10 40

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 40"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,1,12,20 X5,13,6,12 X7,17,8,16 X15,19,16,18 X19,15,20,14 X13,7,14,6 X17,9,18,8 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 16 2 20 6 18 8 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[222112]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_40.

A1 Invariants.

Weight	Invariant
1	−q5 + 2q3−3q + 4q−1−q−3 + 2q−5 + q−7−3q−9 + 3q−11−3q−13 + 2q−15−q−17
2	q16−2q14−q12 + 7q10−7q8−8q6 + 20q4−7q2−21 + 28q−2 + 2q−4−26q−6 + 20q−8 + 11q−10−18q−12 + 12q−16−20q−20 + 9q−22 + 21q−24−27q−26−q−28 + 27q−30−20q−32−8q−34 + 19q−36−7q−38−6q−40 + 7q−42−q−44−2q−46 + q−48
3	−q33 + 2q31 + q29−3q27−4q25 + 7q23 + 11q21−14q19−23q17 + 17q15 + 44q13−15q11−74q9 + 3q7 + 105q5 + 20q3−126q−61q−1 + 143q−3 + 98q−5−131q−7−132q−9 + 112q−11 + 153q−13−71q−15−156q−17 + 32q−19 + 143q−21 + 9q−23−115q−25−55q−27 + 80q−29 + 87q−31−37q−33−124q−35−2q−37 + 141q−39 + 54q−41−151q−43−95q−45 + 141q−47 + 126q−49−114q−51−145q−53 + 78q−55 + 145q−57−40q−59−127q−61 + 8q−63 + 98q−65 + 12q−67−67q−69−18q−71 + 39q−73 + 17q−75−20q−77−12q−79 + 10q−81 + 6q−83−4q−85−3q−87 + q−89 + 2q−91−q−93
4	q56−2q54−q52 + 3q50 + 4q46−10q44−6q42 + 15q40 + 8q38 + 15q36−42q34−40q32 + 38q30 + 57q28 + 77q26−101q24−166q22−q20 + 157q18 + 296q16−86q14−399q12−244q10 + 172q8 + 676q6 + 190q4−542q2−701−119q−2 + 964q−4 + 712q−6−341q−8−1061q−10−648q−12 + 852q−14 + 1117q−16 + 152q−18−1013q−20−1053q−22 + 391q−24 + 1103q−26 + 585q−28−608q−30−1069q−32−119q−34 + 735q−36 + 764q−38−96q−40−788q−42−516q−44 + 247q−46 + 763q−48 + 394q−50−390q−52−834q−54−262q−56 + 665q−58 + 836q−60 + 82q−62−1017q−64−772q−66 + 381q−68 + 1117q−70 + 624q−72−891q−74−1112q−76−97q−78 + 1016q−80 + 1014q−82−427q−84−1042q−86−533q−88 + 545q−90 + 1003q−92 + 64q−94−605q−96−617q−98 + 57q−100 + 631q−102 + 250q−104−159q−106−389q−108−142q−110 + 244q−112 + 163q−114 + 34q−116−141q−118−106q−120 + 58q−122 + 48q−124 + 40q−126−31q−128−37q−130 + 13q−132 + 4q−134 + 13q−136−5q−138−9q−140 + 4q−142 + 3q−146−q−148−2q−150 + q−152
5	−q85 + 2q83 + q81−3q79−q73 + 5q71 + 5q69−11q67−11q65 + 3q63 + 14q61 + 26q59 + 11q57−37q55−73q53−31q51 + 72q49 + 144q47 + 107q45−80q43−283q41−283q39 + 42q37 + 465q35 + 579q33 + 155q31−620q29−1064q27−594q25 + 656q23 + 1665q21 + 1348q19−380q17−2253q15−2449q13−355q11 + 2611q9 + 3785q7 + 1599q5−2504q3−5039q−3344q−1 + 1729q−3 + 6007q−5 + 5293q−7−362q−9−6264q−11−7097q−13−1569q−15 + 5830q−17 + 8422q−19 + 3582q−21−4630q−23−8958q−25−5426q−27 + 2974q−29 + 8706q−31 + 6700q−33−1125q−35−7733q−37−7320q−39−580q−41 + 6274q−43 + 7284q−45 + 1973q−47−4621q−49−6754q−51−3012q−53 + 2965q−55 + 5952q−57 + 3734q−59−1413q−61−5048q−63−4329q−65−15q−67 + 4204q−69 + 4832q−71 + 1421q−73−3322q−75−5445q−77−2848q−79 + 2460q−81 + 5951q−83 + 4377q−85−1339q−87−6388q−89−5964q−91 + 31q−93 + 6427q−95 + 7401q−97 + 1632q−99−5976q−101−8513q−103−3423q−105 + 4919q−107 + 9000q−109 + 5127q−111−3279q−113−8721q−115−6463q−117 + 1319q−119 + 7658q−121 + 7130q−123 + 625q−125−5937q−127−7025q−129−2212q−131 + 3925q−133 + 6190q−135 + 3164q−137−1998q−139−4835q−141−3419q−143 + 452q−145 + 3317q−147 + 3096q−149 + 512q−151−1942q−153−2396q−155−934q−157 + 892q−159 + 1624q−161 + 955q−163−262q−165−950q−167−741q−169−50q−171 + 466q−173 + 490q−175 + 143q−177−199q−179−271q−181−118q−183 + 60q−185 + 125q−187 + 80q−189−9q−191−57q−193−37q−195 + 2q−197 + 17q−199 + 11q−201 + 6q−203−7q−205−9q−207 + 4q−209 + 4q−211−q−213−3q−219 + q−221 + 2q−223−q−225

