9 34

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8

In[6]:=

DTCode[K]

Out[6]=

6 8 10 16 14 18 4 2 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[8*20]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

BR(4,{−1,2,−1,2,−3,2,−1,2,−3})

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{7, 11}, {10, 2}, {11, 4}, {3, 9}, {5, 10}, {4, 8}, {9, 6}, {1, 5}, {2, 7}, {6, 1}, {8, 3}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_34.

A1 Invariants.

Weight	Invariant
1	−q11 + 3q9−3q7 + 3q5−2q3 + 2q−1−2q−3 + 4q−5−3q−7 + q−9
2	q32−3q30−q28 + 12q26−8q24−16q22 + 25q20 + q18−32q16 + 22q14 + 16q12−28q10 + 6q8 + 19q6−10q4−13q2 + 11 + 14q−2−25q−4−2q−6 + 33q−8−21q−10−15q−12 + 30q−14−7q−16−15q−18 + 10q−20 + q−22−3q−24 + q−26
3	−q63 + 3q61 + q59−8q57−7q55 + 16q53 + 30q51−24q49−64q47 + 6q45 + 107q43 + 40q41−138q39−106q37 + 135q35 + 178q33−96q31−232q29 + 37q27 + 250q25 + 28q23−234q21−86q19 + 198q17 + 119q15−144q13−137q11 + 85q9 + 150q7−25q5−151q3−42q + 150q−1 + 110q−3−132q−5−180q−7 + 98q−9 + 234q−11−44q−13−255q−15−22q−17 + 239q−19 + 81q−21−187q−23−113q−25 + 121q−27 + 105q−29−53q−31−82q−33 + 15q−35 + 48q−37−2q−39−18q−41−3q−43 + 7q−45 + q−47−3q−49 + q−51
4	q104−3q102−q100 + 8q98 + 3q96−q94−30q92−20q90 + 45q88 + 67q86 + 55q84−123q82−211q80−29q78 + 231q76 + 437q74 + 26q72−539q70−606q68−39q66 + 965q64 + 866q62−237q60−1287q58−1138q56 + 669q54 + 1720q52 + 979q50−1012q48−2141q46−546q44 + 1536q42 + 2020q40 + 111q38−2061q36−1539q34 + 565q32 + 2042q30 + 996q28−1229q26−1668q24−264q22 + 1418q20 + 1232q18−399q16−1348q14−721q12 + 758q10 + 1234q8 + 313q6−1013q4−1169q2 + 46 + 1278q−2 + 1203q−4−513q−6−1679q−8−981q−10 + 1003q−12 + 2097q−14 + 482q−16−1631q−18−2013q−20 + 11q−22 + 2191q−24 + 1535q−26−628q−28−2103q−30−1083q−32 + 1160q−34 + 1627q−36 + 495q−38−1100q−40−1210q−42 + 49q−44 + 779q−46 + 690q−48−151q−50−563q−52−227q−54 + 109q−56 + 291q−58 + 71q−60−111q−62−74q−64−24q−66 + 53q−68 + 22q−70−13q−72−6q−74−6q−76 + 7q−78 + q−80−3q−82 + q−84
5	−q155 + 3q153 + q151−8q149−3q147 + 5q145 + 15q143 + 20q141−q139−59q137−86q135−15q133 + 123q131 + 240q129 + 177q127−129q125−545q123−631q121−100q119 + 802q117 + 1419q115 + 982q113−623q111−2384q109−2611q107−542q105 + 2761q103 + 4754q101 + 3135q99−1778q97−6538q95−6794q93−1213q91 + 6706q89 + 10559q87 + 6079q85−4446q83−12971q81−11708q79−320q77 + 12798q75 + 16578q73 + 6682q71−9776q69−19250q67−13013q65 + 4536q63 + 19015q61 + 17848q59 + 1556q57−16212q55−20243q53−7017q51 + 11839q49 + 20042q47 + 10881q45−7083q43−17954q41−12766q39 + 2989q37 + 14851q35 + 12931q33 + 56q31−11650q29−12114q27−1995q25 + 8933q23 + 11006q21 + 3244q19−6807q17−10249q15−4442q13 + 5184q11 + 10107q9 + 6046q7−3522q5−10451q3−8507q + 1299q−1 + 10908q−3 + 11682q−5 + 1980q−7−10674q−9−15162q−11−6449q−13 + 9048q−15 + 18039q−17 + 11723q−19−5574q−21−19230q−23−16847q−25 + 368q−27 + 17919q−29 + 20551q−31 + 5661q−33−13978q−35−21629q−37−11107q−39 + 8117q−41 + 19635q−43 + 14545q−45−1807q−47−15140q−49−15083q−51−3294q−53 + 9385q−55 + 12979q−57 + 6171q−59−4122q−61−9312q−63−6523q−65 + 394q−67 + 5411q−69 + 5270q−71 + 1385q−73−2442q−75−3377q−77−1635q−79 + 677q−81 + 1733q−83 + 1211q−85 + 37q−87−739q−89−650q−91−151q−93 + 224q−95 + 283q−97 + 113q−99−60q−101−110q−103−41q−105 + 20q−107 + 27q−109 + 11q−111−q−113−9q−115−6q−117 + 7q−119 + q−121−3q−123 + q−125

