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2026-04-25T18:56:44Z
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https://katlas.org/index.php?title=A_Sample_KnotTheory%60_Session&diff=1693690
A Sample KnotTheory` Session
2009-05-22T09:52:50Z
<p>TrocbOccch: </p>
<hr />
<div>http://www.textlaouricmonel.com <br />
{{Manual TOC Sidebar}}<br />
<br />
===[[Setup]]===<br />
<br />
The first step is to load <tt>KnotTheory`</tt> as in the [[Setup]] section:<br />
<br />
<!--$$<< KnotTheory`$$--><br />
<!--Robot Land, no human edits to "END"--><br />
<tt><font color=blue>In[1]:=</font></tt><code> << KnotTheory`</code><br />
<br />
<tt>Loading KnotTheory` (version of September 14, 2005, 13:37:36)...</tt><br />
<!--END--><br />
<br />
{{Knot Image Pair|8_17|gif|K11a231|gif}}<br />
<br />
{{Knot Image Pair|L8n6|gif|T(7,5)|jpg}}<br />
<br />
Let us now introduce the four star knots that will accompany us throughout this session:<br />
<br />
<!--$$K = Knot[8, 17];<br />
K11 = Knot[11, Alternating, 231];<br />
L = Link[8, NonAlternating, 6];<br />
TK = TorusKnot[7,5];$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{In|<br />
n = 2 |<br />
in = <nowiki>K = Knot[8, 17];<br />
K11 = Knot[11, Alternating, 231];<br />
L = Link[8, NonAlternating, 6];<br />
TK = TorusKnot[7,5];</nowiki>}}<br />
<!--END--><br />
<br />
===Presentations, Graphical Output and Tube Plots===<br />
<br />
====[[Planar Diagrams]]====<br />
<br />
<!--$$PD[K]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 3 |<br />
in = <nowiki>PD[K]</nowiki> |<br />
out= <nowiki>PD[X[6, 2, 7, 1], X[14, 8, 15, 7], X[8, 3, 9, 4], X[2, 13, 3, 14], <br />
<br />
X[12, 5, 13, 6], X[4, 9, 5, 10], X[16, 12, 1, 11], X[10, 16, 11, 15]]</nowiki>}}<br />
<!--END--><br />
<br />
====[[Gauss Codes]]====<br />
<br />
<!--$${GaussCode[K], GaussCode[L]}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 4 |<br />
in = <nowiki>{GaussCode[K], GaussCode[L]}</nowiki> |<br />
out= <nowiki>{GaussCode[1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 7, -5, 4, -2, 8, -7], <br />
<br />
GaussCode[{1, -7, 2, -8}, {-5, 4, -6, 3}, <br />
<br />
{7, -1, -4, 5, 8, -2, -3, 6}]}</nowiki>}}<br />
<!--END--><br />
<br />
====[[DT (Dowker-Thistlethwaite) Codes]]====<br />
<br />
<!--$$DTCode[K]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 5 |<br />
in = <nowiki>DTCode[K]</nowiki> |<br />
out= <nowiki>DTCode[6, 8, 12, 14, 4, 16, 2, 10]</nowiki>}}<br />
<!--END--><br />
<br />
====[[Braid Representatives]]====<br />
<br />
<!--$$br = BR[K]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 6 |<br />
in = <nowiki>br = BR[K]</nowiki> |<br />
out= <nowiki>BR[3, {-1, -1, 2, -1, 2, -1, 2, 2}]</nowiki>}}<br />
<!--END--><br />
<br />
<!--$${First[br], Crossings[br], BraidIndex[K]}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 7 |<br />
in = <nowiki>{First[br], Crossings[br], BraidIndex[K]}</nowiki> |<br />
out= <nowiki>{3, 8, 3}</nowiki>}}<br />
