10 77

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 77"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X13,17,14,16 X5,15,6,14 X15,7,16,6 X9,19,10,18 X11,1,12,20 X19,11,20,10 X17,13,18,12 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 18 20 16 6 12 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,2++]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_77.

A1 Invariants.

Weight	Invariant
1	−q5 + q3−2q + 4q−1−q−3 + 2q−5 + q−7−2q−9 + 2q−11−3q−13 + 2q−15−q−17
2	q16−q14−q12 + 3q10−4q8−3q6 + 10q4−7q2−9 + 20q−2−4q−4−16q−6 + 19q−8 + 4q−10−14q−12 + 6q−14 + 8q−16−4q−18−11q−20 + 9q−22 + 8q−24−20q−26 + 5q−28 + 16q−30−18q−32−2q−34 + 16q−36−8q−38−5q−40 + 7q−42−q−44−2q−46 + q−48
3	−q33 + q31 + q29−2q25 + 2q23 + 2q21−4q19−4q17 + 8q15 + 8q13−16q11−18q9 + 21q7 + 34q5−26q3−53q + 17q−1 + 83q−3−6q−5−93q−7−18q−9 + 103q−11 + 43q−13−93q−15−60q−17 + 75q−19 + 70q−21−46q−23−69q−25 + 16q−27 + 67q−29 + 11q−31−54q−33−47q−35 + 43q−37 + 66q−39−27q−41−92q−43 + 11q−45 + 102q−47 + 13q−49−104q−51−36q−53 + 95q−55 + 54q−57−73q−59−64q−61 + 46q−63 + 65q−65−20q−67−55q−69 + 3q−71 + 36q−73 + 8q−75−21q−77−9q−79 + 11q−81 + 5q−83−4q−85−3q−87 + q−89 + 2q−91−q−93
4	q56−q54−q52−q48 + 4q46−2q44 + 3q40−6q38 + 4q36−7q34 + 4q32 + 19q30−7q28−5q26−37q24−2q22 + 61q20 + 30q18−2q16−118q14−73q12 + 101q10 + 147q8 + 98q6−205q4−265q2 + 19 + 282q−2 + 365q−4−143q−6−484q−8−253q−10 + 248q−12 + 638q−14 + 125q−16−495q−18−533q−20 + q−22 + 669q−24 + 391q−26−266q−28−575q−30−247q−32 + 436q−34 + 454q−36 + 15q−38−400q−40−353q−42 + 127q−44 + 365q−46 + 226q−48−169q−50−366q−52−164q−54 + 248q−56 + 401q−58 + 54q−60−358q−62−433q−64 + 108q−66 + 535q−68 + 300q−70−271q−72−646q−74−116q−76 + 519q−78 + 515q−80−36q−82−661q−84−356q−86 + 274q−88 + 545q−90 + 251q−92−419q−94−421q−96−42q−98 + 329q−100 + 355q−102−101q−104−257q−106−184q−108 + 65q−110 + 233q−112 + 49q−114−56q−116−119q−118−42q−120 + 77q−122 + 36q−124 + 16q−126−35q−128−28q−130 + 17q−132 + 5q−134 + 10q−136−6q−138−8q−140 + 4q−142 + 3q−146−q−148−2q−150 + q−152
5	−q85 + q83 + q81 + q77−q75−4q73 + 2q69−q67 + 5q65 + 5q63−5q61−7q59−6q57−7q55 + 10q53 + 27q51 + 16q49−13q47−40q45−48q43−2q41 + 71q39 + 106q37 + 40q35−95q33−192q31−137q29 + 83q27 + 312q25 + 319q23 + q21−428q19−597q17−241q15 + 475q13 + 948q11 + 668q9−328q7−1310q5−1302q3−63q + 1503q−1 + 2026q−3 + 824q−5−1426q−7−2748q−9−1748q−11 + 953q−13 + 3162q−15 + 2820q−17−116q−19−3236q−21−3677q−23−899q−25 + 2809q−27 + 4208q−29 + 1925q−31−2061q−33−4257q−35−2715q−37 + 1139q−39 + 3861q−41 + 3146q−43−227q−45−3169q−47−3204q−49−520q−51 + 2345q−53 + 2963q−55 + 1055q−57−1535q−59−2576q−61−1397q−63 + 803q−65 + 2174q−67 + 1648q−69−213q−71−1814q−73−1871q−75−366q−77 + 1546q−79 + 2186q−81 + 883q−83−1300q−85−2505q−87−1536q−89 + 1021q−91 + 2888q−93 + 2206q−95−606q−97−3121q−99−2953q−101−7q−103 + 3176q−105 + 3615q−107 + 782q−109−2875q−111−4075q−113−1659q−115 + 2233q−117 + 4178q−119 + 2483q−121−1309q−123−3848q−125−3048q−127 + 245q−129 + 3108q−131 + 3230q−133 + 729q−135−2110q−137−2971q−139−1404q−141 + 1053q−143 + 2346q−145 + 1681q−147−151q−149−1574q−151−1564q−153−411q−155 + 811q−157 + 1185q−159 + 648q−161−239q−163−753q−165−594q−167−70q−169 + 357q−171 + 417q−173 + 184q−175−115q−177−241q−179−153q−181 + 5q−183 + 102q−185 + 92q−187 + 31q−189−35q−191−50q−193−16q−195 + 10q−197 + 15q−199 + 7q−201 + 2q−203−8q−205−6q−207 + 5q−209 + 3q−211−q−213−3q−219 + q−221 + 2q−223−q−225