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + 2q12−2q10 + 2q8−q6−q4 + 2q2−2 + 3q−2−2q−4 + q−6 + 2q−8−2q−10 + q−12
1,1	q44−6q42 + 20q40−50q38 + 107q36−204q34 + 344q32−514q30 + 701q28−864q26 + 950q24−938q22 + 791q20−502q18 + 98q16 + 374q14−840q12 + 1276q10−1606q8 + 1788q6−1806q4 + 1650q2−1352 + 934q−2−456q−4−16q−6 + 432q−8−726q−10 + 888q−12−918q−14 + 850q−16−712q−18 + 538q−20−376q−22 + 248q−24−148q−26 + 77q−28−38q−30 + 18q−32−6q−34 + q−36
2,0	q42−q40−3q38 + 7q34 + 5q32−10q30−7q28 + 9q26 + 7q24−13q22−7q20 + 13q18 + 8q16−11q14−q12 + 13q10−5q8−3q6 + 4q4−q2−7 + 5q−2 + 7q−4−13q−6−4q−8 + 16q−10 + 6q−12−17q−14 + q−16 + 14q−18−9q−22−3q−24 + 6q−26−2q−30 + q−32

A3 Invariants.

Weight	Invariant
0,1,0	q34−3q32 + q30 + 6q28−12q26 + 6q24 + 13q22−22q20 + 10q18 + 15q16−24q14 + 7q12 + 14q10−13q8−2q6 + 7q4 + 3q2−6−6q−2 + 18q−4−6q−6−17q−8 + 25q−10−5q−12−18q−14 + 20q−16−3q−18−10q−20 + 9q−22−3q−26 + q−28
1,0,0	−q21 + q19 + 2q15−2q13 + 3q11−2q9 + q7−q5 + q3−q−1 + 2q−3−2q−5 + 2q−7−q−9 + 3q−11−2q−13 + q−15

A4 Invariants.

Weight	Invariant
0,1,0,0	q44−q42−3q40 + 2q38 + 4q36−4q34−5q32 + 9q30 + 7q28−14q26−4q24 + 20q22−q20−20q18 + 9q16 + 17q14−14q12−12q10 + 16q8 + 2q6−19q4 + 10q2 + 18−16q−2−5q−4 + 24q−6−5q−8−21q−10 + 8q−12 + 15q−14−12q−16−10q−18 + 14q−20 + 7q−22−10q−24−q−26 + 7q−28−2q−30−2q−32 + q−34
1,0,0,0	−q26 + q24 + 2q18−2q16 + 3q14−q12 + q8−q6 + q4−q2 + 1−2q−2 + 2q−4−2q−6 + 2q−8 + 3q−14−2q−16 + q−18

B2 Invariants.

