10 138

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["10 138"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X7,15,8,14 X15,7,16,6 X20,18,1,17 X18,13,19,14 X12,19,13,20

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 -14 2 16 18 -6 20 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,211,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_138.

A1 Invariants.

Weight	Invariant
1	q7−q5 + 2q3−q + q−1−q−5 + q−7−2q−9 + 2q−11
2	q22−q20−2q18 + 4q16 + q14−6q12 + 4q10 + 5q8−7q6−q4 + 7q2−3−4q−2 + 5q−4 + q−6−5q−8 + q−10 + 6q−12−2q−14−5q−16 + 7q−18 + q−20−8q−22 + 4q−24 + 2q−26−4q−28 + q−30 + q−32
3	q45−q43−2q41 + 5q37 + 3q35−7q33−8q31 + 6q29 + 14q27−19q23−8q21 + 17q19 + 19q17−10q15−26q13 + q11 + 28q9 + 10q7−27q5−19q3 + 22q + 24q−1−15q−3−25q−5 + 11q−7 + 28q−9−6q−11−24q−13−q−15 + 22q−17 + 7q−19−18q−21−17q−23 + 10q−25 + 24q−27 + q−29−28q−31−13q−33 + 28q−35 + 21q−37−21q−39−24q−41 + 14q−43 + 21q−45−3q−47−17q−49 + 8q−53−2q−57−2q−59 + 2q−61

A2 Invariants.

Weight	Invariant
1,0	q10 + q8 + q4−q2−q−4 + 2q−6−q−8 + q−10−q−12−q−14 + q−16 + q−20
2,0	q28 + q26−2q22 + 3q18 + q16−3q14−q12 + 3q10 + 2q8−3q6 + 3q2−2q−2−q−6−2q−8 + q−10 + 2q−16 + 7q−18 + 2q−20−4q−22 + q−26−3q−28−4q−30 + 2q−34 + 2q−36−q−48 + q−52

A3 Invariants.

Weight	Invariant
0,1,0	q20−q18 + 2q16 + 2q14−2q12 + 4q10 + q8−4q6 + 4q4−2q2−5 + 3q−2−3q−6 + 2q−8 + 2q−10 + q−12−q−14 + 4q−18−4q−20 + 5q−24−5q−26−q−28 + 4q−30−3q−32−q−34 + 3q−36
1,0,0	q13 + q11 + 2q9 + q5−2q3−2q−1 + q−7 + 2q−9 + q−13−2q−15−2q−19 + q−21 + q−25 + q−27

B2 Invariants.

Weight	Invariant
0,1	q20−q18 + 4q16−4q14 + 6q12−6q10 + 7q8−6q6 + 4q4−2q2−3 + 5q−2−8q−4 + 11q−6−12q−8 + 14q−10−11q−12 + 9q−14−6q−16 + 2q−18−4q−22 + 5q−24−7q−26 + 7q−28−6q−30 + 5q−32−3q−34 + 3q−36
1,0	q34−q30−q28 + 3q26 + 3q24−2q22−4q20 + q18 + 6q16 + 3q14−6q12−5q10 + 4q8 + 7q6−q4−8q2−2 + 5q−2 + 3q−4−3q−6−4q−8 + 2q−10 + 4q−12−q−14−5q−16 + q−18 + 6q−20 + q−22−5q−24−3q−26 + 5q−28 + 5q−30−3q−32−6q−34 + 2q−36 + 7q−38 + q−40−6q−42−4q−44 + 3q−46 + 5q−48−q−50−4q−52−2q−54 + q−56 + 3q−58

G2 Invariants.

