10 102

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["10 102"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,10,17,9 X10,3,11,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X4,11,5,12 X8,18,9,17 X20,14,1,13 X12,20,13,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 16 4 20 2 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3:2:20]

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,−1,2,−1,−3,2,−1,2,2,3,3})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_102.

A1 Invariants.

Weight	Invariant
1	q9−2q7 + 3q5−3q3 + 3q−q−3 + 2q−5−3q−7 + 3q−9−2q−11 + q−13
2	q26−2q24 + 5q20−7q18 + q16 + 13q14−17q12−3q10 + 25q8−17q6−13q4 + 24q2−2−15q−2 + 6q−4 + 13q−6−9q−8−12q−10 + 20q−12 + q−14−24q−16 + 16q−18 + 13q−20−24q−22 + 4q−24 + 17q−26−12q−28−5q−30 + 8q−32−q−34−2q−36 + q−38
3	q51−2q49 + 2q45 + q43−3q41−q39 + 6q37−5q35−10q33 + 14q31 + 24q29−23q27−53q25 + 28q23 + 91q21−13q19−129q17−24q15 + 148q13 + 70q11−140q9−107q7 + 99q5 + 134q3−46q−131q−1−6q−3 + 111q−5 + 57q−7−87q−9−90q−11 + 58q−13 + 118q−15−35q−17−136q−19 + 5q−21 + 148q−23 + 29q−25−150q−27−67q−29 + 135q−31 + 105q−33−98q−35−133q−37 + 47q−39 + 140q−41 + 4q−43−116q−45−47q−47 + 77q−49 + 65q−51−36q−53−56q−55 + 3q−57 + 36q−59 + 10q−61−17q−63−8q−65 + 5q−67 + 4q−69−q−71−2q−73 + q−75
4	q84−2q82 + 2q78−2q76 + 5q74−5q72−q70 + q68−11q66 + 21q64 + 5q62 + 5q60−18q58−67q56 + 25q54 + 70q52 + 104q50−17q48−250q46−139q44 + 118q42 + 426q40 + 261q38−418q36−636q34−223q32 + 708q30 + 945q28−70q26−1017q24−1010q22 + 343q20 + 1386q18 + 739q16−626q14−1428q12−481q10 + 947q8 + 1150q6 + 222q4−987q2−928 + 87q−2 + 845q−4 + 724q−6−234q−8−813q−10−493q−12 + 369q−14 + 818q−16 + 262q−18−620q−20−779q−22 + 102q−24 + 874q−26 + 588q−28−530q−30−1064q−32−171q−34 + 923q−36 + 1002q−38−243q−40−1256q−42−685q−44 + 591q−46 + 1318q−48 + 423q−50−934q−52−1115q−54−211q−56 + 1022q−58 + 978q−60−69q−62−868q−64−845q−66 + 174q−68 + 789q−70 + 566q−72−105q−74−684q−76−390q−78 + 129q−80 + 441q−82 + 324q−84−144q−86−277q−88−183q−90 + 58q−92 + 199q−94 + 72q−96−25q−98−93q−100−47q−102 + 32q−104 + 26q−106 + 19q−108−11q−110−14q−112 + 2q−114 + q−116 + 4q−118−q−120−2q−122 + q−124

A2 Invariants.

Weight	Invariant
1,0	q12−q10 + q8 + q6−2q4 + 3q2−1 + q−2−2q−6 + 2q−8−2q−10 + q−12 + q−14−q−16 + q−18
1,1	q36−4q34 + 10q32−20q30 + 36q28−58q26 + 90q24−128q22 + 171q20−218q18 + 278q16−348q14 + 407q12−458q10 + 496q8−480q6 + 399q4−242q2 + 24 + 242q−2−534q−4 + 798q−6−1008q−8 + 1140q−10−1165q−12 + 1098q−14−934q−16 + 698q−18−408q−20 + 96q−22 + 188q−24−420q−26 + 583q−28−656q−30 + 638q−32−556q−34 + 445q−36−324q−38 + 208q−40−122q−42 + 66q−44−30q−46 + 12q−48−4q−50 + q−52
2,0	q32−q30−q28 + 3q26−4q22 + 2q20 + 7q18−2q16−10q14 + 4q12 + 12q10−9q8−9q6 + 9q4 + 4q2−5−3q−2 + 8q−4−2q−6−4q−8 + 7q−10 + q−12−7q−14 + 4q−16 + 9q−18−8q−20−5q−22 + 6q−24 + 6q−26−7q−28−7q−30 + 8q−32 + 3q−34−5q−36−3q−38 + 2q−40 + 3q−42−q−44−q−46 + q−48