A2 Invariants.

Weight	Invariant
1,0	−q6−q2−1 + 3q−2 + 4q−6 + 2q−8 + q−12−3q−14 + q−16−q−18−q−20 + q−22−q−24
1,1	q20−2q18 + 6q16−12q14 + 23q12−38q10 + 58q8−92q6 + 128q4−182q2 + 234−300q−2 + 352q−4−382q−6 + 392q−8−338q−10 + 257q−12−96q−14−70q−16 + 282q−18−482q−20 + 652q−22−784q−24 + 832q−26−831q−28 + 740q−30−598q−32 + 410q−34−198q−36−8q−38 + 188q−40−324q−42 + 406q−44−436q−46 + 416q−48−360q−50 + 291q−52−218q−54 + 148q−56−92q−58 + 54q−60−28q−62 + 12q−64−4q−66 + q−68
2,0	q18−q14 + q12 + 2q10−3q8−4q6 + 2q4 + 2q2−8−5q−2 + 9q−4 + q−6−7q−8 + 4q−10 + 12q−12 + 3q−14 + q−16 + 11q−18 + 7q−20−7q−22 + q−26−10q−28−7q−30 + 5q−32−q−34−9q−36 + 7q−40−q−42−7q−44 + 4q−46 + 5q−48−2q−50−2q−52 + q−54 + 2q−56−q−58−q−60 + q−62

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	q16 + q12 + 2q10 + q8−2q6−2q4−3q2−9−12q−2−4q−4 + 3q−6−7q−8 + 4q−10 + 23q−12 + 17q−14 + 3q−16 + 15q−18 + 17q−20−4q−22−10q−24 + 4q−26−q−28−17q−30 + 8q−34−12q−36−8q−38 + 10q−40−3q−42−14q−44 + 2q−46 + 9q−48−5q−50−7q−52 + 7q−54 + 5q−56−5q−58 + 4q−62−q−64−q−66 + q−68
1,0,0,0	−q8−2q4−q2−1−2q−2 + 3q−4 + 5q−8 + 4q−10 + 4q−12 + 3q−14 + q−16−2q−20−3q−24 + q−26−2q−28−2q−34 + q−36−q−38

B2 Invariants.