A2 Invariants.

Weight	Invariant
1,0	−q6 + q4−q2−1 + 3q−2−q−4 + 4q−6 + q−8 + q−12−3q−14 + 2q−16−q−18−q−20 + q−22−q−24
1,1	q20−4q18 + 12q16−28q14 + 58q12−110q10 + 186q8−290q6 + 417q4−566q2 + 704−810q−2 + 856q−4−816q−6 + 678q−8−412q−10 + 73q−12 + 342q−14−758q−16 + 1158q−18−1472q−20 + 1666q−22−1722q−24 + 1616q−26−1389q−28 + 1042q−30−640q−32 + 214q−34 + 181q−36−490q−38 + 708q−40−822q−42 + 835q−44−770q−46 + 652q−48−518q−50 + 385q−52−264q−54 + 170q−56−102q−58 + 56q−60−28q−62 + 12q−64−4q−66 + q−68
2,0	q18−q16−2q14 + 3q12 + 3q10−5q8−6q6 + 6q4 + 6q2−12−6q−2 + 14q−4 + 4q−6−10q−8 + 4q−10 + 15q−12−q−14−4q−16 + 8q−18 + 3q−20−12q−22 + 2q−24 + 6q−26−11q−28−4q−30 + 12q−32 + 2q−34−14q−36−q−38 + 11q−40−2q−42−10q−44 + 3q−46 + 7q−48−2q−50−3q−52 + q−54 + 2q−56−q−58−q−60 + q−62

A3 Invariants.

Weight	Invariant
0,1,0	q14−2q12 + q10 + 4q8−9q6 + 2q4 + 10q2−19 + 3q−2 + 19q−4−21q−6 + 5q−8 + 24q−10−11q−12−2q−14 + 14q−16−2q−18−8q−20−3q−22 + 12q−24−5q−26−17q−28 + 20q−30−q−32−23q−34 + 18q−36 + 3q−38−17q−40 + 11q−42 + 3q−44−8q−46 + 4q−48 + q−50−2q−52 + q−54
1,0,0	−q7 + q5−2q3 + q−2q−1 + 3q−3−q−5 + 4q−7 + 2q−9 + 2q−11 + q−13−q−15 + q−17−3q−19 + 2q−21−2q−23 + q−25−2q−27 + q−29−q−31

