10 88

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 88"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 14 2 16 20 18 10 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.21.21]

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(5,{−1,2,−1,−3,2,−3,2,4,−3,4})

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_88.

A1 Invariants.

Weight	Invariant
1	−q11 + 3q9−4q7 + 5q5−3q3 + q + q−1−3q−3 + 5q−5−4q−7 + 3q−9−q−11
2	q32−3q30 + 11q26−14q24−10q22 + 37q20−18q18−37q16 + 54q14−53q10 + 36q8 + 21q6−37q4−q2 + 29−q−2−37q−4 + 21q−6 + 36q−8−53q−10 + 54q−14−37q−16−18q−18 + 37q−20−10q−22−14q−24 + 11q−26−3q−30 + q−32
3	−q63 + 3q61−7q57−2q55 + 18q53 + 13q51−44q49−36q47 + 70q45 + 93q43−88q41−184q39 + 77q37 + 291q35−15q33−384q31−101q29 + 441q27 + 232q25−426q23−355q21 + 345q19 + 441q17−229q15−461q13 + 93q11 + 432q9 + 41q7−362q5−164q3 + 271q + 271q−1−164q−3−362q−5 + 41q−7 + 432q−9 + 93q−11−461q−13−229q−15 + 441q−17 + 345q−19−355q−21−426q−23 + 232q−25 + 441q−27−101q−29−384q−31−15q−33 + 291q−35 + 77q−37−184q−39−88q−41 + 93q−43 + 70q−45−36q−47−44q−49 + 13q−51 + 18q−53−2q−55−7q−57 + 3q−61−q−63
4	q104−3q102 + 7q98−2q96−2q94−21q92 + 4q90 + 54q88 + 13q86−29q84−144q82−46q80 + 234q78 + 234q76 + 24q74−554q72−518q70 + 359q68 + 966q66 + 786q64−922q62−1844q60−517q58 + 1658q56 + 2747q54 + 35q52−3124q50−2894q48 + 724q46 + 4612q44 + 2698q42−2489q40−5095q38−2009q36 + 4299q34 + 5074q32 + 92q30−5038q28−4396q26 + 1926q24 + 5220q22 + 2525q20−2979q18−4789q16−623q14 + 3557q12 + 3551q10−574q8−3759q6−2380q4 + 1504q2 + 3699 + 1504q−2−2380q−4−3759q−6−574q−8 + 3551q−10 + 3557q−12−623q−14−4789q−16−2979q−18 + 2525q−20 + 5220q−22 + 1926q−24−4396q−26−5038q−28 + 92q−30 + 5074q−32 + 4299q−34−2009q−36−5095q−38−2489q−40 + 2698q−42 + 4612q−44 + 724q−46−2894q−48−3124q−50 + 35q−52 + 2747q−54 + 1658q−56−517q−58−1844q−60−922q−62 + 786q−64 + 966q−66 + 359q−68−518q−70−554q−72 + 24q−74 + 234q−76 + 234q−78−46q−80−144q−82−29q−84 + 13q−86 + 54q−88 + 4q−90−21q−92−2q−94−2q−96 + 7q−98−3q−102 + q−104
5	−q155 + 3q153−7q149 + 2q147 + 6q145 + 5q143 + 4q141−14q139−41q137−10q135 + 70q133 + 105q131 + 43q129−135q127−298q125−225q123 + 222q121 + 732q119 + 682q117−159q115−1354q113−1787q111−496q109 + 2095q107 + 3755q105 + 2268q103−2201q101−6450q99−5989q97 + 635q95 + 9175q93 + 11773q91 + 3865q89−10226q87−18885q85−12188q83 + 7671q81 + 25415q79 + 23813q77−4q75−28588q73−36643q71−13038q69 + 26035q67 + 47593q65 + 29531q63−16857q61−53210q59−46175q57 + 1965q55 + 51702q53 + 59244q51 + 15546q49−43098q47−65778q45−32065q43 + 29307q41 + 65010q39 + 44420q37−13624q35−57872q33−50838q31−1066q29 + 46663q27 + 51692q25 + 12623q23−34147q21−48336q19−20550q17 + 22334q15 + 42967q13 + 25616q11−12308q9−37451q7−29220q5 + 3904q3 + 32846q + 32846q−1 + 3904q−3−29220q−5−37451q−7−12308q−9 + 25616q−11 + 42967q−13 + 22334q−15−20550q−17−48336q−19−34147q−21 + 12623q−23 + 51692q−25 + 46663q−27−1066q−29−50838q−31−57872q−33−13624q−35 + 44420q−37 + 65010q−39 + 29307q−41−32065q−43−65778q−45−43098q−47 + 15546q−49 + 59244q−51 + 51702q−53 + 1965q−55−46175q−57−53210q−59−16857q−61 + 29531q−63 + 47593q−65 + 26035q−67−13038q−69−36643q−71−28588q−73−4q−75 + 23813q−77 + 25415q−79 + 7671q−81−12188q−83−18885q−85−10226q−87 + 3865q−89 + 11773q−91 + 9175q−93 + 635q−95−5989q−97−6450q−99−2201q−101 + 2268q−103 + 3755q−105 + 2095q−107−496q−109−1787q−111−1354q−113−159q−115 + 682q−117 + 732q−119 + 222q−121−225q−123−298q−125−135q−127 + 43q−129 + 105q−131 + 70q−133−10q−135−41q−137−14q−139 + 4q−141 + 5q−143 + 6q−145 + 2q−147−7q−149 + 3q−153−q−155