<!--END--><br />
<br />
====[[Drawing Braids]]====<br />
<br />
<!--$$Show[BraidPlot[br]]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{Graphics|<br />
n = 8 |<br />
in = <nowiki>Show[BraidPlot[br]]</nowiki> |<br />
img= A_Sample_KnotTheory_Session_Out_8.gif |<br />
out= <nowiki>-Graphics-</nowiki>}}<br />
<!--END--><br />
<br />
====[[Drawing MorseLink Presentations]]====<br />
<br />
<!--$$Show[DrawMorseLink[K]]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{Graphics|<br />
n = 9 |<br />
in = <nowiki>Show[DrawMorseLink[K]]</nowiki> |<br />
img= A_Sample_KnotTheory_Session_Out_9.gif |<br />
out= <nowiki>-Graphics-</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$Show[DrawMorseLink[L]]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{Graphics|<br />
n = 10 |<br />
in = <nowiki>Show[DrawMorseLink[L]]</nowiki> |<br />
img= A_Sample_KnotTheory_Session_Out_10.gif |<br />
out= <nowiki>-Graphics-</nowiki>}}<br />
<!--END--><br />
<br />
====[[Drawing with TubePlot]]====<br />
<br />
<!--$$Show[TubePlot[TK]]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{Graphics|<br />
n = 11 |<br />
in = <nowiki>Show[TubePlot[TK]]</nowiki> |<br />
img= A_Sample_KnotTheory_Session_Out_11.gif |<br />
out= <nowiki>-Graphics3D-</nowiki>}}<br />
<!--END--><br />
<br />
===[[Three Dimensional Invariants]]===<br />
<br />
<!--$$(#[K]&) /@ {<br />
SymmetryType, UnknottingNumber, ThreeGenus,<br />
BridgeIndex, SuperBridgeIndex, NakanishiIndex<br />
}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 12 |<br />
in = <nowiki>(#[K]&) /@ {<br />
SymmetryType, UnknottingNumber, ThreeGenus,<br />
BridgeIndex, SuperBridgeIndex, NakanishiIndex<br />
}</nowiki> |<br />
out= <nowiki>{NegativeAmphicheiral, 1, 3, 3, 4, 1}</nowiki>}}<br />
<!--END--><br />
<br />
===Polynomial Invariants===<br />
<br />
====[[The Alexander-Conway Polynomial]]====<br />
<br />
<!--$$alex = Alexander[K11][t]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 13 |<br />
in = <nowiki>alex = Alexander[K11][t]</nowiki> |<br />
out= <nowiki> -4 5 12 20 2 3 4<br />
-23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t<br />
3 2 t<br />
t t</nowiki>}}<br />
<!--END--><br />
<br />
<!--$$Conway[K11][t]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 14 |<br />
in = <nowiki>Conway[K11][t]</nowiki> |<br />
out= <nowiki> 2 4 6 8<br />
1 + t - 2 t - 3 t - t</nowiki>}}<br />
<!--END--><br />
<br />
====="Similar" Knots (within the Atlas)=====<br />
<br />
<!--$$Select[AllKnots[], (alex === Alexander[#][t])&]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 15 |<br />
in = <nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki> |<br />
out= <nowiki>{Knot[11, Alternating, 57], Knot[11, Alternating, 108], <br />
<br />
Knot[11, Alternating, 139], Knot[11, Alternating, 231]}</nowiki>}}<br />
<!--END--><br />
<br />
{{Knot Image Quadruple|K11a57|gif|K11a108|gif|K11a139|gif|K11a231|gif}}<br />
<br />