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q14 + 3q10−q8−5q6 + 2q4 + 3q2−9−8q−2 + 8q−4 + 5q−6−15q−8 + 2q−10 + 23q−12 + 3q−14−9q−16 + 18q−18 + 15q−20−11q−22−4q−24 + 13q−26−5q−28−16q−30 + 10q−32 + 8q−34−18q−36−4q−38 + 16q−40−8q−42−17q−44 + 7q−46 + 10q−48−9q−50−7q−52 + 9q−54 + 4q−56−6q−58 + 4q−62−q−64−q−66 + q−68
1,0,0,0	−q8 + q6−2q4−2q−2 + 3q−4−q−6 + 4q−8 + 2q−10 + 3q−12 + 2q−14 + q−16−q−20 + q−22−3q−24 + 2q−26−2q−28−2q−34 + q−36−q−38

B2 Invariants.

Weight	Invariant
0,1	−q14 + 2q12−5q10 + 8q8−13q6 + 18q4−22q2 + 25−25q−2 + 23q−4−15q−6 + 7q−8 + 6q−10−17q−12 + 32q−14−40q−16 + 48q−18−48q−20 + 47q−22−40q−24 + 29q−26−17q−28 + 2q−30 + 7q−32−17q−34 + 22q−36−25q−38 + 25q−40−21q−42 + 17q−44−12q−46 + 8q−48−5q−50 + 2q−52−q−54
1,0	q24−2q20−2q18 + 3q16 + 6q14−q12−11q10−7q8 + 11q6 + 16q4−5q2−24−9q−2 + 22q−4 + 22q−6−12q−8−26q−10 + 27q−14 + 13q−16−15q−18−14q−20 + 13q−22 + 17q−24−5q−26−17q−28 + 2q−30 + 16q−32 + q−34−18q−36−6q−38 + 17q−40 + 10q−42−17q−44−18q−46 + 13q−48 + 24q−50−3q−52−28q−54−11q−56 + 21q−58 + 21q−60−9q−62−23q−64−4q−66 + 16q−68 + 11q−70−6q−72−10q−74−q−76 + 6q−78 + 3q−80−2q−82−2q−84 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−2q16 + 3q14−4q12 + 7q10−11q8 + 11q6−15q4 + 17q2−22 + 17q−2−19q−4 + 20q−6−13q−8 + 9q−10 + q−12 + 4q−14 + 16q−16−17q−18 + 25q−20−27q−22 + 37q−24−38q−26 + 34q−28−38q−30 + 36q−32−28q−34 + 21q−36−20q−38 + 9q−40−6q−44 + 6q−46−16q−48 + 19q−50−18q−52 + 18q−54−20q−56 + 18q−58−13q−60 + 11q−62−10q−64 + 7q−66−4q−68 + 3q−70−2q−72 + q−74

G2 Invariants.

Weight

Invariant

1,0

q32−2q30 + 5q28−8q26 + 8q24−7q22−2q20 + 16q18−33q16 + 47q14−51q12 + 33q10 + 2q8−50q6 + 101q4−129q2 + 121−72q−2−17q−4 + 104q−6−167q−8 + 180q−10−128q−12 + 42q−14 + 60q−16−129q−18 + 140q−20−83q−22−4q−24 + 84q−26−117q−28 + 87q−30 + 6q−32−105q−34 + 189q−36−204q−38 + 146q−40−22q−42−125q−44 + 230q−46−268q−48 + 222q−50−104q−52−35q−54 + 148q−56−201q−58 + 176q−60−92q−62−19q−64 + 94q−66−114q−68 + 68q−70 + 21q−72−98q−74 + 142q−76−123q−78 + 47q−80 + 49q−82−139q−84 + 179q−86−159q−88 + 92q−90−5q−92−71q−94 + 115q−96−120q−98 + 93q−100−47q−102−q−104 + 31q−106−47q−108 + 43q−110−30q−112 + 17q−114−2q−116−6q−118 + 8q−120−8q−122 + 5q−124−2q−126 + q−128

.

