10 92

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["10 92"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X18,7,19,8 X12,18,13,17 X6,19,7,20 X8,14,9,13 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4

In[6]:=

DTCode[K]

Out[6]=

4 10 14 18 2 16 8 20 12 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.21.2.20]

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,{1,1,1,2,2,−3,2,−1,2,−3,2})

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, 11, 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[{3, 11}, {2, 6}, {1, 3}, {12, 8}, {10, 7}, {8, 5}, {6, 4}, {11, 9}, {5, 10}, {9, 2}, {4, 12}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_92.

A1 Invariants.

Weight	Invariant
1	q−2q−1 + 4q−3−3q−5 + 4q−7−q−9−q−11 + 2q−13−4q−15 + 4q−17−3q−19 + q−21
2	q6−2q4−q2 + 9−6q−2−13q−4 + 24q−6 + q−8−34q−10 + 27q−12 + 21q−14−41q−16 + 11q−18 + 30q−20−26q−22−11q−24 + 22q−26 + 3q−28−25q−30 + 2q−32 + 33q−34−25q−36−21q−38 + 43q−40−12q−42−28q−44 + 28q−46 + q−48−15q−50 + 8q−52 + q−54−3q−56 + q−58
3	q15−2q13−q11 + 4q9 + 6q7−9q5−19q3 + 13q + 44q−1−3q−3−77q−5−33q−7 + 111q−9 + 92q−11−114q−13−172q−15 + 85q−17 + 247q−19−14q−21−291q−23−75q−25 + 293q−27 + 168q−29−258q−31−233q−33 + 192q−35 + 269q−37−116q−39−282q−41 + 44q−43 + 259q−45 + 34q−47−229q−49−106q−51 + 176q−53 + 181q−55−110q−57−243q−59 + 20q−61 + 288q−63 + 77q−65−295q−67−168q−69 + 262q−71 + 233q−73−192q−75−251q−77 + 108q−79 + 229q−81−42q−83−174q−85−q−87 + 109q−89 + 19q−91−60q−93−16q−95 + 30q−97 + 5q−99−10q−101−3q−103 + 5q−105 + q−107−3q−109 + q−111
4	q28−2q26−q24 + 4q22 + q20 + 3q18−15q16−13q14 + 21q12 + 30q10 + 38q8−61q6−117q4−19q2 + 116 + 269q−2 + 28q−4−330q−6−394q−8−75q−10 + 664q−12 + 656q−14−103q−16−949q−18−1039q−20 + 406q−22 + 1467q−24 + 1122q−26−627q−28−2170q−30−1028q−32 + 1205q−34 + 2480q−36 + 964q−38−2076q−40−2550q−42−370q−44 + 2572q−46 + 2568q−48−683q−50−2837q−52−1928q−54 + 1465q−56 + 2993q−58 + 748q−60−2054q−62−2479q−64 + 263q−66 + 2457q−68 + 1487q−70−1081q−72−2308q−74−596q−76 + 1663q−78 + 1884q−80−133q−82−1952q−84−1480q−86 + 643q−88 + 2233q−90 + 1164q−92−1199q−94−2451q−96−951q−98 + 2029q−100 + 2574q−102 + 342q−104−2637q−106−2630q−108 + 727q−110 + 2948q−112 + 2022q−114−1453q−116−3072q−118−863q−120 + 1790q−122 + 2450q−124 + 104q−126−1962q−128−1354q−130 + 315q−132 + 1520q−134 + 670q−136−631q−138−778q−140−264q−142 + 507q−144 + 394q−146−67q−148−204q−150−174q−152 + 96q−154 + 98q−156−q−158−15q−160−44q−162 + 17q−164 + 13q−166−6q−168 + 2q−170−6q−172 + 5q−174 + q−176−3q−178 + q−180
5	q45−2q43−q41 + 4q39 + q37−2q35−3q33−9q31−5q29 + 24q27 + 36q25 + 8q23−40q21−97q19−89q17 + 41q15 + 232q13 + 284q11 + 74q9−338q7−670q5−512q3 + 244q + 1159q−1 + 1393q−3 + 420q−5−1360q−7−2638q−9−2020q−11 + 671q−13 + 3748q−15 + 4487q−17 + 1516q−19−3666q−21−7192q−23−5449q−25 + 1493q−27 + 8853q−29 + 10312q−31 + 3344q−33−7926q−35−14714q−37−10269q−39 + 3624q−41 + 16694q−43 + 17663q−45 + 3934q−47−14899q−49−23458q−51−13256q−53 + 9106q−55 + 25834q−57 + 22167q−59−386q−61−23990q−63−28639q−65−9350q−67 + 18542q−69 + 31428q−71 + 17884q−73−10903q−75−30419q−77−23837q−79 + 2987q−81 + 26597q−83 + 26569q−85 + 3629q−87−21302q−89−26442q−91−8267q−93 + 15967q−95 + 24539q−97 + 10786q−99−11497q−101−21849q−103−11951q−105 + 8041q−107 + 19517q−109 + 12632q−111−5369q−113−17777q−115−13776q−117 + 2565q−119 + 16606q−121 + 15936q−123 + 1127q−125−15198q−127−19017q−129−6419q−131 + 12642q−133 + 22214q−135 + 13326q−137−7954q−139−24321q−141−21034q−143 + 851q−145 + 23869q−147 + 28018q−149 + 8190q−151−20034q−153−32512q−155−17554q−157 + 12908q−159 + 32986q−161 + 25248q−163−3741q−165−29118q−167−29445q−169−5295q−171 + 21820q−173 + 29170q−175 + 12189q−177−13005q−179−25026q−181−15577q−183 + 4974q−185 + 18497q−187 + 15350q−189 + 806q−191−11613q−193−12610q−195−3736q−197 + 5992q−199 + 8795q−201 + 4272q−203−2239q−205−5256q−207−3493q−209 + 328q−211 + 2711q−213 + 2263q−215 + 317q−217−1146q−219−1230q−221−399q−223 + 416q−225 + 592q−227 + 226q−229−119q−231−217q−233−121q−235 + 23q−237 + 83q−239 + 48q−241−15q−243−24q−245−5q−247 + 2q−251 + 6q−253−q−255−6q−257 + 5q−259 + q−261−3q−263 + q−265