=====[[The Determinant and the Signature]]=====<br />
<br />
<!--$${KnotDet[K], KnotSignature[K]}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 16 |<br />
in = <nowiki>{KnotDet[K], KnotSignature[K]}</nowiki> |<br />
out= <nowiki>{37, 0}</nowiki>}}<br />
<!--END--><br />
<br />
====[[The Jones Polynomial]]====<br />
<br />
<!--$$J=Jones[K11][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 17 |<br />
in = <nowiki>J=Jones[K11][q]</nowiki> |<br />
out= <nowiki> -7 3 7 12 14 16 16 2 3 4<br />
-12 + q - -- + -- - -- + -- - -- + -- + 10 q - 5 q + 2 q - q<br />
6 5 4 3 2 q<br />
q q q q q</nowiki>}}<br />
<!--END--><br />
<br />
====="Similar" Knots (within the Atlas)=====<br />
<br />
<!--$$Select[AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q])&]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 18 |<br />
in = <nowiki>Select[AllKnots[], (J === Jones[#][q] &#124;&#124; (J /. q -> 1/q) === Jones[#][q])&]</nowiki> |<br />
out= <nowiki>{Knot[11, Alternating, 57], Knot[11, Alternating, 231]}</nowiki>}}<br />
<!--END--><br />
<br />
====[[The A2 Invariant]]====<br />
<br />
<!--$$A2Invariant[L][q]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 19 |<br />
in = <nowiki>A2Invariant[L][q]</nowiki> |<br />
out= <nowiki> -30 3 4 5 4 3 2 -16 -14 -10 -8<br />
q + --- + --- + --- + --- + --- + --- + q + q + q + q + <br />
28 26 24 22 20 18<br />
q q q q q q<br />
<br />
-6<br />
q</nowiki>}}<br />
<!--END--><br />
<br />
====[[The HOMFLY-PT Polynomial]]====<br />
<br />
<!--$$HOMFLYPT[L][a, z]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 20 |<br />
in = <nowiki>HOMFLYPT[L][a, z]</nowiki> |<br />
out= <nowiki> 6 8 10<br />
4 8 a 2 a a 4 2 4 4<br />
2 a - 2 a + -- - ---- + --- + 4 a z + a z<br />
2 2 2<br />
z z z</nowiki>}}<br />
<!--END--><br />
<br />
====[[The Kauffman Polynomial]]====<br />
<br />
<!--$$Kauffman[L][a, z]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 21 |<br />
in = <nowiki>Kauffman[L][a, z]</nowiki> |<br />
out= <nowiki> 6 8 10 7 9<br />
4 6 8 10 a 2 a a 2 a 2 a 7<br />
2 a - 2 a - 9 a - 6 a + -- + ---- + --- - ---- - ---- + 8 a z + <br />
2 2 2 z z<br />
z z z<br />
<br />
9 4 2 8 2 10 2 7 3 9 3 4 4<br />
8 a z - 4 a z + 14 a z + 10 a z - 6 a z - 6 a z + a z - <br />
<br />
8 4 10 4 7 5 9 5 8 6 10 6<br />
7 a z - 6 a z + a z + a z + a z + a z</nowiki>}}<br />
<!--END--><br />
<br />
<br />
===[[Finite Type (Vassiliev) Invariants]]===<br />
<br />
<!--$${Vassiliev[2][K], Vassiliev[3][K]}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 22 |<br />
in = <nowiki>{Vassiliev[2][K], Vassiliev[3][K]}</nowiki> |<br />
out= <nowiki>{-1, 0}</nowiki>}}<br />
<!--END--><br />
<br />
===[[Khovanov Homology]]===<br />
<br />
<!--$$Kh[TK][q, t]$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 23 |<br />
in = <nowiki>Kh[TK][q, t]</nowiki> |<br />
out= <nowiki> 23 25 27 2 31 3 29 4 31 4 33 5 35 5<br />
q + q + q t + q t + q t + q t + q t + q t + <br />