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 40"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−8t2 + 17t−21 + 17t−1−8t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 4z4 + 3z2 + 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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 3q7−6q6 + 9q5−12q4 + 13q3−11q2 + 10q−6 + 3q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + z6a−4 + 3z4a−2 + 3z4a−4−z4a−6−z4 + 4z2a−2 + 3z2a−4−2z2a−6−2z2 + 3a−2−a−6−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z9a−3 + z9a−5 + 3z8a−2 + 6z8a−4 + 3z8a−6 + 4z7a−1 + 7z7a−3 + 7z7a−5 + 4z7a−7 + z6a−2−5z6a−4 + 3z6a−8 + 3z6 + az5−5z5a−1−12z5a−3−13z5a−5−6z5a−7 + z5a−9−9z4a−2−2z4a−4−5z4a−6−6z4a−8−6z4−2az3−z3a−1 + 3z3a−3 + 6z3a−5 + 2z3a−7−2z3a−9 + 7z2a−2 + z2a−4 + z2a−6 + 3z2a−8 + 4z2 + az + 2za−1 + 2za−3 + za−9−3a−2 + a−6−1

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["10 40"];

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 + 17t−21 + 17t−1−8t−2 + 2t−3, −q8 + 3q7−6q6 + 9q5−12q4 + 13q3−11q2 + 10q−6 + 3q−1−q−2 }

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_103,}

In[6]:=

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

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

Out[6]=

{10_103,}

n	Jn
2	q23−3q22 + q21 + 9q20−16q19 + 35q17−43q16−12q15 + 82q14−71q13−38q12 + 130q11−83q10−67q9 + 150q8−71q7−79q6 + 132q5−42q4−70q3 + 86q2−14q−44 + 37q−1−17q−3 + 9q−4 + q−5−3q−6 + q−7
3	−q45 + 3q44−q43−4q42−2q41 + 13q40 + 3q39−26q38−10q37 + 50q36 + 25q35−83q34−59q33 + 129q32 + 111q31−173q30−194q29 + 216q28 + 296q27−240q26−417q25 + 247q24 + 536q23−225q22−653q21 + 191q20 + 741q19−138q18−796q17 + 69q16 + 828q15−14q14−803q13−66q12 + 768q11 + 110q10−669q9−177q8 + 580q7 + 195q6−445q5−218q4 + 336q3 + 196q2−216q−173 + 132q−1 + 131q−2−70q−3−88q−4 + 30q−5 + 54q−6−11q−7−29q−8 + 3q−9 + 14q−10−2q−11−4q−12−q−13 + 3q−14−q−15
4	q74−3q73 + q72 + 4q71−3q70 + 5q69−16q68 + 5q67 + 22q66−12q65 + 14q64−66q63 + 11q62 + 93q61−4q60 + 24q59−230q58−24q57 + 268q56 + 125q55 + 105q54−616q53−271q52 + 498q51 + 534q50 + 486q49−1190q48−945q47 + 510q46 + 1203q45 + 1425q44−1648q43−2023q42 + q41 + 1818q40 + 2866q39−1646q38−3136q37−1014q36 + 2039q35 + 4381q34−1153q33−3872q32−2167q31 + 1794q30 + 5480q29−399q28−4043q27−3094q26 + 1222q25 + 5924q24 + 385q23−3674q22−3610q21 + 459q20 + 5652q19 + 1077q18−2814q17−3639q16−395q15 + 4702q14 + 1538q13−1621q12−3121q11−1107q10 + 3258q9 + 1578q8−456q7−2156q6−1372q5 + 1758q4 + 1165q3 + 264q2−1103q−1120 + 675q−1 + 583q−2 + 423q−3−371q−4−634q−5 + 171q−6 + 167q−7 + 268q−8−58q−9−252q−10 + 32q−11 + 9q−12 + 103q−13 + 7q−14−74q−15 + 12q−16−10q−17 + 25q−18 + 5q−19−17q−20 + 5q−21−3q−22 + 4q−23 + q−24−3q−25 + q−26