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	q34−q32−q30 + 3q28−4q24 + q22 + 7q20−9q16 + 3q14 + 13q12−7q10−12q8 + 14q6 + 7q4−12q2 + 1 + 15q−2−3q−4−12q−6 + 8q−8 + 5q−10−19q−12−3q−14 + 17q−16−11q−18−12q−20 + 17q−22 + 9q−24−12q−26−q−28 + 14q−30−10q−34 + 3q−36 + 6q−38−6q−40−2q−42 + 4q−44−q−46−q−48 + q−50
1,0,0,0	q18−q16 + 2q14 + 2q8−2q6 + 3q4−q2 + 1−q−6−q−8−2q−12 + 2q−14−2q−16 + 2q−18 + 2q−24−q−26 + q−28

B2 Invariants.

Weight	Invariant
0,1	q28−2q26 + 4q24−7q22 + 10q20−14q18 + 19q16−21q14 + 24q12−22q10 + 18q8−10q6 + 12q2−24 + 34q−2−42q−4 + 46q−6−46q−8 + 41q−10−32q−12 + 22q−14−9q−16−3q−18 + 13q−20−20q−22 + 23q−24−24q−26 + 23q−28−19q−30 + 14q−32−9q−34 + 5q−36−2q−38 + q−40
1,0	q46−2q42−2q40 + 2q38 + 6q36 + q34−8q32−8q30 + 5q28 + 16q26 + 4q24−16q22−16q20 + 9q18 + 24q16 + 3q14−23q12−13q10 + 17q8 + 20q6−9q4−20q2 + 3 + 20q−2 + 3q−4−17q−6−7q−8 + 13q−10 + 9q−12−12q−14−12q−16 + 10q−18 + 15q−20−7q−22−21q−24 + 24q−28 + 12q−30−20q−32−22q−34 + 11q−36 + 26q−38 + 3q−40−21q−42−12q−44 + 13q−46 + 14q−48−4q−50−11q−52−2q−54 + 6q−56 + 3q−58−2q−60−2q−62 + q−66

D4 Invariants.

Weight	Invariant
1,0,0,0	q38−2q36 + 2q34−3q32 + 6q30−8q28 + 8q26−10q24 + 16q22−16q20 + 15q18−17q16 + 21q14−15q12 + 11q10−9q8 + 5q6 + 7q4−10q2 + 17−24q−2 + 31q−4−32q−6 + 33q−8−39q−10 + 33q−12−32q−14 + 26q−16−24q−18 + 14q−20−7q−22 + 3q−24 + 5q−26−9q−28 + 18q−30−16q−32 + 19q−34−19q−36 + 20q−38−17q−40 + 13q−42−12q−44 + 8q−46−5q−48 + 3q−50−2q−52 + q−54

G2 Invariants.

Weight

Invariant

1,0

q66−2q64 + 4q62−6q60 + 5q58−3q56−2q54 + 11q52−18q50 + 26q48−28q46 + 19q44−4q42−19q40 + 42q38−59q36 + 68q34−61q32 + 33q30 + 15q28−64q26 + 110q24−123q22 + 100q20−42q18−40q16 + 108q14−134q12 + 108q10−29q8−57q6 + 110q4−105q2 + 39 + 57q−2−141q−4 + 164q−6−116q−8 + 14q−10 + 107q−12−193q−14 + 216q−16−165q−18 + 60q−20 + 55q−22−151q−24 + 192q−26−166q−28 + 88q−30 + 14q−32−100q−34 + 135q−36−108q−38 + 29q−40 + 61q−42−125q−44 + 126q−46−65q−48−34q−50 + 131q−52−171q−54 + 148q−56−68q−58−33q−60 + 111q−62−142q−64 + 125q−66−69q−68 + 6q−70 + 42q−72−63q−74 + 57q−76−36q−78 + 16q−80 + q−82−10q−84 + 10q−86−9q−88 + 5q−90−2q−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["10 102"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t3 + 8t2−16t + 21−16t−1 + 8t−2−2t−3

In[5]:=

Conway[K][z]

