11a
32
(K11a
32
)
1
Arc Sequences
4 1 7 2 11 10 3 5 6 9 8
Solving Sequence
1,4
2 3
5,8
9 7 11 6 10
c
1
c
2
c
4
c
8
c
7
c
11
c
5
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
6
b
5
b
4
+ 2b
3
b + 1, a, u 1i
I
u
2
= hu
75
+ 7u
74
+ ··· 5u 1, 73u
74
402u
73
+ ··· + 4a + 14,
2667u
74
16330u
73
+ ··· + 32b + 2491i
There are 2 irreducible components with 81 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
= hb
6
b
5
b
4
+ 2b
3
b + 1, a, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
3
=
0
1
a
5
=
1
0
a
8
=
0
b
a
9
=
b
b
a
7
=
0
b
a
11
=
1
b
2
a
6
=
b
2
1
b
4
a
10
=
b
4
b
2
+ 1
b
4
a
10
=
b
4
b
2
+ 1
b
4
(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 = 1.00000
a = 0
b = 1.002193 0.295542I
3.53554 0.92430I 10.03026 + 0.88960I
u = 1.00000
a = 0
b = 1.002193 + 0.295542I
3.53554 + 0.92430I 10.03026 0.88960I
u = 1.00000
a = 0
b = 0.428243 0.664531I
0.245672 0.924305I 5.20252 + 1.68215I
u = 1.00000
a = 0
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 5.20252 1.68215I
u = 1.00000
a = 0
b = 1.073950 0.558752I
1.64493 + 5.69302I 6.76721 6.15196I
u = 1.00000
a = 0
b = 1.073950 + 0.558752I
1.64493 5.69302I 6.76721 + 6.15196I
3
II. I
u
2
= hu
75
+ 7u
74
+ · · · 5u 1, 73u
74
402u
73
+ · · · + 4a +
14, 2667u
74
16330u
73
+ · · · + 32b + 2491i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
8
=
73
4
u
74
+
201
2
u
73
+ ··· 28u
7
2
83.3438u
74
+ 510.313u
73
+ ··· 333.875u 77.8438
a
9
=
46.8438u
74
314.063u
73
+ ··· + 281.125u + 72.8438
139.344u
74
+ 836.563u
73
+ ··· 502.625u 113.094
a
7
=
45
2
u
74
161u
73
+ ··· + 171u +
185
4
127.594u
74
+ 769.063u
73
+ ··· 466.625u 105.094
a
11
=
u
7
+ 2u
5
+ 2u
4
2u
3
2u
2
+ 2
1
32
u
74
+
3
16
u
73
+ ···
9
8
u
1
32
a
6
=
3
32
u
74
+
9
16
u
73
+ ···
43
8
u
3
32
1.28125u
74
+ 7.75000u
73
+ ··· 4.31250u 1.34375
a
10
=
1.06250u
74
+ 6.43750u
73
+ ··· 2.43750u 0.125000
1.34375u
74
8.12500u
73
+ ··· + 5.56250u + 1.40625
a
10
=
1.06250u
74
+ 6.43750u
73
+ ··· 2.43750u 0.125000
1.34375u
74
8.12500u
73
+ ··· + 5.56250u + 1.40625
(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 = 1.155508 0.649574I
a = 1.01483 1.47364I
b = 0.79889 + 1.59357I
3.1307 16.0547I 2.96756 + 10.68477I
u = 1.155508 + 0.649574I
a = 1.01483 + 1.47364I
b = 0.79889 1.59357I
3.1307 + 16.0547I 2.96756 10.68477I
u = 1.145665 0.642227I
a = 0.97178 + 1.41886I
b = 0.83020 1.50674I
0.76093 10.98052I 6.06132 + 7.06114I
u = 1.145665 + 0.642227I
a = 0.97178 1.41886I
b = 0.83020 + 1.50674I
0.76093 + 10.98052I 6.06132 7.06114I
u = 1.129102 0.660193I
a = 1.06700 1.32077I
b = 0.66904 + 1.42604I
5.41191 8.13992I 0.20766 + 5.17511I
u = 1.129102 + 0.660193I
a = 1.06700 + 1.32077I
b = 0.66904 1.42604I
5.41191 + 8.13992I 0.20766 5.17511I
u = 1.120127 0.610216I
a = 0.79865 + 1.28631I
b = 0.99237 1.25830I
0.87148 8.74235I 7.38582 + 7.98051I
u = 1.120127 + 0.610216I
a = 0.79865 1.28631I
b = 0.99237 + 1.25830I