Weight

Invariant

1,0

q46−q44 + 4q42−5q40 + 5q38−2q36−4q34 + 14q32−18q30 + 20q28−11q26−4q24 + 19q22−27q20 + 29q18−16q16 + 17q12−25q10 + 19q8−5q6−13q4 + 20q2−21 + 7q−2 + 8q−4−25q−6 + 33q−8−29q−10 + 14q−12 + 5q−14−25q−16 + 35q−18−34q−20 + 24q−22−4q−24−12q−26 + 28q−28−27q−30 + 17q−32 + q−34−15q−36 + 22q−38−17q−40 + 16q−44−24q−46 + 25q−48−15q−50−5q−52 + 18q−54−26q−56 + 22q−58−13q−60 + q−62 + 9q−64−12q−66 + 11q−68−8q−70 + 5q−72 + q−74−3q−76 + q−78−2q−80 + 2q−82−q−84 + 2q−86 + q−88

.

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["10 138"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−5t2 + 8t−7 + 8t−1−5t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + z4−3z2 + 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]=

{ 35, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

2q5−4q4 + 5q3−6q2 + 6q−5 + 4q−1−2q−2 + q−3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + 4z4a−2−z4a−4−2z4 + a2z2 + 5z2a−2−3z2a−4−6z2 + 2a2 + 3a−2−2a−4 + a−6−3

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−2 + z8 + 2az7 + 5z7a−1 + 3z7a−3 + a2z6 + 3z6a−2 + 3z6a−4 + z6−7az5−14z5a−1−6z5a−3 + z5a−5−4a2z4−13z4a−2−5z4a−4−12z4 + 6az3 + 8z3a−1 + 5z3a−3 + 3z3a−5 + 5a2z2 + 10z2a−2 + 6z2a−4 + 3z2a−6 + 12z2−az−za−1−2za−3−2za−5−2a2−3a−2−2a−4−a−6−3

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["10 138"];