Out[5]=

−2z6−4z4−2z2 + 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]=

{ 73, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q6−3q5 + 6q4−9q3 + 11q2−12q + 12−9q−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 + a2z4−3z4a−2 + z4a−4−3z4 + 2a2z2−3z2a−2 + 2z2a−4−3z2 + a2−a−2 + a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−1 + 2z9a−3 + 9z8a−2 + 4z8a−4 + 5z8 + 6az7 + 4z7a−1 + z7a−3 + 3z7a−5 + 5a2z6−24z6a−2−11z6a−4 + z6a−6−7z6 + 3a3z5−9az5−17z5a−1−14z5a−3−9z5a−5 + a4z4−6a2z4 + 21z4a−2 + 8z4a−4−3z4a−6 + 3z4−3a3z3 + 7az3 + 16z3a−1 + 13z3a−3 + 7z3a−5−a4z2 + 3a2z2−8z2a−2−4z2a−4 + 2z2a−6 + 2z2−2az−4za−1−4za−3−2za−5−a2 + a−2 + a−4

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["10 102"];

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 + 8t2−16t + 21−16t−1 + 8t−2−2t−3, q6−3q5 + 6q4−9q3 + 11q2−12q + 12−9q−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]=

{}

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	q18−3q17 + q16 + 10q15−16q14−6q13 + 39q12−29q11−34q10 + 76q9−26q8−74q7 + 101q6−7q5−106q4 + 104q3 + 15q2−113q + 83 + 28q−1−87q−2 + 46q−3 + 24q−4−45q−5 + 18q−6 + 10q−7−15q−8 + 6q−9 + 2q−10−3q−11 + q−12
3	q36−3q35 + q34 + 5q33 + 2q32−16q31−8q30 + 32q29 + 28q28−49q27−67q26 + 52q25 + 129q24−37q23−191q22−17q21 + 249q20 + 99q19−284q18−197q17 + 284q16 + 302q15−254q14−399q13 + 201q12 + 481q11−135q10−542q9 + 60q8 + 582q7 + 18q6−602q5−88q4 + 585q3 + 162q2−548q−205 + 460q−1 + 247q−2−368q−3−240q−4 + 254q−5 + 214q−6−158q−7−162q−8 + 82q−9 + 109q−10−42q−11−58q−12 + 19q−13 + 28q−14−12q−15−11q−16 + 9q−17 + 4q−18−7q−19 + 2q−21 + 2q−22−3q−23 + q−24
4	q60−3q59 + q58 + 5q57−3q56 + 2q55−19q54 + 4q53 + 35q52 + 4q51 + 8q50−98q49−42q48 + 103q47 + 101q46 + 135q45−239q44−283q43 + 9q42 + 234q41 + 603q40−122q39−595q38−510q37−60q36 + 1182q35 + 549q34−372q33−1125q32−1079q31 + 1159q30 + 1348q29 + 675q28−1081q27−2312q26 + 255q25 + 1529q24 + 2032q23−186q22−3039q21−1021q20 + 958q19 + 3045q18 + 1059q17−3118q16−2115q15 + 65q14 + 3579q13 + 2177q12−2832q11−2887q10−816q9 + 3738q8 + 3059q7−2276q6−3336q5−1678q4 + 3418q3 + 3638q2−1318q−3215−2436q−1 + 2403q−2 + 3579q−3−109q−4−2287q−5−2639q−6 + 975q−7 + 2632q−8 + 693q−9−922q−10−1992q−11−68q−12 + 1279q−13 + 686q−14 + 25q−15−977q−16−305q−17 + 348q−18 + 273q−19 + 243q−20−298q−21−140q−22 + 40q−23 + 16q−24 + 132q−25−65q−26−19q−27 + 6q−28−29q−29 + 40q−30−16q−31 + 4q−32 + 6q−33−13q−34 + 8q−35−4q−36 + 2q−37 + 2q−38−3q−39 + q−40
5	q90−3q89 + q88 + 5q87−3q86−3q85−q84−7q83 + 6q82 + 30q81 + 8q80−30q79−45q78−50q77 + 16q76 + 128q75 + 157q74 + 17q73−192q72−335q71−235q70 + 179q69 + 605q68 + 637q67 + 82q66−758q65−1232q64−757q63 + 550q62 + 1791q61 + 1853q60 + 283q59−1920q58−3045q57−1897q56 + 1146q55 + 3922q54 + 3977q53 + 714q52−3730q51−5956q50−3646q49 + 2126q48 + 7099q47 + 6977q46 + 1015q45−6689q44−9976q43−5332q42 + 4510q41 + 11851q40 + 10028q39−659q38−12077q37−14368q36−4326q35 + 10632q34 + 