0.87148 + 8.74235I 7.38582 7.98051I
u = 1.096560 0.590139I
a = 0.699149 1.173112I
b = 1.06597 + 1.05792I
0.07882 3.89973I 5.99014 + 2.18775I
u = 1.096560 + 0.590139I
a = 0.699149 + 1.173112I
b = 1.06597 1.05792I
0.07882 + 3.89973I 5.99014 2.18775I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.050219 0.693988I
a = 1.17073 + 0.86818I
b = 0.326464 0.944045I
6.86340 7.55493I 1.60275 + 6.55880I
u = 1.050219 + 0.693988I
a = 1.17073 0.86818I
b = 0.326464 + 0.944045I
6.86340 + 7.55493I 1.60275 6.55880I
u = 1.014218 0.671104I
a = 1.008366 0.729025I
b = 0.428776 + 0.699364I
3.13966 4.03622I 3.07312 + 3.45198I
u = 1.014218 + 0.671104I
a = 1.008366 + 0.729025I
b = 0.428776 0.699364I
3.13966 + 4.03622I 3.07312 3.45198I
u = 0.996956 0.705289I
a = 1.135138 + 0.591809I
b = 0.214156 0.626907I
5.93451 + 0.41079I 0.573935 0.408107I
u = 0.996956 + 0.705289I
a = 1.135138 0.591809I
b = 0.214156 + 0.626907I
5.93451 0.41079I 0.573935 + 0.408107I
u = 0.993853 0.440511I
a = 0.166101 0.783333I
b = 1.56989 + 0.06150I
1.89293 7.72902I 5.92850 + 9.55510I
u = 0.993853 + 0.440511I
a = 0.166101 + 0.783333I
b = 1.56989 0.06150I
1.89293 + 7.72902I 5.92850 9.55510I
u = 0.959321 0.422476I
a = 0.127985 + 0.694740I
b = 1.52139 + 0.15181I
3.43299 2.76594I 8.24177 + 4.46128I
u = 0.959321 + 0.422476I
a = 0.127985 0.694740I
b = 1.52139 0.15181I
3.43299 + 2.76594I 8.24177 4.46128I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.891387 0.371717I
a = 0.033242 + 0.565045I
b = 1.42939 + 0.58093I
3.04790 0.34112I 6.76351 + 4.26460I
u = 0.891387 + 0.371717I
a = 0.033242 0.565045I
b = 1.42939 0.58093I
3.04790 + 0.34112I 6.76351 4.26460I
u = 0.868237 0.338548I
a = 0.012725 0.536185I
b = 1.41093 0.77808I
1.13942 + 4.67222I 3.07204 0.08862I
u = 0.868237 + 0.338548I
a = 0.012725 + 0.536185I
b = 1.41093 + 0.77808I
1.13942 4.67222I 3.07204 + 0.08862I
u = 0.843169 0.484968I
a = 0.203680 0.416849I
b = 0.961078 0.362588I
1.69304 2.02908I 0.14928 + 3.81049I
u = 0.843169 + 0.484968I
a = 0.203680 + 0.416849I
b = 0.961078 + 0.362588I
1.69304 + 2.02908I 0.14928 3.81049I
u = 0.648496 0.836558I
a = 0.351520 0.975110I
b = 0.006205 + 0.945446I
6.99192 6.13645I 1.80334 + 5.81380I
u = 0.648496 + 0.836558I
a = 0.351520 + 0.975110I
b = 0.006205 0.945446I
6.99192 + 6.13645I 1.80334 5.81380I
u = 0.616299 0.807630I
a = 0.160572 + 0.892832I
b = 0.141279 0.953665I
4.34347 1.49968I 1.34301 + 2.14167I
u = 0.616299 + 0.807630I
a = 0.160572 0.892832I
b = 0.141279 + 0.953665I
4.34347 + 1.49968I 1.34301 2.14167I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.582384 0.856983I
a = 0.062854 1.166115I
b = 0.127560 + 1.134839I
8.27872 + 1.80903I 3.80655 1.24949I
u = 0.582384 + 0.856983I
a = 0.062854 + 1.166115I
b = 0.127560 1.134839I
8.27872 1.80903I 3.80655 + 1.24949I
u = 0.454473 0.894363I
a = 0.56950 + 1.42313I
b = 0.45430 1.38825I
7.45504 + 2.41458I 3.12915 0.77897I