<br />
31 6 33 6 35 7 37 7 33 8 35 8 37 9<br />
q t + q t + q t + q t + q t + 2 q t + q t + <br />
<br />
39 9 37 10 41 11 39 12 41 12 45 12<br />
2 q t + 2 q t + 3 q t + q t + 2 q t + q t + <br />
<br />
43 13 45 13 43 14 47 14 47 15 47 16<br />
2 q t + 2 q t + q t + q t + 2 q t + q t + <br />
<br />
51 16 51 17<br />
q t + q t</nowiki>}}<br />
<!--END--><br />
<br />
===[[The Coloured Jones Polynomials]]===<br />
<br />
<!--$$ColouredJones[K, #][q]& /@ {2, 3, 4, 5, 6, 7}$$--><br />
<!--Robot Land, no human edits to "END"--><br />
{{InOut|<br />
n = 24 |<br />
in = <nowiki>ColouredJones[K, #][q]& /@ {2, 3, 4, 5, 6, 7}</nowiki> |<br />
out= <nowiki> -12 3 -10 9 14 3 28 25 14 47 29 25<br />
{55 + q - --- + q + -- - -- - -- + -- - -- - -- + -- - -- - -- - <br />
11 9 8 7 6 5 4 3 2 q<br />
q q q q q q q q q<br />
<br />
2 3 4 5 6 7 8 9<br />
25 q - 29 q + 47 q - 14 q - 25 q + 28 q - 3 q - 14 q + 9 q + <br />
<br />
10 11 12 -24 3 -22 5 -20 14<br />
q - 3 q + q , 225 + q - --- + q + --- + q - --- - <br />
23 21 19<br />
q q q<br />
<br />
6 29 17 43 40 55 73 64 108 61 146<br />
--- + --- + --- - --- - --- + --- + --- - --- - --- + -- + --- - <br />
18 17 16 15 14 13 12 11 10 9 8<br />
q q q q q q q q q q q<br />
<br />
53 177 38 205 26 216 6 2 3 4<br />
-- - --- + -- + --- - -- - --- + - + 6 q - 216 q - 26 q + 205 q + <br />
7 6 5 4 3 2 q<br />
q q q q q q<br />
<br />
5 6 7 8 9 10 11<br />
38 q - 177 q - 53 q + 146 q + 61 q - 108 q - 64 q + <br />
<br />
12 13 14 15 16 17 18<br />
73 q + 55 q - 40 q - 43 q + 17 q + 29 q - 6 q - <br />
<br />
19 20 21 22 23 24<br />
14 q + q + 5 q + q - 3 q + q , <br />
<br />
-40 3 -38 5 3 -35 17 6 31<br />
1233 + q - --- + q + --- - --- + q - --- + --- + --- + <br />
39 37 36 34 33 32<br />
q q q q q q<br />
<br />
-31 82 16 96 69 52 216 146 120 216 260<br />
q - --- - --- + --- + --- + --- - --- - --- + --- + --- + --- - <br />
29 28 27 26 25 24 23 22 21 20<br />
q q q q q q q q q q<br />
<br />
323 393 7 340 605 292 631 265 347 945 149<br />
--- - --- - --- + --- + --- - --- - --- - --- + --- + --- - --- - <br />
19 18 17 16 15 14 13 12 11 10 9<br />
q q q q q q q q q q q<br />
<br />
759 522 261 1161 11 771 694 144 2<br />
--- - --- + --- + ---- + -- - --- - --- + --- + 144 q - 694 q - <br />
8 7 6 5 4 3 2 q<br />
q q q q q q q<br />
<br />
3 4 5 6 7 8 9<br />
771 q + 11 q + 1161 q + 261 q - 522 q - 759 q - 149 q + <br />
<br />
10 11 12 13 14 15<br />
945 q + 347 q - 265 q - 631 q - 292 q + 605 q + <br />
<br />
16 17 18 19 20 21 22<br />
340 q - 7 q - 393 q - 323 q + 260 q + 216 q + 120 q - <br />
<br />
23 24 25 26 27 28 29<br />
146 q - 216 q + 52 q + 69 q + 96 q - 16 q - 82 q + <br />
<br />
31 32 33 34 35 36 37 38 39<br />
q + 31 q + 6 q - 17 q + q - 3 q + 5 q + q - 3 q + <br />
<br />
40 -60 3 -58 5 3 3 2 5 8<br />
q , 4841 + q - --- + q + --- - --- - --- - --- - --- + --- + <br />
59 57 56 55 54 53 52<br />
q q q q q q q<br />