A2 Invariants.

Weight	Invariant
1,0	1−q−2 + q−4 + 2q−6−2q−8 + 4q−10−q−12 + q−14 + q−16−3q−18 + 2q−20−3q−22 + q−24 + q−26−2q−28 + q−30
1,1	q4−4q2 + 14−36q−2 + 82q−4−162q−6 + 298q−8−484q−10 + 726q−12−994q−14 + 1244q−16−1420q−18 + 1479q−20−1354q−22 + 1034q−24−528q−26−104q−28 + 814q−30−1524q−32 + 2132q−34−2575q−36 + 2796q−38−2780q−40 + 2510q−42−2033q−44 + 1402q−46−700q−48 + 16q−50 + 570q−52−998q−54 + 1256q−56−1340q−58 + 1276q−60−1116q−62 + 916q−64−706q−66 + 506q−68−344q−70 + 226q−72−136q−74 + 73q−76−38q−78 + 18q−80−6q−82 + q−84
2,0	q4−q2−2 + 3q−2 + 5q−4−2q−6−8q−8 + 4q−10 + 14q−12−7q−14−14q−16 + 10q−18 + 15q−20−10q−22−9q−24 + 14q−26 + 7q−28−11q−30 + q−32 + 10q−34−10q−36−6q−38 + 10q−40−8q−42−14q−44 + 8q−46 + 14q−48−12q−50−10q−52 + 16q−54 + 8q−56−13q−58−5q−60 + 11q−62 + 2q−64−5q−66−2q−68 + 3q−70−2q−74 + q−76

A3 Invariants.

Weight	Invariant
0,1,0	1−2q−2 + 2q−4 + 4q−6−9q−8 + 8q−10 + 9q−12−21q−14 + 18q−16 + 13q−18−28q−20 + 20q−22 + 14q−24−29q−26 + 7q−28 + 11q−30−14q−32−6q−34 + 5q−36 + 11q−38−13q−40−8q−42 + 29q−44−16q−46−17q−48 + 33q−50−13q−52−17q−54 + 22q−56−5q−58−10q−60 + 9q−62−3q−66 + q−68
1,0,0	q−1−q−3 + 2q−5−q−7 + 3q−9−2q−11 + 4q−13−q−15 + 2q−17−3q−25 + 2q−27−3q−29 + 2q−31−2q−33 + 2q−35−2q−37 + q−39

A4 Invariants.