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + 2q12−3q10 + 3q8−2q4 + 3q2−3 + 3q−2−2q−4 + 3q−8−3q−10 + 2q−12 + q−14−q−16
2,0	q42−q40−3q38 + q36 + 7q34 + q32−14q30−3q28 + 21q26 + 4q24−25q22−5q20 + 28q18 + 8q16−32q14 + q12 + 26q10−6q8−15q6 + 7q4 + 7q2−10 + 7q−2 + 7q−4−15q−6−6q−8 + 26q−10 + q−12−32q−14 + 8q−16 + 28q−18−5q−20−25q−22 + 4q−24 + 21q−26−3q−28−14q−30 + q−32 + 7q−34 + q−36−3q−38−q−40 + q−42

A3 Invariants.

Weight	Invariant
0,1,0	q34−3q32 + q30 + 7q28−14q26 + 4q24 + 22q22−30q20 + 5q18 + 35q16−41q14 + 2q12 + 35q10−30q8−6q6 + 22q4−2q2−10−2q−2 + 22q−4−6q−6−30q−8 + 35q−10 + 2q−12−41q−14 + 35q−16 + 5q−18−30q−20 + 22q−22 + 4q−24−14q−26 + 7q−28 + q−30−3q−32 + q−34
1,0,0	−q21 + q19 + 2q15−3q13 + 4q11−2q9 + 2q7−2q5 + 2q3−q−q−1 + 2q−3−2q−5 + 2q−7−2q−9 + 4q−11−3q−13 + 2q−15 + q−19−q−21

B2 Invariants.

Weight	Invariant
0,1	−q34 + 3q32−7q30 + 13q28−22q26 + 32q24−40q22 + 48q20−49q18 + 45q16−33q14 + 16q12 + 7q10−32q8 + 56q6−76q4 + 90q2−96 + 90q−2−76q−4 + 56q−6−32q−8 + 7q−10 + 16q−12−33q−14 + 45q−16−49q−18 + 48q−20−40q−22 + 32q−24−22q−26 + 13q−28−7q−30 + 3q−32−q−34
1,0	q56−3q52−3q50 + 4q48 + 10q46−18q42−13q40 + 19q38 + 31q36−4q34−43q32−19q30 + 40q28 + 42q26−21q24−53q22−4q20 + 49q18 + 22q16−36q14−31q12 + 22q10 + 33q8−11q6−33q4 + 4q2 + 35 + 4q−2−33q−4−11q−6 + 33q−8 + 22q−10−31q−12−36q−14 + 22q−16 + 49q−18−4q−20−53q−22−21q−24 + 42q−26 + 40q−28−19q−30−43q−32−4q−34 + 31q−36 + 19q−38−13q−40−18q−42 + 10q−46 + 4q−48−3q−50−3q−52 + q−56

G2 Invariants.