<br />
26 4 30 43 34 35 112 107 31 197 237<br />
--- + --- - --- - --- - --- + --- + --- + --- - --- - --- - --- - <br />
51 50 49 48 47 46 45 44 43 42 41<br />
q q q q q q q q q q q<br />
<br />
60 270 462 264 285 728 603 141 976 1094 186<br />
--- + --- + --- + --- - --- - --- - --- + --- + --- + ---- + --- - <br />
40 39 38 37 36 35 34 33 32 31 30<br />
q q q q q q q q q q q<br />
<br />
1134 1650 699 1099 2200 1387 888 2662 2125 494<br />
---- - ---- - --- + ---- + ---- + ---- - --- - ---- - ---- + --- + <br />
29 28 27 26 25 24 23 22 21 20<br />
q q q q q q q q q q<br />
<br />
2955 2877 9 3114 3506 568 3121 4033 1086 3040<br />
---- + ---- + --- - ---- - ---- - --- + ---- + ---- + ---- - ---- - <br />
19 18 17 16 15 14 13 12 11 10<br />
q q q q q q q q q q<br />
<br />
4387 1560 2881 4660 1920 2707 4762 2247 2479<br />
---- - ---- + ---- + ---- + ---- - ---- - ---- - ---- + ---- + <br />
9 8 7 6 5 4 3 2 q<br />
q q q q q q q q<br />
<br />
2 3 4 5 6 7<br />
2479 q - 2247 q - 4762 q - 2707 q + 1920 q + 4660 q + 2881 q - <br />
<br />
8 9 10 11 12 13<br />
1560 q - 4387 q - 3040 q + 1086 q + 4033 q + 3121 q - <br />
<br />
14 15 16 17 18 19<br />
568 q - 3506 q - 3114 q + 9 q + 2877 q + 2955 q + <br />
<br />
20 21 22 23 24 25<br />
494 q - 2125 q - 2662 q - 888 q + 1387 q + 2200 q + <br />
<br />
26 27 28 29 30 31<br />
1099 q - 699 q - 1650 q - 1134 q + 186 q + 1094 q + <br />
<br />
32 33 34 35 36 37<br />
976 q + 141 q - 603 q - 728 q - 285 q + 264 q + <br />
<br />
38 39 40 41 42 43 44<br />
462 q + 270 q - 60 q - 237 q - 197 q - 31 q + 107 q + <br />
<br />
45 46 47 48 49 50 51<br />
112 q + 35 q - 34 q - 43 q - 30 q + 4 q + 26 q + <br />
<br />
52 53 54 55 56 57 58 59 60<br />
8 q - 5 q - 2 q - 3 q - 3 q + 5 q + q - 3 q + q , <br />
<br />
-84 3 -82 5 3 3 6 10 3 3<br />
26111 + q - --- + q + --- - --- - --- - --- + --- - --- + --- + <br />
83 81 80 79 78 77 76 75<br />
q q q q q q q q<br />
<br />
29 15 31 49 14 16 61 153 14 117 273<br />
--- - --- - --- - --- + --- + --- + --- + --- + --- - --- - --- - <br />
74 73 72 71 70 69 68 67 66 65 64<br />
q q q q q q q q q q q<br />
<br />
149 92 203 641 463 57 691 870 1005 189 1343<br />
--- - --- + --- + --- + --- + --- - --- - --- - ---- - --- + ---- + <br />
63 62 61 60 59 58 57 56 55 54 53<br />
q q q q q q q q q q q<br />
<br />
1882 1543 247 1725 3355 2544 658 3470 4894 2812<br />
---- + ---- - --- - ---- - ---- - ---- + --- + ---- + ---- + ---- - <br />
52 51 50 49 48 47 46 45 44 43<br />
q q q q q q q q q q<br />
<br />
590 5842 7188 3328 2689 8288 8456 4312 5674<br />
--- - ---- - ---- - ---- + ---- + ---- + ---- + ---- - ---- - <br />
42 41 40 39 38 37 36 35 34<br />
q q q q q q q q q<br />
<br />
11801 10070 1954 8878 14013 11845 1776 13643 16608<br />
----- - ----- - ---- + ---- + ----- + ----- - ---- - ----- - ----- - <br />
33 32 31 30 29 28 27 26 25<br />
q q q q q q q q q<br />
<br />