Weight	Invariant
0,1,0,0	q−2−q−4 + 4q−8−q−10−4q−12 + 6q−14 + 6q−16−8q−18−2q−20 + 18q−22 + 4q−24−15q−26 + 11q−28 + 26q−30−17q−32−19q−34 + 20q−36 + 3q−38−32q−40−3q−42 + 20q−44−14q−46−15q−48 + 23q−50 + 9q−52−24q−54 + 9q−56 + 23q−58−17q−60−15q−62 + 20q−64 + 7q−66−20q−68−2q−70 + 16q−72−2q−74−12q−76 + 4q−78 + 7q−80−3q−82−2q−84 + q−86
1,0,0,0	q−2−q−4 + 2q−6 + 3q−12−2q−14 + 4q−16−q−18 + 2q−20 + q−22−q−28−3q−32 + 2q−34−3q−36 + 2q−38−q−40−q−42 + 2q−44−2q−46 + q−48

B2 Invariants.

Weight	Invariant
0,1	1−2q−2 + 6q−4−10q−6 + 17q−8−24q−10 + 31q−12−35q−14 + 38q−16−33q−18 + 26q−20−12q−22−4q−24 + 23q−26−41q−28 + 57q−30−68q−32 + 72q−34−69q−36 + 59q−38−45q−40 + 26q−42−7q−44−10q−46 + 23q−48−33q−50 + 37q−52−37q−54 + 32q−56−25q−58 + 18q−60−11q−62 + 6q−64−3q−66 + q−68
1,0	q2−2q−2−2q−4 + 4q−6 + 7q−8−2q−10−13q−12−5q−14 + 18q−16 + 19q−18−13q−20−30q−22−q−24 + 38q−26 + 21q−28−28q−30−34q−32 + 13q−34 + 40q−36 + 6q−38−34q−40−17q−42 + 22q−44 + 20q−46−16q−48−23q−50 + 9q−52 + 23q−54−6q−56−27q−58 + 28q−62 + 9q−64−28q−66−19q−68 + 25q−70 + 30q−72−15q−74−38q−76−q−78 + 38q−80 + 19q−82−25q−84−30q−86 + 7q−88 + 27q−90 + 8q−92−15q−94−14q−96 + 3q−98 + 10q−100 + 3q−102−3q−104−3q−106 + q−110

D4 Invariants.

Weight	Invariant
1,0,0,0	q−2−2q−4 + 4q−6−5q−8 + 10q−10−13q−12 + 18q−14−20q−16 + 27q−18−28q−20 + 30q−22−26q−24 + 30q−26−19q−28 + 14q−30−3q−32−2q−34 + 14q−36−31q−38 + 33q−40−44q−42 + 48q−44−59q−46 + 53q−48−52q−50 + 52q−52−42q−54 + 31q−56−23q−58 + 17q−60−10q−64 + 13q−66−21q−68 + 30q−70−29q−72 + 25q−74−29q−76 + 27q−78−18q−80 + 14q−82−14q−84 + 10q−86−4q−88 + 3q−90−3q−92 + q−94

G2 Invariants.

Weight

Invariant

1,0

q−2−2q−4 + 6q−6−10q−8 + 13q−10−12q−12 + 3q−14 + 19q−16−45q−18 + 75q−20−88q−22 + 65q−24−7q−26−85q−28 + 181q−30−229q−32 + 207q−34−97q−36−66q−38 + 227q−40−319q−42 + 300q−44−165q−46−29q−48 + 202q−50−276q−52 + 226q−54−72q−56−100q−58 + 223q−60−234q−62 + 120q−64 + 62q−66−250q−68 + 358q−70−329q−72 + 175q−74 + 56q−76−283q−78 + 422q−80−428q−82 + 287q−84−63q−86−176q−88 + 333q−90−352q−92 + 236q−94−38q−96−143q−98 + 230q−100−197q−102 + 55q−104 + 115q−106−235q−108 + 256q−110−158q−112−6q−114 + 173q−116−275q−118 + 278q−120−193q−122 + 61q−124 + 66q−126−157q−128 + 184q−130−156q−132 + 99q−134−28q−136−26q−138 + 54q−140−65q−142 + 54q−144−34q−146 + 16q−148 + q−150−8q−152 + 10q−154−10q−156 + 6q−158−3q−160 + q−162

.