u = 0.454473 + 0.894363I
a = 0.56950 1.42313I
b = 0.45430 + 1.38825I
7.45504 2.41458I 3.12915 + 0.77897I
u = 0.428137 0.690269I
a = 0.668511 + 0.521270I
b = 0.588098 0.950437I
1.89347 1.09587I 2.44221 + 2.55300I
u = 0.428137 + 0.690269I
a = 0.668511 0.521270I
b = 0.588098 + 0.950437I
1.89347 + 1.09587I 2.44221 2.55300I
u = 0.408706 0.889849I
a = 0.80491 1.38796I
b = 0.59555 + 1.38804I
2.98486 + 5.33708I 2.97378 3.18856I
u = 0.408706 + 0.889849I
a = 0.80491 + 1.38796I
b = 0.59555 1.38804I
2.98486 5.33708I 2.97378 + 3.18856I
u = 0.406046 0.914346I
a = 0.83062 + 1.51440I
b = 0.60234 1.46500I
5.40227 + 10.31975I 0.10143 6.81318I
u = 0.406046 + 0.914346I
a = 0.83062 1.51440I
b = 0.60234 + 1.46500I
5.40227 10.31975I 0.10143 + 6.81318I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.395112 0.797470I
a = 0.829656 0.944652I
b = 0.635642 + 1.147259I
1.26952 + 3.45416I 4.11948 4.13749I
u = 0.395112 + 0.797470I
a = 0.829656 + 0.944652I
b = 0.635642 1.147259I
1.26952 3.45416I 4.11948 + 4.13749I
u = 0.215453 0.258244I
a = 0.988925 0.959286I
b = 0.402835 0.578892I
1.62489 1.27832I 1.31582 + 1.24706I
u = 0.215453 + 0.258244I
a = 0.988925 + 0.959286I
b = 0.402835 + 0.578892I
1.62489 + 1.27832I 1.31582 1.24706I
u = 0.147490 0.463827I
a = 2.03597 0.59027I
b = 0.787606 0.109338I
0.40746 + 4.84347I 2.88894 6.95560I
u = 0.147490 + 0.463827I
a = 2.03597 + 0.59027I
b = 0.787606 + 0.109338I
0.40746 4.84347I 2.88894 + 6.95560I
u = 0.267477 0.336990I
a = 2.04829 + 0.66333I
b = 0.581312 0.049909I
1.335431 + 0.455559I 7.14666 1.75479I
u = 0.267477 + 0.336990I
a = 2.04829 0.66333I
b = 0.581312 + 0.049909I
1.335431 0.455559I 7.14666 + 1.75479I
u = 0.625476 0.471165I
a = 1.67010 + 1.18891I
b = 0.365304 0.668182I
0.750144 0.597052I 5.79846 0.41792I
u = 0.625476 + 0.471165I
a = 1.67010 1.18891I
b = 0.365304 + 0.668182I
0.750144 + 0.597052I 5.79846 + 0.41792I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.625774 0.553199I
a = 1.82685 1.38552I
b = 0.488149 + 0.843348I
1.56352 5.23293I 2.28055 + 3.34796I
u = 0.625774 + 0.553199I
a = 1.82685 + 1.38552I
b = 0.488149 0.843348I
1.56352 + 5.23293I 2.28055 3.34796I
u = 0.744498 0.529322I
a = 1.44577 1.56624I
b = 0.149282 + 0.958064I
3.05917 + 2.25044I 0.26910 3.63492I
u = 0.744498 + 0.529322I
a = 1.44577 + 1.56624I
b = 0.149282 0.958064I
3.05917 2.25044I 0.26910 + 3.63492I
u = 0.777815
a = 0.931575
b = 0.0235617
1.12557 9.38806
u = 0.921822 0.543679I
a = 0.80291 + 1.93730I
b = 0.450724 1.151159I
2.49508 + 2.11040I 0.89981 2.52852I
u = 0.921822 + 0.543679I
a = 0.80291 1.93730I
b = 0.450724 + 1.151159I
2.49508 2.11040I 0.89981 + 2.52852I
u = 0.976476 0.533746I
a = 0.54173 1.96861I
b = 0.682122 + 1.116876I
1.82400 + 4.84040I 7.60479 4.63850I
u = 0.976476 + 0.533746I
a = 0.54173 + 1.96861I
b = 0.682122 1.116876I
1.82400 4.84040I 7.60479 + 4.63850I
u = 0.980018 0.559884I
a = 0.55911 + 2.09753I
b = 0.69820 1.23354I
