10 87

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 87"];

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 X16,7,17,8 X6,19,7,20 X8,12,9,11 X18,13,19,14 X12,17,13,18 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 2 8 18 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]=

[.22.20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_87.

A1 Invariants.

Weight	Invariant
1	q9−2q7 + 3q5−4q3 + 3q + 3q−5−3q−7 + 3q−9−3q−11 + q−13
2	q26−2q24 + 5q20−8q18 + 2q16 + 14q14−21q12 + 29q8−25q6−12q4 + 31q2−7−18q−2 + 12q−4 + 14q−6−14q−8−11q−10 + 25q−12−2q−14−28q−16 + 23q−18 + 12q−20−32q−22 + 9q−24 + 20q−26−18q−28−4q−30 + 12q−32−2q−34−3q−36 + q−38
3	q51−2q49 + 2q45 + q43−4q41 + 8q37−8q35−13q33 + 20q31 + 30q29−37q27−63q25 + 52q23 + 112q21−50q19−165q17 + 17q15 + 209q13 + 35q11−221q9−93q7 + 184q5 + 150q3−125q−170q−1 + 48q−3 + 172q−5 + 27q−7−150q−9−90q−11 + 123q−13 + 138q−15−93q−17−173q−19 + 56q−21 + 202q−23−14q−25−215q−27−41q−29 + 215q−31 + 94q−33−181q−35−151q−37 + 127q−39 + 178q−41−57q−43−172q−45−11q−47 + 137q−49 + 57q−51−85q−53−68q−55 + 34q−57 + 55q−59−34q−63−8q−65 + 11q−67 + 7q−69−2q−71−3q−73 + q−75
4	q84−2q82 + 2q78−2q76 + 5q74−6q72 + 3q68−13q66 + 19q64 + 4q62 + 12q60−15q58−83q56 + 22q54 + 88q52 + 132q50−34q48−331q46−150q44 + 224q42 + 560q40 + 211q38−706q36−781q34 + 17q32 + 1165q30 + 1047q28−620q26−1580q24−907q22 + 1141q20 + 1985q18 + 322q16−1569q14−1876q12 + 142q10 + 1940q8 + 1336q6−512q4−1853q2−942 + 881q−2 + 1504q−4 + 615q−6−987q−8−1323q−10−232q−12 + 1059q−14 + 1177q−16−147q−18−1272q−20−906q−22 + 660q−24 + 1430q−26 + 427q−28−1216q−30−1424q−32 + 304q−34 + 1636q−36 + 1063q−38−965q−40−1898q−42−392q−44 + 1450q−46 + 1759q−48−130q−50−1855q−52−1277q−54 + 510q−56 + 1858q−58 + 947q−60−910q−62−1516q−64−680q−66 + 983q−68 + 1295q−70 + 281q−72−763q−74−1064q−76−96q−78 + 660q−80 + 664q−82 + 112q−84−544q−86−408q−88−24q−90 + 293q−92 + 297q−94−35q−96−153q−98−144q−100−5q−102 + 102q−104 + 45q−106 + 6q−108−35q−110−24q−112 + 7q−114 + 6q−116 + 7q−118−2q−120−3q−122 + q−124
5	q125−2q123 + 2q119−2q117 + 2q115 + 3q113−6q111−5q109 + 4q107 + q105 + 11q103 + 18q101−11q99−41q97−41q95 + 6q93 + 84q91 + 130q89 + 42q87−186q85−334q83−161q81 + 310q79 + 708q77 + 531q75−384q73−1380q71−1309q69 + 243q67 + 2250q65 + 2730q63 + 523q61−3118q59−4910q57−2293q55 + 3477q53 + 7555q51 + 5356q49−2612q47−10011q45−9571q43 + 6q41 + 11339q39 + 14087q37 + 4365q35−10550q33−17711q31−9846q29 + 7368q27 + 19232q25 + 15034q23−2228q21−17901q19−18557q17−3730q15 + 13964q13 + 19550q11 + 8890q9−8366q7−17767q5−12339q3 + 2466q + 14108q−1 + 13564q−3 + 2461q−5−9526q−7−12990q−9−5920q−11 + 5290q−13 + 11443q−15 + 7873q−17−2014q−19−9787q−21−8885q−23−157q−25 + 8681q−27 + 9624q−29 + 1544q−31−8304q−33−10621q−35−2745q−37 + 8381q−39 + 12159q−41 + 4325q−43−8384q−45−14072q−47−6688q−49 + 7672q−51 + 15854q−53 + 9828q−55−5647q−57−16837q−59−13368q−61 + 2183q−63 + 16255q−65 + 16452q−67 + 2547q−69−13634q−71−18265q−73−7599q−75 + 9072q−77 + 17831q−79 + 11923q−81−3195q−83−15007q−85−14338q−87−2660q−89 + 10124q−91 + 14150q−93 + 7201q−95−4357q−97−11487q−99−9419q−101−818q−103 + 7240q−105 + 9053q−107 + 4251q−109−2750q−111−6767q−113−5465q−115−676q−117 + 3703q−119 + 4749q−121 + 2441q−123−981q−125−3064q−127−2673q−129−629q−131 + 1324q−133 + 1921q−135 + 1164q−137−116q−139−1001q−141−981q−143−347q−145 + 291q−147 + 535q−149 + 389q−151 + 47q−153−216q−155−226q−157−100q−159 + 31q−161 + 86q−163 + 73q−165 + 15q−167−29q−169−25q−171−9q−173 + 2q−175 + 6q−177 + 7q−179−2q−181−3q−183 + q−185