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["10 92"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 10t2−20t + 25−20t−1 + 10t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−2z4 + 2z2 + 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]=

{ 89, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q10−4q9 + 8q8−12q7 + 14q6−15q5 + 14q4−10q3 + 7q2−3q + 1

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6a−4−z6a−6 + z4a−2−2z4a−4−2z4a−6 + z4a−8 + 2z2a−2−z2a−6 + z2a−8 + a−2 + a−4−a−6

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−5 + 2z9a−7 + 4z8a−4 + 11z8a−6 + 7z8a−8 + 3z7a−3 + 5z7a−5 + 12z7a−7 + 10z7a−9 + z6a−2−8z6a−4−22z6a−6−5z6a−8 + 8z6a−10−8z5a−3−22z5a−5−32z5a−7−14z5a−9 + 4z5a−11−3z4a−2 + 2z4a−4 + 10z4a−6−4z4a−8−8z4a−10 + z4a−12 + 6z3a−3 + 18z3a−5 + 21z3a−7 + 7z3a−9−2z3a−11 + 3z2a−2 + z2a−4−2z2a−6 + 2z2a−8 + 2z2a−10−za−3−5za−5−5za−7−za−9−a−2 + a−4 + a−6

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["10 92"];

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]=

{ −2t3 + 10t2−20t + 25−20t−1 + 10t−2−2t−3, q10−4q9 + 8q8−12q7 + 14q6−15q5 + 14q4−10q3 + 7q2−3q + 1 }

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]=

{K11a153, K11a224, K11n35, K11n43,}

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	q28−4q27 + 4q26 + 8q25−27q24 + 20q23 + 35q22−83q21 + 36q20 + 90q19−147q18 + 32q17 + 148q16−178q15 + 5q14 + 176q13−159q12−28q11 + 161q10−103q9−47q8 + 109q7−41q6−41q5 + 48q4−6q3−18q2 + 11q + 1−3q−1 + q−2
3	q54−4q53 + 4q52 + 4q51−7q50−11q49 + 19q48 + 29q47−53q46−55q45 + 98q44 + 119q43−163q42−228q41 + 230q40 + 390q39−284q38−587q37 + 289q36 + 815q35−255q34−1017q33 + 162q32 + 1187q31−44q30−1285q29−101q28 + 1320q27 + 247q26−1290q25−383q24 + 1197q23 + 510q22−1065q21−598q20 + 871q19 + 676q18−680q17−675q16 + 446q15 + 651q14−254q13−550q12 + 78q11 + 435q10 + 23q9−289q8−84q7 + 178q6 + 81q5−83q4−65q3 + 34q2 + 37q−9−18q−1 + 3q−2 + 5q−3 + q−4−3q−5 + q−6
4	q88−4q87 + 4q86 + 4q85−11q84 + 9q83−12q82 + 23q81 + 8q80−72q79 + 38q78 + 2q77 + 122q76 + 6q75−342q74 + 8q73 + 139q72 + 583q71 + 119q70−1113q69−506q68 + 286q67 + 1884q66 + 969q65−2318q64−2175q63−322q62 + 3950q61 + 3315q60−2978q59−4828q58−2531q57 + 5569q56 + 6790q55−2052q54−7049q53−5888q52 + 5562q51 + 9769q50 + 180q49−7594q48−8868q47 + 4062q46 + 11021q45 + 2543q44−6525q43−10458q42 + 1939q41 + 10549q40 + 4362q39−4508q38−10679q37−320q36 + 8837q35 + 5589q34−1940q33−9709q32−2514q31 + 6095q30 + 6014q29 + 862q28−7464q27−4042q26 + 2702q25 + 5105q24 + 3016q23−4213q22−4038q21−240q20 + 2925q19 + 3490q18−1173q17−2522q16−1515q15 + 692q14 + 2348q13 + 370q12−773q11−1170q10−369q9 + 903q8 + 460q7 + 73q6−411q5−361q4 + 164q3 + 141q2 + 137q−53−120q−1 + 11q−2 + 6q−3 + 39q−4 + 3q−5−21q−6 + 3q−7−3q−8 + 5q−9 + q−10−3q−11 + q−12