Weight	Invariant
0,1	−q34 + 3q32−7q30 + 12q28−18q26 + 24q24−25q22 + 26q20−22q18 + 15q16−4q14−9q12 + 22q10−35q8 + 44q6−49q4 + 49q2−44 + 34q−2−20q−4 + 8q−6 + 5q−8−15q−10 + 23q−12−26q−14 + 26q−16−23q−18 + 18q−20−11q−22 + 6q−24−3q−26 + q−28
1,0	q56−3q52−3q50 + 4q48 + 9q46−q44−15q42−8q40 + 18q38 + 19q36−10q34−27q32−3q30 + 28q28 + 15q26−21q24−22q22 + 10q20 + 23q18−q16−20q14−3q12 + 18q10 + 7q8−16q6−11q4 + 13q2 + 16−10q−2−20q−4 + 7q−6 + 23q−8 + q−10−25q−12−10q−14 + 23q−16 + 22q−18−14q−20−27q−22 + q−24 + 23q−26 + 10q−28−13q−30−14q−32 + 3q−34 + 10q−36 + 3q−38−3q−40−3q−42 + q−46

D4 Invariants.

Weight	Invariant
1,0,0,0	q46−3q44 + 4q42−6q40 + 10q38−15q36 + 17q34−18q32 + 22q30−21q28 + 18q26−15q24 + 14q22−5q20−5q18 + 10q16−14q14 + 25q12−33q10 + 33q8−35q6 + 40q4−37q2 + 31−29q−2 + 25q−4−12q−6 + 4q−8−2q−10−5q−12 + 16q−14−17q−16 + 18q−18−22q−20 + 23q−22−16q−24 + 14q−26−14q−28 + 10q−30−4q−32 + 3q−34−3q−36 + q−38

G2 Invariants.

Weight

Invariant

1,0

q80−3q78 + 7q76−13q74 + 14q72−12q70−q68 + 27q66−55q64 + 83q62−84q60 + 44q58 + 24q56−112q54 + 181q52−188q50 + 127q48−11q46−114q44 + 205q42−213q40 + 135q38−7q36−116q34 + 167q32−131q30 + 24q28 + 103q26−178q24 + 183q22−102q20−37q18 + 174q16−269q14 + 272q12−182q10 + 32q8 + 130q6−244q4 + 280q2−217 + 85q−2 + 58q−4−170q−6 + 191q−8−117q−10−6q−12 + 123q−14−165q−16 + 125q−18−16q−20−113q−22 + 195q−24−206q−26 + 141q−28−26q−30−88q−32 + 159q−34−165q−36 + 126q−38−55q−40−10q−42 + 48q−44−68q−46 + 60q−48−38q−50 + 18q−52 + q−54−8q−56 + 10q−58−10q−60 + 6q−62−3q−64 + q−66

.

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

−t3 + 6t2−16t + 23−16t−1 + 6t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6−z2 + 1

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

{1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 69, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

q4−4q3 + 8q2−10q + 12−12q−1 + 10q−2−7q−3 + 4q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

−z6 + 2a2z4 + z4a−2−3z4−a4z2 + 3a2z2 + z2a−2−4z2 + a2 + a−2−1

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

3a2z8 + 3z8 + 6a3z7 + 14az7 + 8z7a−1 + 4a4z6 + 5a2z6 + 8z6a−2 + 9z6 + a5z5−11a3z5−26az5−10z5a−1 + 4z5a−3−7a4z4−19a2z4−10z4a−2 + z4a−4−23z4−a5z3 + 5a3z3 + 12az3 + 4z3a−1−2z3a−3 + 3a4z2 + 10a2z2 + 4z2a−2 + 11z2−az−za−1−a2−a−2−1