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]=

{ t3−5t2 + 8t−7 + 8t−1−5t−2 + t−3, 2q5−4q4 + 5q3−6q2 + 6q−5 + 4q−1−2q−2 + q−3 }

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]=

{K11n117,}

n	Jn
2	q15−5q13 + 7q12 + 2q11−17q10 + 16q9 + 8q8−29q7 + 19q6 + 16q5−34q4 + 13q3 + 22q2−30q + 4 + 23q−1−20q−2−4q−3 + 17q−4−8q−5−5q−6 + 7q−7−q−8−2q−9 + q−10
3	2q29−4q28 + 2q26 + 10q25−12q24−17q23 + 16q22 + 34q21−19q20−55q19 + 19q18 + 76q17−12q16−96q15 + 4q14 + 105q13 + 11q12−110q11−23q10 + 104q9 + 36q8−95q7−46q6 + 81q5 + 54q4−61q3−63q2 + 45q + 64−22q−1−65q−2 + 4q−3 + 56q−4 + 15q−5−47q−6−23q−7 + 29q−8 + 31q−9−18q−10−25q−11 + 4q−12 + 20q−13 + q−14−11q−15−4q−16 + 6q−17 + 2q−18−q−19−2q−20 + q−21
4	q48−5q46 + 11q44 + 3q43−6q42−29q41−7q40 + 56q39 + 34q38−17q37−113q36−58q35 + 145q34 + 140q33 + 4q32−253q31−201q30 + 215q29 + 311q28 + 107q27−367q26−400q25 + 200q24 + 440q23 + 259q22−376q21−547q20 + 120q19 + 456q18 + 371q17−302q16−582q15 + 38q14 + 376q13 + 414q12−193q11−538q10−31q9 + 257q8 + 413q7−72q6−448q5−96q4 + 114q3 + 378q2 + 57q−315−140q−1−40q−2 + 285q−3 + 153q−4−145q−5−117q−6−156q−7 + 136q−8 + 159q−9 + 2q−10−25q−11−170q−12 + 78q−14 + 56q−15 + 59q−16−96q−17−46q−18−3q−19 + 26q−20 + 68q−21−21q−22−22q−23−23q−24−6q−25 + 32q−26 + 2q−27−9q−29−8q−30 + 7q−31 + q−32 + 2q−33−q−34−2q−35 + q−36
5	2q71−4q70 + 2q67 + 12q66 + 2q65−28q64−20q63 + 8q62 + 38q61 + 73q60 + 6q59−123q58−140q57−16q56 + 187q55 + 299q54 + 100q53−316q52−530q51−235q50 + 404q49 + 833q48 + 502q47−451q46−1189q45−861q44 + 414q43 + 1507q42 + 1294q41−244q40−1769q39−1743q38−q37 + 1904q36 + 2127q35 + 318q34−1915q33−2422q32−619q31 + 1817q30 + 2580q29 + 893q28−1660q27−2628q26−1082q25 + 1461q24 + 2577q23 + 1219q22−1262q21−2476q20−1292q19 + 1064q18 + 2326q17 + 1350q16−856q15−2166q14−1398q13 + 639q12 + 1983q11 + 1437q10−384q9−1769q8−1486q7 + 117q6 + 1520q5 + 1487q4 + 173q3−1204q2−1465q−441 + 865q−1 + 1334q−2 + 675q−3−484q−4−1144q−5−811q−6 + 128q−7 + 853q−8 + 845q−9 + 180q−10−542q−11−745q−12−375q−13 + 209q−14 + 569q−15 + 458q−16 + 29q−17−328q−18−402q−19−207q−20 + 113q−21 + 295q−22 + 237q−23 + 48q−24−133q−25−213q−26−127q−27 + 23q−28 + 123q−29 + 133q−30 + 53q−31−49q−32−93q−33−71q−34−9q−35 + 51q−36 + 60q−37 + 22q−38−11q−39−33q−40−30q−41−2q−42 + 19q−43 + 13q−44 + 6q−45−11q−47−6q−48 + 3q−49 + 2q−50 + q−51 + 2q−52−q−53−2q−54 + q−55
6	q99−5q97 + 7q95 + 4q94−q92−10q91−33q90−10q89 + 59q88 + 67q87 + 20q86−31q85−122q84−195q83−62q82 + 257q81 + 407q80 + 252q79−79q78−560q77−895q76−468q75 + 628q74 + 1434q73 + 1313q72 + 304q71−1360q70−2689q69−2062q68 + 516q67 + 3106q66 + 3836q65 + 2149q64−1635q63−5303q62−5388q61−1267q60 + 4189q59 + 7192q58 + 5816q57−106q56−7123q55−9375q54−4840q53 + 3324q52 + 9467q51 + 9864q50 + 3090q49−6812q48−11955q47−8532q46 + 850q45 + 9496q44 + 12299q43 + 6222q42−4911q41−12255q40−10591q39−1587q38 + 8003q37 + 12603q36 + 7896q35−2919q34−11135q33−10849q32−2970q31 + 6298q30 + 11702q29 + 8245q28−1516q27−9694q26−10261q25−3638q24 + 4811q23 + 10513q22 + 8180q21−286q20−8186q19−9566q18−4371q17 + 3130q16 + 9158q15 + 8220q14 + 1363q13−6217q12−8719q11−5424q10 + 830q9 + 7204q8 + 8087q7 + 3427q6−3442q5−7140q4−6256q3−1911q2 + 4286q + 7006 + 5123q−1−154q−2−4390q−3−5880q−4−4121q−5 + 780q−6 + 4481q−7 + 5297q−8 + 2493q−9−950q−10−3787q−11−4539q−12−1980q−13 + 1169q−14 + 3515q−15 + 3167q−16 + 1655q−17−857q−18−2896q−19−2648q−20−1220q−21 + 916q−22 + 1813q−23 + 2148q−24 + 1081q−25−601q−26−1432q−27−1556q−28−641q−29 + 27q−30 + 1024q−31 + 1156q−32 + 582q−33−26q−34−604q−35−600q−36−665q−37−45q−38 + 337q−39 + 442q−40 + 379q−41 + 118q−42−21q−43−382q−44−249q−45−127q−46 + 25q−47 + 135q−48 + 168q−49 + 188q−50−49q−51−62q−52−105q−53−77q−54−40q−55 + 26q−56 + 102q−57 + 21q−58 + 24q−59−13q−60−24q−61−39q−62−16q−63 + 24q−64 + 3q−65 + 14q−66 + 5q−67 + 3q−68−11q−69−8q−70 + 5q−71−2q−72 + 2q−73 + q−74 + 2q−75−q−76−2q−77 + q−78
7	2q131−4q130 + 10q126 + 4q125−4q124−16q123−22q122−4q121 + 12q120 + 34q119 + 78q118 + 53q117−55q116−161q115−201q114−82q113 + 110q112 + 332q111 + 519q110 + 367q109−196q108−861q107−1194q106−828q105 + 220q104 + 1498q103 + 2463q102 + 2111q101 + 138q100−2584q99−4616q98−4296q97−1200q96 + 3502q95 + 7564q94 + 8110q93 + 3737q92−3940q91−11278q90−13524q89−8126q88 + 3039q87 + 14879q86 + 20312q85 + 14871q84 + 11q83−17596q82−27866q81−23558q80−5459q79 + 18330q78 + 34791q77 + 33434q76 + 13450q75−16471q74−40128q73−43282q72−22929q71 + 11993q70 + 42776q69 + 51651q68 + 32875q67−5362q66−42614q65−57642q64−41859q63−2210q62 + 39922q61 + 60703q60 + 48891q59 + 9628q58−35597q57−61109q56−53422q55−15874q54 + 30643q53 + 59505q52 + 55554q51 + 20410q50−25991q49−56715q48−55711q47−23237q46 + 22058q45 + 53549q44 + 54696q43 + 24672q42−19075q41−50470q40−53065q39−25281q38 + 16697q37 + 47676q36 + 51393q35 + 25669q34−14603q33−45126q32−49896q31−26220q30 + 12340q29 + 42479q28 + 48603q27 + 27295q26−9483q25−39537q24−47428q23−28846q22 + 5877q21 + 35879q20 + 45981q19 + 30851q18−1327q17−31309q16−44046q15−32974q14−3902q13 + 25631q12 + 41104q11 + 34720q10 + 9706q9−18807q8−36959q7−35645q6−15403q5 + 11148q4 + 31224q3 + 35068q2 + 20477q−2969−24086q−1−32665q−2−24040q−3−4890q−4 + 15793q−5 + 28117q−6 + 25535q−7 + 11601q−8−7124q−9−21699q−10−24446q−11−16263q−12−1023q−13 + 14079q−14 + 20880q−15 + 18220q−16 + 7466q−17−6217q−18−15283q−19−17356q−20−11524q−21−640q−22 + 8867q−23 + 14043q−24 + 12633q−25 + 5489q−26−2628q−27−9241q−28−11298q−29−7856q−30−2056q−31 + 4272q−32 + 8104q−33 + 7715q−34 + 4766q−35−93q−36−4413q−37−5942q−38−5309q−39−2389q−40 + 1154q−41 + 3337q−42 + 4322q−43 + 3255q−44 + 957q−45−1012q−46−2621q−47−2798q−48−1757q−49−543q−50 + 966q−51 + 1729q−52 + 1595q−53 + 1179q−54 + 141q−55−678q−56−979q−57−1080q−58−580q−59−48q−60 + 299q−61 + 691q−62 + 603q−63 + 327q−64 + 83q−65−282q−66−339q−67−312q−68−270q−69 + 2q−70 + 146q−71 + 203q−72 + 222q−73 + 72q−74 + 9q−75−48q−76−149q−77−99q−78−53q−79 + 6q−80 + 73q−81 + 40q−82 + 40q−83 + 37q−84−15q−85−27q−86−34q−87−22q−88 + 13q−89 + q−90 + 5q−91 + 16q−92 + 5q−93 + 2q−94−8q−95−8q−96 + 3q−97−2q−99 + 2q−100 + q−101 + 2q−102−q−103−2q−104 + q−105