Weight	Invariant
0,1	−q14 + q12−3q10 + 4q8−7q6 + 9q4−13q2 + 15−16q−2 + 18q−4−13q−6 + 11q−8−q−10−4q−12 + 16q−14−22q−16 + 30q−18−33q−20 + 34q−22−33q−24 + 26q−26−19q−28 + 9q−30−q−32−7q−34 + 13q−36−17q−38 + 18q−40−18q−42 + 15q−44−11q−46 + 8q−48−5q−50 + 2q−52−q−54
1,0	q24−q20−q18 + 2q16 + 3q14−q12−6q10−4q8 + 4q6 + 8q4−3q2−15−8q−2 + 11q−4 + 17q−6−4q−8−17q−10−3q−12 + 20q−14 + 15q−16−6q−18−12q−20 + 7q−22 + 15q−24 + q−26−13q−28−3q−30 + 10q−32 + 4q−34−11q−36−9q−38 + 8q−40 + 9q−42−8q−44−14q−46 + 4q−48 + 16q−50 + 2q−52−18q−54−12q−56 + 13q−58 + 18q−60−4q−62−18q−64−6q−66 + 13q−68 + 10q−70−5q−72−9q−74−q−76 + 6q−78 + 3q−80−2q−82−2q−84 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−q16 + 2q14−2q12 + 4q10−6q8 + 4q6−10q4 + 7q2−16 + 8q−2−14q−4 + 15q−6−10q−8 + 13q−10 + q−12 + 13q−14 + 10q−16−3q−18 + 14q−20−14q−22 + 22q−24−26q−26 + 21q−28−29q−30 + 26q−32−24q−34 + 19q−36−20q−38 + 11q−40−6q−42 + q−44−q−46−8q−48 + 11q−50−12q−52 + 13q−54−15q−56 + 15q−58−12q−60 + 10q−62−9q−64 + 7q−66−4q−68 + 3q−70−2q−72 + q−74

G2 Invariants.

Weight

Invariant

1,0

q32−q30 + 3q28−4q26 + 3q24−3q22−3q20 + 8q18−14q16 + 17q14−19q12 + 12q10−q8−16q6 + 34q4−51q2 + 53−43q−2 + 11q−4 + 26q−6−65q−8 + 95q−10−88q−12 + 60q−14−5q−16−49q−18 + 88q−20−85q−22 + 56q−24−42q−28 + 65q−30−46q−32 + 2q−34 + 59q−36−95q−38 + 96q−40−54q−42−20q−44 + 95q−46−142q−48 + 146q−50−104q−52 + 28q−54 + 54q−56−116q−58 + 135q−60−109q−62 + 47q−64 + 19q−66−69q−68 + 78q−70−50q−72−q−74 + 52q−76−77q−78 + 62q−80−17q−82−47q−84 + 94q−86−108q−88 + 84q−90−33q−92−27q−94 + 73q−96−90q−98 + 81q−100−48q−102 + 9q−104 + 20q−106−40q−108 + 39q−110−29q−112 + 17q−114−3q−116−5q−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 77"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−7t2 + 14t−17 + 14t−1−7t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 5z4 + 4z2 + 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]=

{ 63, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 3q7−6q6 + 8q5−10q4 + 11q3−9q2 + 8q−4 + 2q−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 + 4z4a−2 + 3z4a−4−z4a−6−z4 + 7z2a−2 + 2z2a−4−2z2a−6−3z2 + 5a−2−a−4−a−6−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 77"];

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]=

{ 2t3−7t2 + 14t−17 + 14t−1−7t−2 + 2t−3, −q8 + 3q7−6q6 + 8q5−10q4 + 11q3−9q2 + 8q−4 + 2q−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_65, K11n71, K11n75,}