A2 Invariants.

Weight	Invariant
1,0	q12−q10 + q8 + q6−3q4 + 2q2−2 + q−2 + 2q−4 + 4q−8−2q−10−2q−16 + q−18
1,1	q36−4q34 + 10q32−20q30 + 38q28−64q26 + 100q24−148q22 + 207q20−278q18 + 362q16−464q14 + 572q12−668q10 + 742q8−762q6 + 697q4−520q2 + 238 + 136q−2−566q−4 + 996q−6−1372q−8 + 1646q−10−1774q−12 + 1752q−14−1566q−16 + 1260q−18−851q−20 + 388q−22 + 70q−24−474q−26 + 771q−28−952q−30 + 994q−32−922q−34 + 773q−36−586q−38 + 402q−40−244q−42 + 133q−44−62q−46 + 22q−48−6q−50 + q−52
2,0	q32−q30−q28 + 3q26−5q22 + 2q20 + 7q18−3q16−11q14 + 5q12 + 14q10−12q8−9q6 + 12q4 + 7q2−8−4q−2 + 11q−4−3q−6−6q−8 + 6q−10 + 3q−12−8q−14 + 7q−16 + 12q−18−8q−20−7q−22 + 6q−24 + 5q−26−12q−28−8q−30 + 11q−32 + 4q−34−7q−36−2q−38 + 5q−40 + 4q−42−3q−44−2q−46 + q−48

A3 Invariants.

Weight	Invariant
0,1,0	q28−2q26 + 5q22−7q20−3q18 + 15q16−9q14−10q12 + 25q10−8q8−17q6 + 23q4−7q2−16 + 10q−2 + 2q−4−4q−6−3q−8 + 13q−10 + 10q−12−17q−14 + 8q−16 + 14q−18−24q−20 + 2q−22 + 16q−24−19q−26 + 5q−28 + 10q−30−11q−32 + 4q−34 + 2q−36−3q−38 + q−40
1,0,0	q15−q13 + 2q11−q9 + 2q7−3q5 + 2q3−3q + q−5 + 3q−7 + q−9 + 4q−11−2q−13 + q−15−3q−17 + q−19−2q−21 + q−23

A4 Invariants.

Weight	Invariant
0,1,0,0	q34−q32−q30 + 3q28−5q24−q22 + 7q20 + q18−11q16 + 2q14 + 18q12−4q10−14q8 + 16q6 + 12q4−17q2−8 + 12q−2−7q−4−21q−6 + 8q−8 + 15q−10−14q−12 + 4q−14 + 31q−16−3q−18−16q−20 + 17q−22 + 9q−24−21q−26−10q−28 + 13q−30−14q−34 + 4q−36 + 10q−38−5q−40−3q−42 + 5q−44−q−46−2q−48 + q−50
1,0,0,0	q18−q16 + 2q14 + 2q8−3q6 + 2q4−3q2−1−q−2 + q−6 + 2q−8 + 4q−10 + q−12 + 4q−14−2q−16 + q−18−2q−20−2q−22 + q−24−2q−26 + q−28

B2 Invariants.