0.49666 + 9.74933I 4.18987 8.27234I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.980018 + 0.559884I
a = 0.55911 2.09753I
b = 0.69820 + 1.23354I
0.49666 9.74933I 4.18987 + 8.27234I
u = 1.004559 0.452388I
a = 0.33920 1.62466I
b = 0.761658 + 0.767228I
3.12121 + 3.06401I 9.90946 4.96157I
u = 1.004559 + 0.452388I
a = 0.33920 + 1.62466I
b = 0.761658 0.767228I
3.12121 3.06401I 9.90946 + 4.96157I
u = 1.046729 0.394188I
a = 0.135208 + 1.388778I
b = 0.855036 0.498982I
2.04922 1.40109I 8.18227 + 2.01664I
u = 1.046729 + 0.394188I
a = 0.135208 1.388778I
b = 0.855036 + 0.498982I
2.04922 + 1.40109I 8.18227 2.01664I
u = 1.170606 0.255451I
a = 0.271560 + 0.787523I
b = 0.928676 + 0.247240I
2.42380 + 3.72637I 7.61863 2.36821I
u = 1.170606 + 0.255451I
a = 0.271560 0.787523I
b = 0.928676 0.247240I
2.42380 3.72637I 7.61863 + 2.36821I
u = 1.195246 0.198323I
a = 0.271492 0.579940I
b = 0.788651 0.477728I
3.76765 0.74515I 9.58698 + 4.43983I
u = 1.195246 + 0.198323I
a = 0.271492 + 0.579940I
b = 0.788651 + 0.477728I
3.76765 + 0.74515I 9.58698 4.43983I
u = 1.263612 0.138529I
a = 0.321540 0.309132I
b = 0.669397 0.885758I
2.79598 2.38150I 6.60298 + 3.87453I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.263612 + 0.138529I
a = 0.321540 + 0.309132I
b = 0.669397 + 0.885758I
2.79598 + 2.38150I 6.60298 3.87453I
u = 1.267989 0.074126I
a = 0.176506 + 0.199334I
b = 0.364909 + 0.990702I
1.311432 + 0.241237I 0.973926 + 0.823194I
u = 1.267989 + 0.074126I
a = 0.176506 0.199334I
b = 0.364909 0.990702I
1.311432 0.241237I 0.973926 0.823194I
u = 1.292749 0.134584I
a = 0.368918 + 0.227788I
b = 0.699572 + 1.035677I
0.53723 7.20125I 3.16725 + 7.28273I
u = 1.292749 + 0.134584I
a = 0.368918 0.227788I
b = 0.699572 1.035677I
0.53723 + 7.20125I 3.16725 7.28273I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u 1)
6
(u
75
+ 7u
74
+ ··· 5u 1)
c
2
(u + 1)
6
(u
75
+ 35u
74
+ ··· + 5u + 1)
c
3
, c
7
u
6
(u
75
+ u
74
+ ··· + 128u + 64)
c
5
(u
6
+ 3u
5
+ ··· + u + 1)(u
75
+ 6u
74
+ ··· 164u 77)
c
6
(u
6
+ u
5
+ ··· + u + 1)(u
75
+ 2u
74
+ ··· + 6u
2
1)
c
8
(u
6
+ u
5
+ ··· + u + 1)(u
75
+ 2u
74
+ ··· + 126u + 9)
c
9
(u
6
+ u
5
+ ··· + u + 1)(u
75
+ 2u
74
+ ··· + 6u
2
1)
c
10
(u
6
+ 3u
5
+ ··· + u + 1)(u
75
+ 36u
74
+ ··· + 12u + 1)
c
11
(u
6
+ u
5
+ ··· + u + 1)(u
75
+ 8u
74
+ ··· + 26798u + 565)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
75
35y
74
+ ··· + 5y 1)
c
2
(y 1)
6
(y
75
+ 17y
74
+ ··· 207y 1)
c
3
, c
7
y
6
(y
75
+ 39y
74
+ ··· 61440y 4096)
c
5
(y
6
+ y
5
+ ··· + 3y + 1)(y
75
+ 20y
74
+ ··· 398760y 5929)
c
6
(y
6
3y
5
+ ··· y + 1)(y
75
36y
74
+ ··· + 12y 1)
c
8
(y
6
3y
5
+ ··· y + 1)(y
75
12y
74
+ ··· + 2088y 81)
c
9
(y
6
3y
5
+ ··· y + 1)(y
75
36y
74
+ ··· + 12y 1)
c
10
(y
6
+ y
5
+ ··· + 3y + 1)(y
75
+ 8y
74
+ ··· + 56y 1)
c
11
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
75
+ 24y
74
+ ··· + 571587624y 319225)
14