8872 6032 16953 18906 4000 12400 20628 15121 1645<br />
---- + ---- + ----- + ----- + ---- - ----- - ----- - ----- + ---- + <br />
24 23 22 21 20 19 18 17 16<br />
q q q q q q q q q<br />
<br />
17146 23466 9075 9763 22071 19122 2210 15962 25565<br />
----- + ----- + ---- - ---- - ----- - ----- - ---- + ----- + ----- + <br />
15 14 13 12 11 10 9 8 7<br />
q q q q q q q q q<br />
<br />
12389 7226 22014 21134 4957 14414 2<br />
----- - ---- - ----- - ----- - ---- + ----- + 14414 q - 4957 q - <br />
6 5 4 3 2 q<br />
q q q q q<br />
<br />
3 4 5 6 7 8<br />
21134 q - 22014 q - 7226 q + 12389 q + 25565 q + 15962 q - <br />
<br />
9 10 11 12 13 14<br />
2210 q - 19122 q - 22071 q - 9763 q + 9075 q + 23466 q + <br />
<br />
15 16 17 18 19<br />
17146 q + 1645 q - 15121 q - 20628 q - 12400 q + <br />
<br />
20 21 22 23 24 25<br />
4000 q + 18906 q + 16953 q + 6032 q - 8872 q - 16608 q - <br />
<br />
26 27 28 29 30 31<br />
13643 q - 1776 q + 11845 q + 14013 q + 8878 q - 1954 q - <br />
<br />
32 33 34 35 36 37<br />
10070 q - 11801 q - 5674 q + 4312 q + 8456 q + 8288 q + <br />
<br />
38 39 40 41 42 43<br />
2689 q - 3328 q - 7188 q - 5842 q - 590 q + 2812 q + <br />
<br />
44 45 46 47 48 49<br />
4894 q + 3470 q + 658 q - 2544 q - 3355 q - 1725 q - <br />
<br />
50 51 52 53 54 55<br />
247 q + 1543 q + 1882 q + 1343 q - 189 q - 1005 q - <br />
<br />
56 57 58 59 60 61 62<br />
870 q - 691 q + 57 q + 463 q + 641 q + 203 q - 92 q - <br />
<br />
63 64 65 66 67 68 69<br />
149 q - 273 q - 117 q + 14 q + 153 q + 61 q + 16 q + <br />
<br />
70 71 72 73 74 75 76<br />
14 q - 49 q - 31 q - 15 q + 29 q + 3 q - 3 q + <br />
<br />
77 78 79 80 81 82 83 84<br />
10 q - 6 q - 3 q - 3 q + 5 q + q - 3 q + q , <br />
<br />
-112 3 -110 5 3 3 6 6<br />
127145 + q - ---- + q + ---- - ---- - ---- - ---- + ---- + <br />
111 109 108 107 106 105<br />
q q q q q q<br />
<br />
12 8 6 10 16 26 41 7 74 44<br />
---- - ---- + ---- + ---- - ---- - --- - --- + --- + --- + --- + <br />
104 103 102 101 100 99 98 97 96 95<br />
q q q q q q q q q q<br />
<br />
71 43 78 159 283 154 143 317 550 516 93<br />
--- + --- - --- - --- - --- - --- + --- + --- + --- + --- + --- - <br />
94 93 92 91 90 89 88 87 86 85 84<br />
q q q q q q q q q q q<br />
<br />
417 1159 1332 683 256 1725 2573 2216 836 1934<br />
--- - ---- - ---- - --- + --- + ---- + ---- + ---- + --- - ---- - <br />
83 82 81 80 79 78 77 76 75 74<br />
q q q q q q q q q q<br />
<br />
4278 4774 3301 913 5542 8189 7815 2389 5364<br />
---- - ---- - ---- + --- + ---- + ---- + ---- + ---- - ---- - <br />
73 72 71 70 69 68 67 66 65<br />
q q q q q q q q q<br />
<br />
11792 14143 8598 2327 13890 21402 18063 4775 12979<br />
----- - ----- - ---- + ---- + ----- + ----- + ----- + ---- - ----- - <br />
64 63 62 61 60 59 58 57 56<br />
q q q q q q q q q<br />
<br />
28137 29695 16138 7462 32099 41960 31319 3221<br />
----- - ----- - ----- + ---- + ----- + ----- + ----- + ---- - <br />