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	q23−3q22 + q21 + 9q20−15q19−2q18 + 33q17−33q16−18q15 + 67q14−44q13−43q12 + 95q11−43q10−63q9 + 102q8−31q7−65q6 + 82q5−13q4−50q3 + 47q2−q−26 + 18q−1 + q−2−9q−3 + 5q−4−2q−6 + q−7
3	−q45 + 3q44−q43−4q42−2q41 + 12q40 + 5q39−24q38−14q37 + 41q36 + 33q35−57q34−72q33 + 76q32 + 118q31−76q30−182q29 + 67q28 + 245q27−35q26−313q25−q24 + 362q23 + 54q22−404q21−104q20 + 427q19 + 147q18−427q17−194q16 + 420q15 + 212q14−371q13−245q12 + 335q11 + 235q10−255q9−240q8 + 200q7 + 202q6−119q5−180q4 + 79q3 + 127q2−32q−91 + 13q−1 + 57q−2−5q−3−31q−4 + 18q−6−3q−7−7q−8 + 6q−10−3q−11−q−12 + 2q−14−q−15
4	q74−3q73 + q72 + 4q71−3q70 + 5q69−15q68 + 3q67 + 20q66−8q65 + 17q64−60q63−4q62 + 71q61 + 12q60 + 58q59−179q58−81q57 + 134q56 + 117q55 + 242q54−347q53−330q52 + 61q51 + 273q50 + 698q49−373q48−701q47−318q46 + 275q45 + 1368q44−79q43−972q42−948q41−30q40 + 1993q39 + 472q38−968q37−1583q36−560q35 + 2368q34 + 1043q33−733q32−2010q31−1101q30 + 2443q29 + 1455q28−386q27−2163q26−1513q25 + 2241q24 + 1652q23 + 9q22−2024q21−1751q20 + 1761q19 + 1605q18 + 424q17−1585q16−1769q15 + 1078q14 + 1277q13 + 733q12−928q11−1491q10 + 410q9 + 743q8 + 771q7−308q6−978q5 + 20q4 + 242q3 + 540q2 + 33q−470−63q−1−21q−2 + 256q−3 + 93q−4−167q−5−14q−6−67q−7 + 82q−8 + 48q−9−51q−10 + 18q−11−36q−12 + 19q−13 + 13q−14−19q−15 + 16q−16−10q−17 + 4q−18 + 2q−19−8q−20 + 6q−21−q−22 + q−23−2q−25 + q−26
5	−q110 + 3q109−q108−4q107 + 3q106−2q104 + 7q103 + q102−15q101 + q100 + 10q99 + 3q98 + 15q97−4q96−41q95−33q94 + 25q93 + 69q92 + 76q91 + 6q90−138q89−191q88−63q87 + 195q86 + 375q85 + 239q84−198q83−618q82−587q81 + 36q80 + 889q79 + 1126q78 + 339q77−992q76−1809q75−1117q74 + 879q73 + 2549q72 + 2171q71−327q70−3102q69−3574q68−688q67 + 3410q66 + 5010q65 + 2174q64−3224q63−6437q62−3981q61 + 2610q60 + 7549q59 + 5966q58−1529q57−8382q56−7873q55 + 194q54 + 8749q53 + 9623q52 + 1304q51−8821q50−11056q49−2752q48 + 8581q47 + 12138q46 + 4116q45−8139q44−12923q43−5309q42 + 7612q41 + 13343q40 + 6299q39−6836q38−13563q37−7221q36 + 6107q35 + 13400q34 + 7900q33−4975q32−13037q31−8592q30 + 3907q29 + 12221q28 + 8941q27−2385q26−11129q25−9210q24 + 1042q23 + 9537q22 + 8976q21 + 557q20−7756q19−8495q18−1680q17 + 5683q16 + 7434q15 + 2753q14−3770q13−6212q12−3079q11 + 1975q10 + 4656q9 + 3194q8−650q7−3276q6−2737q5−234q4 + 1955q3 + 2194q2 + 672q−1026−1535q−1−757q−2 + 389q−3 + 955q−4 + 664q−5−44q−6−539q−7−477q−8−84q−9 + 239q−10 + 308q−11 + 125q−12−110q−13−159q−14−91q−15 + 10q−16 + 88q−17 + 70q−18−13q−19−24q−20−25q−21−25q−22 + 15q−23 + 24q−24−5q−25 + 3q−26 + 4q−27−14q−28−2q−29 + 7q−30−4q−31 + 2q−32 + 6q−33−4q−34−2q−35 + q−36−q−37 + 2q−39−q−40