330722I
5.05351 5.86110I 1.95645 + 4.15344I
u = 0.686258 + 1.016054I
a = 0.319897 + 0.653343I
b = 0.318257 + 0.330722I
5.05351 + 5.86110I 1.95645 4.15344I
u = 0.757822 0.403687I
a = 0.17642 + 1.66371I
b = 0.0272654 0.1267258I
5.06249 + 5.26813I 6.22611 3.60097I
u = 0.757822 + 0.403687I
a = 0.17642 1.66371I
b = 0.0272654 + 0.1267258I
5.06249 5.26813I 6.22611 + 3.60097I
u = 0.777689 0.592166I
a = 0.870243 0.677476I
b = 0.680131 0.243099I
3.41598 + 5.21199I 0.12705 3.56799I
u = 0.777689 + 0.592166I
a = 0.870243 + 0.677476I
b = 0.680131 + 0.243099I
3.41598 5.21199I 0.12705 + 3.56799I
u = 0.813463 0.649972I
a = 0.732771 + 0.231234I
b = 0.769790 + 0.312153I
6.17950 + 0.24137I 3.95341 + 1.60515I
u = 0.813463 + 0.649972I
a = 0.732771 0.231234I
b = 0.769790 0.312153I
6.17950 0.24137I 3.95341 1.60515I
u = 1.121411 0.484988I
a = 0.516744 + 0.760920I
b = 0.515081 0.609971I
0.91613 5.10341I 4.10325 + 10.19626I
u = 1.121411 + 0.484988I
a = 0.516744 0.760920I
b = 0.515081 + 0.609971I
0.91613 + 5.10341I 4.10325 10.19626I
13
V. I
v
1
= ha, v 1, b
3
+ b + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
1
=
1
0
a
3
=
1
0
a
10
=
0
b
a
4
=
1
b
2
a
7
=
b
2
+ 1
b
2
b
a
9
=
b
b
a
8
=
0
b
a
11
=
b
2
+ 1
b
2
a
11
=
b
2
+ 1
b
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 0.682328
1.64493 6.00000
v = 1.00000
a = 0
b = 0.341164 1.161541I
1.64493 6.00000
v = 1.00000
a = 0
b = 0.341164 + 1.161541I
1.64493 6.00000
15
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u
3
(u
2
u + 1)
3
(u
2
+ u + 1)
5
(u
50
+ 4u
49
+ ··· + 255u + 62)
c
2
u
3
(u
2
+ u + 1)
8
(u
50
+ 20u
49
+ ··· + 27851u + 3844)
c
3
(u
2
u + 1)
2
(u
3
+ u + 1)(u
4
+ u
3
+ ··· u + 1)(1 + 2u + 2u
2
+ u
3
+ u
4
)
2
(9u
50
+ 9u
49
+ ··· 6u + 1)
c
4
(u
2
u + 1)
2
(u
3
+ u + 1)(u
4
+ u
3
+ ··· u + 1)(1 + 2u + 2u
2
+ u
3
+ u
4
)
2
(9u
50
+ 9u
49
+ ··· + 8u + 1)
c
5
u
3
(u
2
u + 1)
8
(u
50
+ 4u
49
+ ··· + 255u + 62)
c
6
(u
2
3u + 3)(u
2
+ u + 1)(u
3
2u
2
+ u + 1)(u
4
u
3
+ ··· 2u + 1)
(u
4
+ u
3
u
2
u + 1)(13u
4
+ 13u
3
+ 8u
2
+ 2u + 1)
(16u
50
+ 32u
49
+ ··· + 26298u + 5463)
c
7
u
6
(u 1)
13
(u
50
+ 6u
49
+ ··· + 7031u + 1274)
c
8
(u
2
u + 1)(u
2
+ u + 1)(u
3
+ u + 1)(u
4
u
3
u
2
+ u + 1)
(u
4
u
3
+ 2u
2
2u + 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
(9u
50
+ 9u
49
+ ··· 6u + 1)
c
9
(u
2
u + 1)(u
2
+ u + 1)(u
3
+ u + 1)(u
4
u
3
u
2
+ u + 1)
(u
4
u
3
+ 2u
2
2u + 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
(9u
50
+ 9u
49
+ ··· + 8u + 1)
c
10
u
6
(u 1)
3
(u + 1)
10
(u
50
+ 6u
49
+ ··· + 7031u + 1274)
c
11
(u
2
+ u + 1)
3
(u
2
+ 2u + 4)(u
3
+ u + 1)(u
4
+ 3u
3
+ 2u
2
+ 1)
(13u
4
+ 26u
3
+ ··· + 5u + 1)(16u
50
16u
49
+ ··· + 612u + 63)
16
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y
3
(y
2
+ y + 1)
8
(y
50
+ 20y
49
+ ··· + 27851y + 3844)
c
2
y
3
(y
2
+ y + 1)
8
(y
50
+ 20y
49
+ ··· + 9.72711 × 10
7
y + 1.47763 × 10
7
)
c
3
, c
8
(y
2
+ y + 1)
2
(y
3
+ 2y
2
+ y 1)(y
4
3y
3
+ 5y
2
3y + 1)
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
(81y
50
+ 2457y
49
+ ··· 2y + 1)
c
4
, c
9
(y
2
+ y + 1)
2
(y
3
+ 2y
2
+ y 1)(y
4
3y
3
+ 5y
2
3y + 1)
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
(81y
50
+ 2781y
49
+ ··· 2y + 1)
c
6
(y
2
3y + 9)(y
2
+ y + 1)(y
3
2y
2
+ 5y 1)(y
4
3y
3
+ ··· 3y + 1)
(y
4
+ 3y
3
+ 2y
2
+ 1)(169y
4
+ 39y
3
+ 38y
2
+ 12y + 1)
(256y
50
1408y
49
+ ··· 86273478y + 29844369)
c
7
, c
10
y
6
(y 1)
13
(y
50
34y
49
+ ··· + 1.59798 × 10
7
y + 1623076)
c
11
(y
2
+ y + 1)
3
(y
2
+ 4y + 16)(y
3
+ 2y
2
+ y 1)(y
4
5y
3
+ ··· + 4y + 1)
(169y
4
208y
3
+ 90y
2
+ 11y + 1)
(256y
50
1920y
49
+ ··· + 153522y + 3969)
17