10 121

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["10 121"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X7,20,8,1 X9,19,10,18 X3,11,4,10 X17,5,18,4 X5,12,6,13 X11,16,12,17 X19,14,20,15 X13,8,14,9 X15,2,16,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 12 20 18 16 8 2 4 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[9*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,−2,3,−2,1,−2,3,−2,3,−2})

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[{5, 3}, {2, 4}, {3, 1}, {6, 13}, {10, 5}, {7, 11}, {9, 6}, {8, 10}, {12, 9}, {11, 2}, {13, 7}, {4, 8}, {1, 12}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_121.

A1 Invariants.

Weight	Invariant
1	−q17 + 3q15−5q13 + 5q11−4q9 + 2q7 + 2q5−3q3 + 5q−5q−1 + 4q−3−q−5
2	q48−3q46 + q44 + 10q42−18q40−5q38 + 44q36−30q34−40q32 + 71q30−9q28−68q26 + 53q24 + 24q22−53q20 + 4q18 + 40q16−9q14−45q12 + 35q10 + 38q8−71q6 + 8q4 + 68q2−53−21q−2 + 53q−4−14q−6−21q−8 + 15q−10 + q−12−4q−14 + q−16
3	−q93 + 3q91−q89−6q87 + 3q85 + 17q83−4q81−53q79−q77 + 115q75 + 51q73−191q71−179q69 + 243q67 + 366q65−204q63−573q61 + 55q59 + 731q57 + 168q55−768q53−409q51 + 679q49 + 597q47−501q45−688q43 + 279q41 + 681q39−47q37−619q35−151q33 + 509q31 + 336q29−384q27−507q25 + 236q23 + 655q21−53q19−761q17−159q15 + 782q13 + 388q11−694q9−577q7 + 507q5 + 671q3−262q−634q−1 + 29q−3 + 499q−5 + 116q−7−317q−9−154q−11 + 150q−13 + 127q−15−49q−17−76q−19 + 14q−21 + 28q−23 + q−25−11q−27−q−29 + 4q−31−q−33

A2 Invariants.

Weight	Invariant
1,0	−q24 + 2q22−2q20−2q18 + 4q16−3q14 + 3q12−q8 + 3q6−4q4 + 4q2−1−q−2 + 3q−4−q−6
1,1	q68−6q66 + 20q64−50q62 + 111q60−224q58 + 412q56−712q54 + 1155q52−1734q50 + 2410q48−3106q46 + 3677q44−3940q42 + 3770q40−3072q38 + 1810q36−94q34−1888q32 + 3880q30−5664q28 + 7012q26−7730q24 + 7754q22−7064q20 + 5760q18−3990q16 + 1976q14 + 17q12−1768q10 + 3064q8−3800q6 + 3997q4−3742q2 + 3190−2472q−2 + 1767q−4−1174q−6 + 712q−8−388q−10 + 194q−12−88q−14 + 32q−16−8q−18 + q−20
2,0	q62−2q60 + 5q56−3q54−9q52 + 22q48 + 2q46−29q44−2q42 + 29q40 + q38−36q36 + 4q34 + 29q32−4q30−23q28 + 13q26 + 13q24−18q22 + 8q20 + 8q18−14q16−6q14 + 25q12−q10−33q8 + 9q6 + 34q4−10q2−30 + 16q−2 + 24q−4−9q−6−16q−8 + 2q−10 + 10q−12−2q−14−3q−16 + q−18

A3 Invariants.

Weight	Invariant
0,1,0	q54−3q52 + q50 + 8q48−14q46 + 2q44 + 26q42−35q40 + q38 + 45q36−50q34−q32 + 46q30−39q28−7q26 + 29q24−6q22−12q20−q18 + 24q16−5q14−35q12 + 41q10 + 7q8−53q6 + 43q4 + 9q2−41 + 29q−2 + 5q−4−19q−6 + 11q−8 + 2q−10−4q−12 + q−14
1,0,0	−q31 + 2q29−3q27 + q25−3q23 + 4q21−3q19 + 4q17 + q13−q9 + 2q7−4q5 + 4q3−2q + 3q−1−2q−3 + 3q−5−q−7

A4 Invariants.

Weight	Invariant
0,1,0,0	q68−2q66−2q64 + 8q62−16q58 + 7q56 + 23q54−15q52−25q50 + 26q48 + 23q46−41q44−20q42 + 46q40−q38−48q36 + 23q34 + 39q32−34q30−15q28 + 42q26−7q24−41q22 + 27q20 + 35q18−41q16−14q14 + 51q12−2q10−45q8 + 12q6 + 31q4−17q2−19 + 18q−2 + 12q−4−12q−6−3q−8 + 9q−10−q−12−3q−14 + q−16
1,0,0,0	−q38 + 2q36−3q34−3q28 + 4q26−3q24 + 4q22 + q20 + q18 + q16−2q10 + 2q8−4q6 + 4q4−2q2 + 2 + 2q−2−2q−4 + 3q−6−q−8

B2 Invariants.