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ −t3 + 6t2−16t + 23−16t−1 + 6t−2−t−3, q4−4q3 + 8q2−10q + 12−12q−1 + 10q−2−7q−3 + 4q−4−q−5 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n32, K11n119,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

n	Jn
2	q12−4q11 + 4q10 + 10q9−29q8 + 12q7 + 47q6−74q5 + 6q4 + 101q3−109q2−17q + 140−112q−1−41q−2 + 143q−3−83q−4−54q−5 + 109q−6−39q−7−48q−8 + 55q−9−6q−10−24q−11 + 14q−12 + 2q−13−4q−14 + q−15
3	q24−4q23 + 4q22 + 6q21−9q20−19q19 + 20q18 + 56q17−42q16−116q15 + 49q14 + 214q13−26q12−350q11−25q10 + 482q9 + 132q8−611q7−258q6 + 693q5 + 410q4−747q3−536q2 + 741q + 652−707q−1−728q−2 + 632q−3 + 778q−4−532q−5−793q−6 + 410q−7 + 771q−8−269q−9−714q−10 + 126q−11 + 623q−12−7q−13−492q−14−87q−15 + 354q−16 + 129q−17−218q−18−130q−19 + 113q−20 + 97q−21−40q−22−63q−23 + 12q−24 + 27q−25−9q−27−2q−28 + 4q−29−q−30
4	q40−4q39 + 4q38 + 6q37−13q36 + q35−11q34 + 39q33 + 37q32−90q31−49q30−48q29 + 221q28 + 257q27−272q26−385q25−384q24 + 633q23 + 1098q22−183q21−1115q20−1643q19 + 743q18 + 2693q17 + 949q16−1582q15−3886q14−277q13 + 4168q12 + 3112q11−926q10−6066q9−2301q8 + 4550q7 + 5225q6 + 689q5−7160q4−4285q3 + 3852q2 + 6391q + 2405−7085q−1−5517q−2 + 2637q−3 + 6547q−4 + 3731q−5−6164q−6−5993q−7 + 1158q−8 + 5920q−9 + 4680q−10−4533q−11−5807q−12−524q−13 + 4516q−14 + 5119q−15−2308q−16−4761q−17−2001q−18 + 2412q−19 + 4597q−20−136q−21−2852q−22−2485q−23 + 330q−24 + 3002q−25 + 993q−26−861q−27−1744q−28−721q−29 + 1195q−30 + 844q−31 + 189q−32−641q−33−622q−34 + 191q−35 + 277q−36 + 256q−37−76q−38−211q−39−15q−40 + 17q−41 + 74q−42 + 12q−43−33q−44−3q−45−5q−46 + 9q−47 + 2q−48−4q−49 + q−50
5	q60−4q59 + 4q58 + 6q57−13q56−3q55 + 9q54 + 8q53 + 20q52−q51−74q50−72q49 + 59q48 + 181q47 + 190q46−60q45−449q44−571q43−30q42 + 957q41 + 1364q40 + 462q39−1505q38−2883q37−1772q36 + 1892q35 + 5191q34 + 4347q33−1364q32−7900q31−8689q30−897q29 + 10381q28 + 14640q27 + 5444q26−11494q25−21368q24−12686q23 + 10324q22 + 27973q21 + 21796q20−6404q19−32886q18−31910q17−198q16 + 35624q15 + 41435q14 + 8486q13−35518q12−49461q11−17413q10 + 33241q9 + 55091q8 + 25783q7−29202q6−58452q5−32910q4 + 24528q3 + 59579q2 + 38437q−19500−59226q−1−42519q−2 + 14722q−3 + 57635q−4 + 45366q−5−9932q−6−55165q−7−47442q−8 + 5096q−9 + 51828q−10 + 48808q−11 + 119q−12−47403q−13−49515q−14−5832q−15 + 41709q−16 + 49272q−17 + 11825q−18−34528q−19−47595q−20−17694q−21 + 25954q−22 + 44084q−23 + 22696q−24−16564q−25−38434q−26−25897q−27 + 7098q−28 + 30858q−29 + 26727q−30 + 1204q−31−22142q−32−24730q−33−7381q−34 + 13309q−35 + 20490q−36 + 10678q−37−5684q−38−14834q−39−11161q−40 + 191q−41 + 9102q−42 + 9415q−43 + 2841q−44−4309q−45−6681q−46−3662q−47 + 1183q−48 + 3834q−49 + 3097q−50 + 451q−51−1768q−52−2043q−53−810q−54 + 531q−55 + 1028q−56 + 678q−57−7q−58−438q−59−373q−60−86q−61 + 126q−62 + 147q−63 + 79q−64−22q−65−67q−66−23q−67 + 9q−68 + 9q−69 + 8q−70 + 5q−71−9q−72−2q−73 + 4q−74−q−75