55 54 53 52 51 50 49 48<br />
q q q q q q q q<br />
<br />
31814 52797 48414 18546 26269 60101 65456 37209<br />
----- - ----- - ----- - ----- + ----- + ----- + ----- + ----- - <br />
47 46 45 44 43 42 41 40<br />
q q q q q q q q<br />
<br />
15734 62982 80416 56819 1416 61028 91744 75657<br />
----- - ----- - ----- - ----- + ---- + ----- + ----- + ----- + <br />
39 38 37 36 35 34 33 32<br />
q q q q q q q q<br />
<br />
14955 55290 98994 91866 31357 46837 102372 104758<br />
----- - ----- - ----- - ----- - ----- + ----- + ------ + ------ + <br />
31 30 29 28 27 26 25 24<br />
q q q q q q q q<br />
<br />
46339 37376 102713 113992 58991 28020 101063 120209<br />
----- - ----- - ------ - ------ - ----- + ----- + ------ + ------ + <br />
23 22 21 20 19 18 17 16<br />
q q q q q q q q<br />
<br />
68876 19792 98300 123826 76313 12722 95254 125943<br />
----- - ----- - ----- - ------ - ----- + ----- + ----- + ------ + <br />
15 14 13 12 11 10 9 8<br />
q q q q q q q q<br />
<br />
81699 7156 92151 126720 85773 2177 89089<br />
----- - ---- - ----- - ------ - ----- + ---- + ----- + 89089 q + <br />
7 6 5 4 3 2 q<br />
q q q q q q<br />
<br />
2 3 4 5 6 7<br />
2177 q - 85773 q - 126720 q - 92151 q - 7156 q + 81699 q + <br />
<br />
8 9 10 11 12<br />
125943 q + 95254 q + 12722 q - 76313 q - 123826 q - <br />
<br />
13 14 15 16 17<br />
98300 q - 19792 q + 68876 q + 120209 q + 101063 q + <br />
<br />
18 19 20 21 22<br />
28020 q - 58991 q - 113992 q - 102713 q - 37376 q + <br />
<br />
23 24 25 26 27<br />
46339 q + 104758 q + 102372 q + 46837 q - 31357 q - <br />
<br />
28 29 30 31 32<br />
91866 q - 98994 q - 55290 q + 14955 q + 75657 q + <br />
<br />
33 34 35 36 37<br />
91744 q + 61028 q + 1416 q - 56819 q - 80416 q - <br />
<br />
38 39 40 41 42<br />
62982 q - 15734 q + 37209 q + 65456 q + 60101 q + <br />
<br />
43 44 45 46 47<br />
26269 q - 18546 q - 48414 q - 52797 q - 31814 q + <br />
<br />
48 49 50 51 52<br />
3221 q + 31319 q + 41960 q + 32099 q + 7462 q - <br />
<br />
53 54 55 56 57<br />
16138 q - 29695 q - 28137 q - 12979 q + 4775 q + <br />
<br />
58 59 60 61 62<br />
18063 q + 21402 q + 13890 q + 2327 q - 8598 q - <br />
<br />
63 64 65 66 67 68<br />
14143 q - 11792 q - 5364 q + 2389 q + 7815 q + 8189 q + <br />
<br />
69 70 71 72 73 74<br />
5542 q + 913 q - 3301 q - 4774 q - 4278 q - 1934 q + <br />
<br />
75 76 77 78 79 80<br />
836 q + 2216 q + 2573 q + 1725 q + 256 q - 683 q - <br />
<br />
81 82 83 84 85 86<br />
1332 q - 1159 q - 417 q + 93 q + 516 q + 550 q + <br />
<br />
87 88 89 90 91 92 93<br />
317 q + 143 q - 154 q - 283 q - 159 q - 78 q + 43 q + <br />
<br />
94 95 96 97 98 99 100<br />
71 q + 44 q + 74 q + 7 q - 41 q - 26 q - 16 q + <br />
<br />
101 102 103 104 105 106 107<br />
10 q + 6 q - 8 q + 12 q + 6 q - 6 q - 3 q - <br />
<br />
108 109 110 111 112<br />
3 q + 5 q + q - 3 q + q }</nowiki>}}<br />
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