Weight	Invariant
0,1	q28−2q26 + 4q24−7q22 + 11q20−15q18 + 21q16−25q14 + 28q12−29q10 + 24q8−17q6 + 5q4 + 9q2−24 + 38q−2−48q−4 + 56q−6−57q−8 + 55q−10−44q−12 + 33q−14−16q−16 + 2q−18 + 12q−20−22q−22 + 28q−24−31q−26 + 29q−28−26q−30 + 19q−32−14q−34 + 8q−36−3q−38 + q−40
1,0	q46−2q42−2q40 + 2q38 + 6q36 + q34−9q32−9q30 + 5q28 + 18q26 + 6q24−18q22−19q20 + 9q18 + 30q16 + 6q14−27q12−19q10 + 18q8 + 26q6−9q4−28q2−2 + 24q−2 + 7q−4−20q−6−12q−8 + 16q−10 + 16q−12−10q−14−15q−16 + 12q−18 + 21q−20−4q−22−25q−24−3q−26 + 27q−28 + 16q−30−24q−32−30q−34 + 9q−36 + 32q−38 + 6q−40−27q−42−18q−44 + 16q−46 + 21q−48−4q−50−15q−52−4q−54 + 9q−56 + 5q−58−3q−60−3q−62 + q−66

D4 Invariants.

Weight	Invariant
1,0,0,0	q38−2q36 + 2q34−3q32 + 6q30−9q28 + 8q26−11q24 + 18q22−18q20 + 18q18−20q16 + 27q14−19q12 + 16q10−15q8 + 8q6 + q4−11q2 + 13−29q−2 + 33q−4−37q−6 + 42q−8−44q−10 + 48q−12−36q−14 + 41q−16−29q−18 + 24q−20−13q−22 + 6q−24−q−26−12q−28 + 16q−30−21q−32 + 21q−34−25q−36 + 26q−38−22q−40 + 19q−42−16q−44 + 12q−46−8q−48 + 5q−50−3q−52 + q−54

G2 Invariants.

Weight

Invariant

1,0

q66−2q64 + 4q62−6q60 + 5q58−4q56−2q54 + 11q52−19q50 + 28q48−31q46 + 25q44−8q42−16q40 + 46q38−69q36 + 84q34−82q32 + 51q30 + 2q28−68q26 + 135q24−165q22 + 149q20−83q18−21q16 + 122q14−185q12 + 174q10−92q8−27q6 + 124q4−159q2 + 104 + 16q−2−140q−4 + 207q−6−188q−8 + 76q−10 + 86q−12−229q−14 + 300q−16−263q−18 + 141q−20 + 32q−22−184q−24 + 272q−26−261q−28 + 172q−30−32q−32−105q−34 + 188q−36−179q−38 + 96q−40 + 31q−42−140q−44 + 177q−46−128q−48 + 5q−50 + 128q−52−219q−54 + 225q−56−142q−58 + 4q−60 + 126q−62−201q−64 + 202q−66−135q−68 + 37q−70 + 46q−72−96q−74 + 99q−76−70q−78 + 35q−80−18q−84 + 20q−86−16q−88 + 8q−90−3q−92 + q−94

.

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 87"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 9t2−18t + 23−18t−1 + 9t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−3z4 + 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]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q6−4q5 + 7q4−10q3 + 13q2−13q + 13−10q−1 + 6q−2−3q−3 + q−4

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6a−2−z6 + a2z4−2z4a−2 + z4a−4−3z4 + 2a2z2 + z2a−2 + z2a−4−4z2 + a2 + 3a−2−a−4−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−1 + 2z9a−3 + 10z8a−2 + 5z8a−4 + 5z8 + 6az7 + 7z7a−1 + 5z7a−3 + 4z7a−5 + 5a2z6−21z6a−2−12z6a−4 + z6a−6−3z6 + 3a3z5−6az5−21z5a−1−23z5a−3−11z5a−5 + a4z4−5a2z4 + 8z4a−2 + 5z4a−4−2z4a−6−5z4−3a3z3 + 2az3 + 13z3a−1 + 15z3a−3 + 7z3a−5−a4z2 + 3a2z2 + 3z2a−2 + z2a−4 + z2a−6 + 7z2 + a3z + az−za−1−za−3−a2−3a−2−a−4−2

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 87"];

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 + 9t2−18t + 23−18t−1 + 9t−2−2t−3, q6−4q5 + 7q4−10q3 + 13q2−13q + 13−10q−1 + 6q−2−3q−3 + q−4 }