Weight	Invariant
0,1	−q54 + 3q52−7q50 + 14q48−24q46 + 36q44−50q42 + 61q40−65q38 + 61q36−50q34 + 29q32−2q30−31q28 + 65q26−93q24 + 116q22−124q20 + 123q18−108q16 + 83q14−51q12 + 17q10 + 13q8−39q6 + 57q4−65q2 + 65−57q−2 + 47q−4−31q−6 + 19q−8−10q−10 + 4q−12−q−14
1,0	q88−3q84−3q82 + 4q80 + 11q78 + q76−19q74−16q72 + 18q70 + 38q68−q66−53q64−29q62 + 47q60 + 57q58−21q56−71q54−11q52 + 64q50 + 35q48−46q46−47q44 + 26q42 + 48q40−10q38−47q36 + 45q32 + 11q30−42q28−21q26 + 40q24 + 35q22−35q20−50q18 + 23q16 + 65q14 + 2q12−68q10−33q8 + 54q6 + 57q4−24q2−59−7q−2 + 43q−4 + 26q−6−20q−8−25q−10 + q−12 + 15q−14 + 6q−16−4q−18−4q−20 + q−24

D4 Invariants.

Weight	Invariant
1,0,0,0	q74−3q72 + 4q70−6q68 + 12q66−19q64 + 23q62−28q60 + 41q58−49q56 + 48q54−49q52 + 52q50−45q48 + 27q46−22q44 + 7q42 + 17q40−40q38 + 49q36−65q34 + 89q32−91q30 + 93q28−95q26 + 96q24−76q22 + 64q20−55q18 + 33q16−7q14−7q12 + 16q10−36q8 + 49q6−49q4 + 51q2−52 + 48q−2−35q−4 + 31q−6−25q−8 + 16q−10−8q−12 + 6q−14−4q−16 + q−18

G2 Invariants.

Weight

Invariant

1,0

q128−3q126 + 7q124−13q122 + 16q120−16q118 + 7q116 + 17q114−48q112 + 88q110−120q108 + 119q106−76q104−33q102 + 190q100−339q98 + 424q96−367q94 + 144q92 + 189q90−524q88 + 710q86−650q84 + 336q82 + 111q80−519q78 + 707q76−574q74 + 195q72 + 258q70−566q68 + 567q66−274q64−195q62 + 623q60−813q58 + 696q56−280q54−274q52 + 766q50−1016q48 + 928q46−540q44−20q42 + 548q40−851q38 + 845q36−520q34 + 30q32 + 417q30−637q28 + 517q26−141q24−311q22 + 622q20−634q18 + 356q16 + 94q14−517q12 + 736q10−680q8 + 389q6−3q4−331q2 + 497−471q−2 + 323q−4−115q−6−58q−8 + 155q−10−181q−12 + 143q−14−81q−16 + 29q−18 + 10q−20−25q−22 + 26q−24−20q−26 + 10q−28−4q−30 + q−32

.

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["10 121"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−11t2 + 27t−35 + 27t−1−11t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + z4 + 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]=

{ 115, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q2 + 5q−10 + 15q−1−18q−2 + 20q−3−18q−4 + 14q−5−9q−6 + 4q−7−q−8

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4a6−z2a6−a6 + z6a4 + 2z4a4 + 3z2a4 + 2a4 + z6a2 + z4a2−z2a2−a2−z4 + 1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z5a9−z3a9 + 4z6a8−5z4a8 + z2a8 + 8z7a7−13z5a7 + 8z3a7−2za7 + 9z8a6−14z6a6 + 9z4a6−3z2a6 + a6 + 4z9a5 + 9z7a5−28z5a5 + 19z3a5−3za5 + 19z8a4−36z6a4 + 22z4a4−7z2a4 + 2a4 + 4z9a3 + 11z7a3−30z5a3 + 14z3a3−za3 + 10z8a2−13z6a2 + 3z4a2−3z2a2 + a2 + 10z7a−15z5a + 4z3a + 5z6−5z4 + 1 + z5a−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["10 121"];

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]=

{ 2t3−11t2 + 27t−35 + 27t−1−11t−2 + 2t−3, −q2 + 5q−10 + 15q−1−18q−2 + 20q−3−18q−4 + 14q−5−9q−6 + 4q−7−q−8 }