6	q84−4q83 + 4q82 + 6q81−13q80−3q79 + 5q78 + 28q77−11q76−18q75 + 15q74−87q73−16q72 + 93q71 + 214q70 + 60q69−172q68−226q67−596q66−201q65 + 616q64 + 1477q63 + 1095q62−260q61−1676q60−3767q59−2705q58 + 1083q57 + 6329q56 + 7879q55 + 4218q54−3277q53−14376q52−16750q51−7389q50 + 12518q49 + 28184q48 + 29057q47 + 10529q46−27268q45−53389q44−48973q43−4566q42 + 51710q41 + 87565q40 + 72590q39−6610q38−94578q37−136646q36−85215q35 + 30912q34 + 151409q33 + 191466q32 + 92051q31−82084q30−229420q29−229647q28−78266q27 + 154170q26 + 312752q25 + 257079q24 + 23502q23−255646q22−371665q21−251033q20 + 63988q19 + 363413q18 + 413536q17 + 186123q16−191750q15−441171q14−408894q13−75882q12 + 328123q11 + 497760q10 + 330085q9−83203q8−430541q7−498059q6−197997q5 + 250860q4 + 508808q3 + 414062q2 + 14014q−380135−523707q−1−273997q−2 + 175604q−3 + 482484q−4 + 448797q−5 + 83170q−6−323134q−7−517342q−8−318813q−9 + 109774q−10 + 442844q−11 + 462028q−12 + 141818q−13−259898q−14−495577q−15−356319q−16 + 34442q−17 + 384479q−18 + 463670q−19 + 209578q−20−170599q−21−447370q−22−389077q−23−65372q−24 + 286036q−25 + 435588q−26 + 280278q−27−44606q−28−348258q−29−389585q−30−173593q−31 + 140530q−32 + 348024q−33 + 315023q−34 + 92020q−35−193995q−36−321541q−37−239573q−38−16250q−39 + 199049q−40 + 271427q−41 + 178557q−42−29574q−43−186479q−44−217687q−45−115811q−46 + 42172q−47 + 156660q−48 + 170542q−49 + 71966q−50−45861q−51−123612q−52−118831q−53−48992q−54 + 38852q−55 + 94087q−56 + 80016q−57 + 30049q−58−29789q−59−61282q−60−55241q−61−18946q−62 + 22114q−63 + 38216q−64 + 33982q−65 + 11198q−66−11458q−67−24328q−68−20110q−69−5535q−70 + 5890q−71 + 13020q−72 + 10756q−73 + 4175q−74−3742q−75−6640q−76−4888q−77−2203q−78 + 1492q−79 + 2913q−80 + 2695q−81 + 623q−82−680q−83−1006q−84−1063q−85−320q−86 + 212q−87 + 552q−88 + 242q−89 + 59q−90−27q−91−163q−92−92q−93−26q−94 + 72q−95 + 16q−96 + 2q−97 + 15q−98−14q−99−8q−100−5q−101 + 9q−102 + 2q−103−4q−104 + q−105