Weight

Invariant

1,0

q80−3q78 + 7q76−13q74 + 15q72−15q70 + 4q68 + 21q66−53q64 + 91q62−111q60 + 94q58−33q56−75q54 + 204q52−299q50 + 314q48−218q46 + 18q44 + 223q42−417q40 + 487q38−379q36 + 134q34 + 154q32−373q30 + 418q28−277q26 + 21q24 + 235q22−365q20 + 303q18−66q16−243q14 + 490q12−562q10 + 415q8−96q6−286q4 + 588q2−699 + 588q−2−286q−4−96q−6 + 415q−8−562q−10 + 490q−12−243q−14−66q−16 + 303q−18−365q−20 + 235q−22 + 21q−24−277q−26 + 418q−28−373q−30 + 154q−32 + 134q−34−379q−36 + 487q−38−417q−40 + 223q−42 + 18q−44−218q−46 + 314q−48−299q−50 + 204q−52−75q−54−33q−56 + 94q−58−111q−60 + 91q−62−53q−64 + 21q−66 + 4q−68−15q−70 + 15q−72−13q−74 + 7q−76−3q−78 + q−80

.

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 88"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 8t2−24t + 35−24t−1 + 8t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6 + 2z4−z2 + 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]=

{ 101, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q5 + 4q4−8q3 + 13q2−16q + 17−16q−1 + 13q−2−8q−3 + 4q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6 + 2a2z4 + 2z4a−2−2z4−a4z2 + 2a2z2 + 2z2a−2−z2a−4−3z2 + a2 + a−2−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2az9 + 2z9a−1 + 6a2z8 + 6z8a−2 + 12z8 + 7a3z7 + 14az7 + 14z7a−1 + 7z7a−3 + 4a4z6−2a2z6−2z6a−2 + 4z6a−4−12z6 + a5z5−11a3z5−32az5−32z5a−1−11z5a−3 + z5a−5−6a4z4−10a2z4−10z4a−2−6z4a−4−8z4−a5z3 + 6a3z3 + 19az3 + 19z3a−1 + 6z3a−3−z3a−5 + 3a4z2 + 7a2z2 + 7z2a−2 + 3z2a−4 + 8z2−a3z−4az−4za−1−za−3−a2−a−2−1

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 88"];

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]=

{ −t3 + 8t2−24t + 35−24t−1 + 8t−2−t−3, −q5 + 4q4−8q3 + 13q2−16q + 17−16q−1 + 13q−2−8q−3 + 4q−4−q−5 }