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]=

{10_98, K11a58, K11a165, K11n72,}

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	q18−4q17 + q16 + 15q15−20q14−13q13 + 53q12−31q11−54q10 + 97q9−20q8−105q7 + 123q6 + 7q5−141q4 + 120q3 + 35q2−143q + 90 + 46q−1−105q−2 + 47q−3 + 33q−4−51q−5 + 18q−6 + 12q−7−16q−8 + 6q−9 + 2q−10−3q−11 + q−12
3	q36−4q35 + q34 + 9q33 + 5q32−23q31−25q30 + 43q29 + 60q28−44q27−127q26 + 26q25 + 202q24 + 36q23−275q22−135q21 + 317q20 + 271q19−326q18−413q17 + 287q16 + 546q15−205q14−669q13 + 113q12 + 747q11 + 11q10−815q9−116q8 + 827q7 + 242q6−830q5−329q4 + 767q3 + 419q2−685q−453 + 549q−1 + 464q−2−410q−3−419q−4 + 272q−5 + 336q−6−154q−7−245q−8 + 80q−9 + 154q−10−39q−11−83q−12 + 20q−13 + 39q−14−13q−15−16q−16 + 10q−17 + 6q−18−8q−19 + 2q−21 + 2q−22−3q−23 + q−24
4	q60−4q59 + q58 + 9q57−q56 + 2q55−35q54−10q53 + 50q52 + 38q51 + 59q50−142q49−149q48 + 41q47 + 156q46 + 391q45−146q44−466q43−343q42 + 20q41 + 1047q40 + 406q39−470q38−1099q37−948q36 + 1348q35 + 1450q34 + 544q33−1411q32−2611q31 + 512q30 + 2056q29 + 2401q28−500q27−3959q26−1275q25 + 1478q24 + 4126q23 + 1389q22−4268q21−3117q20−28q19 + 5059q18 + 3417q17−3695q16−4449q15−1756q14 + 5267q13 + 5060q12−2692q11−5219q10−3322q9 + 4901q8 + 6185q7−1368q6−5337q5−4613q4 + 3810q3 + 6521q2 + 234q−4448−5236q−1 + 1987q−2 + 5610q−3 + 1575q−4−2600q−5−4632q−6 + 189q−7 + 3592q−8 + 1882q−9−709q−10−2969q−11−655q−12 + 1544q−13 + 1209q−14 + 251q−15−1302q−16−537q−17 + 396q−18 + 411q−19 + 326q−20−385q−21−188q−22 + 60q−23 + 37q−24 + 145q−25−88q−26−22q−27 + 16q−28−29q−29 + 40q−30−20q−31 + 5q−32 + 8q−33−14q−34 + 8q−35−4q−36 + 2q−37 + 2q−38−3q−39 + q−40
5	q90−4q89 + q88 + 9q87−q86−4q85−10q84−20q83−3q82 + 53q81 + 57q80 + 9q79−65q78−151q77−129q76 + 63q75 + 320q74 + 351q73 + 81q72−395q71−767q70−571q69 + 300q68 + 1236q67 + 1361q66 + 362q65−1364q64−2524q63−1744q62 + 845q61 + 3444q60 + 3784q59 + 944q58−3570q57−6123q56−3944q55 + 2142q54 + 7801q53 + 7945q52 + 1232q51−7936q50−12002q49−6459q48 + 5733q47 + 15075q46 + 12790q45−987q44−16028q43−19243q42−5945q41 + 14406q40 + 24602q39 + 14131q38−10120q37−28002q36−22616q35 + 3740q34 + 29233q33 + 30312q32 + 3785q31−28199q30−36688q29−11811q28 + 25764q27 + 41502q26 + 19260q25−22173q24−44870q23−26171q22 + 18380q21 + 47190q20 + 31969q19−14339q18−48648q17−37297q16 + 10504q15 + 49507q14 + 41817q13−6259q12−49591q11−46135q10 + 1776q9 + 48605q8 + 49590q7 + 3628q6−46021q5−52288q4−9434q3 + 41535q2 + 53054q + 15615−34918q−1−51744q−2−21076q−3 + 26730q−4 + 47626q−5 + 25016q−6−17662q−7−41084q−8−26662q−9 + 9036q−10 + 32799q−11 + 25672q−12−1989q−13−23822q−14−22464q−15−2828q−16 + 15585q−17 + 17807q−18 + 5172q−19−8907q−20−12742q−21−5576q−22 + 4252q−23 + 8230q−24 + 4732q−25−1508q−26−4774q−27−3377q−28 + 174q−29 + 2460q−30 + 2115q−31 + 284q−32−1133q−33−1170q−34−306q−35 + 453q−36 + 563q−37 + 213q−38−137q−39−255q−40−129q−41 + 55q−42 + 92q−43 + 40q−44 + 11q−45−27q−46−41q−47 + 9q−48 + 14q−49−7q−50 + 11q−51 + 3q−52−12q−53 + 2q−54 + 4q−55−4q−56 + 2q−57 + 2q−58−3q−59 + q−60