7	q112−4q111 + 4q110 + 6q109−13q108−3q107 + 5q106 + 24q105 + 9q104−49q103−2q102 + 2q101−31q100 + 28q99 + 75q98 + 158q97 + 48q96−291q95−318q94−260q93−66q92 + 534q91 + 917q90 + 1208q89 + 482q88−1471q87−2844q86−3329q85−1711q84 + 2371q83 + 6470q82 + 9138q81 + 6836q80−2254q79−13194q78−21631q77−19779q76−3481q75 + 20489q74 + 43605q73 + 49176q72 + 25233q71−21814q70−75414q69−103193q68−77368q67 + 217q66 + 106234q65 + 185323q64 + 179387q63 + 71464q62−113375q61−286642q60−341928q59−223202q58 + 56051q57 + 375169q56 + 559280q55 + 480840q54 + 112560q53−400811q52−797484q51−843686q50−430400q49 + 298839q48 + 993820q47 + 1278907q46 + 910274q45−15765q44−1073367q43−1718343q42−1520174q41−470669q40 + 964613q39 + 2073346q38 + 2192067q37 + 1139954q36−633602q35−2263880q34−2829249q33−1922465q32 + 89307q31 + 2235800q30 + 3341119q29 + 2725638q28 + 609765q27−1986923q26−3661296q25−3449927q24−1377286q23 + 1555703q22 + 3768099q21 + 4023424q20 + 2119469q19−1015128q18−3681754q17−4408420q16−2762212q15 + 444233q14 + 3454987q13 + 4609291q12 + 3261430q11 + 87493q10−3149968q9−4658362q8−3610169q7−537899q6 + 2826220q5 + 4604709q4 + 3825894q3 + 890834q2−2522244q−4494135−3945409q−1−1157376q−2 + 2257688q−3 + 4364984q−4 + 4006145q−5 + 1360966q−6−2029292q−7−4235883q−8−4042274q−9−1535296q−10 + 1820136q−11 + 4112104q−12 + 4076227q−13 + 1709872q−14−1604177q−15−3981713q−16−4116003q−17−1909743q−18 + 1351280q−19 + 3823412q−20 + 4156866q−21 + 2147420q−22−1036621q−23−3607912q−24−4177817q−25−2420560q−26 + 641548q−27 + 3304202q−28 + 4148097q−29 + 2710187q−30−164342q−31−2887912q−32−4027779q−33−2978499q−34−377124q−35 + 2346613q−36 + 3778094q−37 + 3175544q−38 + 942327q−39−1691930q−40−3373205q−41−3244729q−42−1469069q−43 + 962632q−44 + 2808633q−45 + 3139347q−46 + 1886766q−47−224497q−48−2115194q−49−2838187q−50−2127516q−51−438709q−52 + 1355500q−53 + 2354958q−54 + 2150543q−55 + 946552q−56−617661q−57−1746377q−58−1954553q−59−1238823q−60−6866q−61 + 1097392q−62 + 1583164q−63 + 1299853q−64 + 447271q−65−504567q−66−1116612q−67−1160225q−68−672613q−69 + 46196q−70 + 648950q−71 + 889664q−72 + 700205q−73 + 237065q−74−260227q−75−574945q−76−587886q−77−349646q−78−1773q−79 + 292569q−80 + 407982q−81 + 333928q−82 + 133949q−83−89221q−84−229710q−85−249493q−86−162516q−87−23369q−88 + 93644q−89 + 149147q−90 + 132781q−91 + 63605q−92−14057q−93−69180q−94−83861q−95−59274q−96−18707q−97 + 19848q−98 + 41277q−99 + 39464q−100 + 23395q−101 + 1684q−102−15056q−103−19946q−104−16215q−105−7040q−106 + 2517q−107 + 7576q−108 + 8655q−109 + 5620q−110 + 985q−111−1979q−112−3381q−113−2854q−114−1311q−115−33q−116 + 1072q−117 + 1289q−118 + 675q−119 + 178q−120−253q−121−344q−122−232q−123−197q−124 + 20q−125 + 141q−126 + 92q−127 + 39q−128−24q−129−21q−130 + 5q−131−26q−132−10q−133 + 14q−134 + 8q−135 + 5q−136−9q−137−2q−138 + 4q−139−q−140