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]=

{K11a41, K11a183, K11a198, K11a331,}

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	q7−5q6 + 5q5 + 15q4−41q3 + 12q2 + 82q−115−20q−1 + 203q−2−175q−3−99q−4 + 312q−5−178q−6−179q−7 + 348q−8−129q−9−215q−10 + 291q−11−52q−12−186q−13 + 170q−14 + 7q−15−106q−16 + 59q−17 + 17q−18−32q−19 + 10q−20 + 4q−21−4q−22 + q−23
3	−q15 + 5q14−5q13−10q12 + 11q11 + 32q10−19q9−100q8 + 38q7 + 208q6 + 4q5−404q4−125q3 + 641q2 + 387q−874−788q−1 + 1013q−2 + 1320q−3−1038q−4−1872q−5 + 896q−6 + 2402q−7−644q−8−2813q−9 + 294q−10 + 3110q−11 + 64q−12−3232q−13−449q−14 + 3233q−15 + 784q−16−3059q−17−1109q−18 + 2765q−19 + 1356q−20−2331q−21−1511q−22 + 1798q−23 + 1543q−24−1233q−25−1429q−26 + 710q−27 + 1184q−28−297q−29−866q−30 + 34q−31 + 556q−32 + 72q−33−296q−34−89q−35 + 134q−36 + 60q−37−54q−38−25q−39 + 18q−40 + 8q−41−5q−42−4q−43 + 4q−44−q−45
4	q26−5q25 + 5q24 + 10q23−16q22−2q21−25q20 + 52q19 + 82q18−106q17−97q16−176q15 + 289q14 + 590q13−173q12−644q11−1236q10 + 481q9 + 2406q8 + 1127q7−1182q6−4720q5−1616q4 + 4894q3 + 5955q2 + 1523q−9653−8735q−1 + 3822q−2 + 12915q−3 + 10561q−4−11015q−5−18889q−6−4083q−7 + 16524q−8 + 23284q−9−5597q−10−26135q−11−15946q−12 + 13781q−13 + 33559q−14 + 3832q−15−27397q−16−26171q−17 + 7018q−18 + 38261q−19 + 12749q−20−24101q−21−32180q−22−530q−23 + 37954q−24 + 19437q−25−18052q−26−34142q−27−7995q−28 + 33219q−29 + 23876q−30−9405q−31−31723q−32−14982q−33 + 23705q−34 + 24646q−35 + 813q−36−23810q−37−18806q−38 + 11021q−39 + 19624q−40 + 8483q−41−12097q−42−16370q−43 + 500q−44 + 10327q−45 + 9553q−46−2283q−47−9179q−48−3362q−49 + 2438q−50 + 5538q−51 + 1493q−52−2867q−53−2180q−54−533q−55 + 1685q−56 + 1111q−57−354q−58−528q−59−477q−60 + 247q−61 + 278q−62−6q−63−26q−64−111q−65 + 25q−66 + 36q−67−10q−68 + 6q−69−13q−70 + 5q−71 + 4q−72−4q−73 + q−74
5	−q40 + 5q39−5q38−10q37 + 16q36 + 7q35−5q34−8q33−34q32−29q31 + 90q30 + 149q29 + 4q28−230q27−408q26−195q25 + 497q24 + 1219q23 + 903q22−872q21−2763q20−2813q19 + 293q18 + 5211q17 + 7390q16 + 2485q15−7674q14−14877q13−10397q12 + 7132q11 + 25362q10 + 25619q9 + 298q8−34927q7−48831q6−19965q5 + 37974q4 + 77330q3 + 54320q2−26923q−104297−102435q−1−3736q−2 + 120534q−3 + 158188q−4 + 55648q−5−118030q−6−211957q−7−124190q−8 + 91957q−9 + 253523q−10 + 201274q−11−44019q−12−275896q−13−275665q−14−19871q−15 + 276333q−16 + 338995q−17 + 90509q−18−257884q−19−385285q−20−159267q−21 + 225845q−22 + 414167q−23 + 219600q−24−187091q−25−427285q−26−269102q−27 + 146528q−28 + 429225q−29 + 307561q−30−107164q−31−422549q−32−337731q−33 + 68646q−34 + 410235q−35 + 361491q−36−29850q−37−390960q−38−380646q−39−12099q−40 + 363719q−41 + 394335q−42 + 57800q−43−325057q−44−399728q−45−106875q−46 + 273351q−47 + 392638q−48 + 155175q−49−208778q−50−368727q−51−196351q−52 + 135017q−53 + 325835q−54 + 223155q−55−59569q−56−265676q−57−229518q−58−7919q−59 + 194245q−60 + 213473q−61 + 58443q−62−121321q−63−178297q−64−86326q−65 + 57493q−66 + 131709q−67 + 91411q−68−10429q−69−83915q−70−79185q−71−16583q−72 + 43794q−73 + 57614q−74 + 26168q−75−15987q−76−35432q−77−24141q−78 + 1065q−79 + 17892q−80 + 16938q−81 + 4466q−82−6938q−83−9673q−84−4710q−85 + 1710q−86 + 4482q−87 + 3074q−88 + 140q−89−1652q−90−1582q−91−417q−92 + 523q−93 + 628q−94 + 230q−95−100q−96−196q−97−131q−98 + 32q−99 + 78q−100 + 10q−101−7q−102−q−103−14q−104−q−105 + 13q−106−5q−107−4q−108 + 4q−109−q−110