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	q15−4q14 + 3q13 + 12q12−29q11 + 7q10 + 59q9−84q8−12q7 + 150q6−138q5−65q4 + 239q3−153q2−123q + 275−123q−1−153q−2 + 239q−3−65q−4−138q−5 + 150q−6−12q−7−84q−8 + 59q−9 + 7q−10−29q−11 + 12q−12 + 3q−13−4q−14 + q−15
3	−q30 + 4q29−3q28−7q27 + 4q26 + 24q25−8q24−64q23 + 12q22 + 130q21 + 15q20−245q19−84q18 + 391q17 + 229q16−551q15−453q14 + 674q13 + 771q12−760q11−1111q10 + 745q9 + 1471q8−664q7−1781q6 + 513q5 + 2025q4−325q3−2172q2 + 110q + 2223 + 110q−1−2172q−2−325q−3 + 2025q−4 + 513q−5−1781q−6−664q−7 + 1471q−8 + 745q−9−1111q−10−760q−11 + 771q−12 + 674q−13−453q−14−551q−15 + 229q−16 + 391q−17−84q−18−245q−19 + 15q−20 + 130q−21 + 12q−22−64q−23−8q−24 + 24q−25 + 4q−26−7q−27−3q−28 + 4q−29−q−30
4	q50−4q49 + 3q48 + 7q47−9q46 + q45−23q44 + 28q43 + 57q42−50q41−41q40−138q39 + 126q38 + 337q37−50q36−251q35−716q34 + 162q33 + 1214q32 + 557q31−431q30−2424q29−760q28 + 2541q27 + 2732q26 + 658q25−5136q24−3919q23 + 2771q22 + 6350q21 + 4546q20−7050q19−9106q18 + 165q17 + 9436q16 + 10854q15−6275q14−14088q13−4965q12 + 10078q11 + 17176q10−2981q9−16783q8−10469q7 + 8268q6 + 21342q5 + 1199q4−16789q3−14594q2 + 5083q + 22721 + 5083q−1−14594q−2−16789q−3 + 1199q−4 + 21342q−5 + 8268q−6−10469q−7−16783q−8−2981q−9 + 17176q−10 + 10078q−11−4965q−12−14088q−13−6275q−14 + 10854q−15 + 9436q−16 + 165q−17−9106q−18−7050q−19 + 4546q−20 + 6350q−21 + 2771q−22−3919q−23−5136q−24 + 658q−25 + 2732q−26 + 2541q−27−760q−28−2424q−29−431q−30 + 557q−31 + 1214q−32 + 162q−33−716q−34−251q−35−50q−36 + 337q−37 + 126q−38−138q−39−41q−40−50q−41 + 57q−42 + 28q−43−23q−44 + q−45−9q−46 + 7q−47 + 3q−48−4q−49 + q−50
5	−q75 + 4q74−3q73−7q72 + 9q71 + 4q70−2q69 + 3q68−21q67−34q66 + 40q65 + 84q64 + 33q63−59q62−199q61−197q60 + 113q59 + 531q58 + 543q57−109q56−1040q55−1392q54−320q53 + 1822q52 + 3134q51 + 1551q50−2527q49−5861q48−4569q47 + 2283q46 + 9758q45 + 10091q44 + 71q43−13769q42−18660q41−6376q40 + 16455q39 + 29950q38 + 17815q37−15371q36−42477q35−34960q34 + 8400q33 + 53555q32 + 56888q31 + 6187q30−60539q29−81348q28−27953q27 + 60590q26 + 105028q25 + 55924q24−52997q23−125046q22−86597q21 + 37910q20 + 138741q19 + 117296q18−17294q17−145636q16−144641q15−6338q14 + 145775q13 + 167068q12 + 30435q11−140607q10−183710q9−53108q8 + 131586q7 + 194854q6 + 73319q5−119974q4−201061q3−91032q2 + 106443q + 203085 + 106443q−1−91032q−2−201061q−3−119974q−4 + 73319q−5 + 194854q−6 + 131586q−7−53108q−8−183710q−9−140607q−10 + 30435q−11 + 167068q−12 + 145775q−13−6338q−14−144641q−15−145636q−16−17294q−17 + 117296q−18 + 138741q−19 + 37910q−20−86597q−21−125046q−22−52997q−23 + 55924q−24 + 105028q−25 + 60590q−26−27953q−27−81348q−28−60539q−29 + 6187q−30 + 56888q−31 + 53555q−32 + 8400q−33−34960q−34−42477q−35−15371q−36 + 17815q−37 + 29950q−38 + 16455q−39−6376q−40−18660q−41−13769q−42 + 71q−43 + 10091q−44 + 9758q−45 + 2283q−46−4569q−47−5861q−48−2527q−49 + 1551q−50 + 3134q−51 + 1822q−52−320q−53−1392q−54−1040q−55−109q−56 + 543q−57 + 531q−58 + 113q−59−197q−60−199q−61−59q−62 + 33q−63 + 84q−64 + 40q−65−34q−66−21q−67 + 3q−68−2q−69 + 4q−70 + 9q−71−7q−72−3q−73 + 4q−74−q−75
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