17699q33 + 9687q32−7753q31−19725q30−14815q29 + 4003q28 + 20514q27 + 19230q26 + 44q25−20330q24−22779q23−3923q22 + 19550q21 + 25525q20 + 7395q19−18533q18−27651q17−10427q16 + 17472q15 + 29344q14 + 13141q13−16275q12−30781q11−15790q10 + 14891q9 + 31811q8 + 18420q7−12795q6−32257q5−21220q4 + 10011q3 + 31695q2 + 23621q−6124−29716q−1−25525q−2 + 1714q−3 + 26177q−4 + 25991q−5 + 2937q−6−21127q−7−24932q−8−6866q−9 + 15224q−10 + 22028q−11 + 9535q−12−9293q−13−17801q−14−10441q−15 + 4184q−16 + 12919q−17 + 9782q−18−549q−19−8329q−20−7902q−21−1488q−22 + 4561q−23 + 5657q−24 + 2170q−25−2047q−26−3534q−27−1961q−28 + 586q−29 + 1934q−30 + 1431q−31 + 42q−32−923q−33−875q−34−207q−35 + 370q−36 + 459q−37 + 187q−38−109q−39−225q−40−124q−41 + 41q−42 + 84q−43 + 50q−44 + 13q−45−32q−46−40q−47 + 8q−48 + 13q−49−3q−50 + 10q−51 + q−52−11q−53 + 2q−54 + 4q−55−4q−56 + 2q−57 + 2q−58−3q−59 + q−60
6	q126−3q125 + q124 + 5q123−3q122−3q121−6q120 + 11q119−5q118 + q117 + 33q116−11q115−30q114−59q113 + 12q112 + 7q111 + 50q110 + 184q109 + 62q108−79q107−318q106−221q105−224q104 + 66q103 + 714q102 + 764q101 + 506q100−475q99−950q98−1696q97−1349q96 + 541q95 + 2130q94 + 3217q93 + 2014q92 + 358q91−3476q90−5774q89−4408q88−654q87 + 4951q86 + 7925q85 + 9048q84 + 2364q83−6422q82−12597q81−13210q80−5607q79 + 5394q78 + 19052q77 + 20077q76 + 10828q75−6186q74−22897q73−28992q72−20834q71 + 6473q70 + 29024q69 + 40296q68 + 28989q67−1052q66−36130q65−56952q64−39274q63−1927q62 + 45337q61 + 70330q60 + 56932q59 + 3884q58−60852q57−87335q56−70063q55−2028q54 + 73356q53 + 113490q52 + 80747q51−9705q50−93783q49−133650q48−84650q47 + 21804q46 + 127695q45 + 151189q44 + 73937q43−48988q42−156602q41−160004q40−58160q39 + 95729q38 + 184571q37 + 150009q36 + 19514q35−139593q34−202709q33−130123q32 + 44450q31 + 184290q30 + 197880q29 + 80015q28−107233q27−217422q26−177844q25 + 117q24 + 171609q23 + 222886q22 + 120894q21−80040q20−221925q19−207773q18−30735q17 + 161460q16 + 240301q15 + 150885q14−59247q13−226308q12−234155q11−60229q10 + 149756q9 + 256463q8 + 184503q7−28946q6−221404q5−260122q4−103337q3 + 116818q2 + 257165q + 221314 + 25412q−1−184231q−2−265980q−3−154497q−4 + 49941q−5 + 216685q−6 + 235317q−7 + 91357q−8−105680q−9−224132q−10−180644q−11−30045q−12 + 131162q−13 + 198022q−14 + 129097q−15−15560q−16−137253q−17−153672q−18−78103q−19 + 38491q−20 + 118814q−21 + 113070q−22 + 38645q−23−49053q−24−89136q−25−73297q−26−15256q−27 + 43494q−28 + 64186q−29 + 42586q−30−315q−31−31353q−32−40384q−33−23473q−34 + 4683q−35 + 22655q−36 + 22982q−37 + 9688q−38−3983q−39−13618q−40−12778q−41−3840q−42 + 4221q−43 + 7427q−44 + 5249q−45 + 1824q−46−2636q−47−4221q−48−2246q−49 + 28q−50 + 1504q−51 + 1457q−52 + 1212q−53−185q−54−1038q−55−619q−56−191q−57 + 202q−58 + 215q−59 + 411q−60 + 51q−61−239q−62−102q−63−52q−64 + 27q−65−11q−66 + 110q−67 + 22q−68−60q−69−4q−70−8q−71 + 11q−72−19q−73 + 23q−74 + 7q−75−16q−76 + 4q−77−2q−78 + 4q−79−4q−80 + 2q−81 + 2q−82−3q−83 + q−84