10 142

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["10 142"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31,3,3-]

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,{1,1,1,2,1,1,1,2,3,−2,3})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_142.

A1 Invariants.

Weight	Invariant
1	q−5 + q−9 + q−13 + q−15−2q−21
2	q−10 + 2q−16 + q−18−q−20 + q−22 + 2q−24−q−28−q−34 + q−36 + 2q−38−q−40−q−46−3q−48−q−50−q−54 + q−56 + q−58 + q−60
3	q−15 + q−21 + 2q−23 + q−25−q−27−q−29 + 2q−31 + 3q−33 + q−35−2q−37−2q−39 + q−41 + 3q−43 + q−45−2q−47−3q−49 + q−51 + 4q−53 + q−55−4q−57−3q−59 + 2q−61 + 2q−63−3q−65−2q−67 + 3q−69 + q−71−2q−73−q−75 + q−77−q−81−3q−83−2q−85 + 3q−89−2q−91−4q−93 + 7q−97 + 4q−99−2q−101−2q−103 + q−105 + 4q−107−2q−111−2q−113

A2 Invariants.

Weight	Invariant
1,0	q−10 + q−14 + q−18 + q−20 + 2q−22 + 3q−24−q−28−3q−30−2q−32−q−34 + q−38
1,1	q−20 + 2q−24−2q−26 + 6q−28−2q−30 + 12q−32−2q−34 + 7q−36 + 6q−42−10q−44 + 10q−46−10q−48 + 8q−50−11q−52 + 2q−54−8q−56−4q−58−3q−60−6q−62 + 6q−64−4q−66 + 10q−68 + 2q−72 + 2q−74−2q−76−2q−78−4q−80−2q−82 + 4q−84 + 2q−88
2,0	q−20 + q−26 + 2q−28 + q−30 + q−32 + 2q−34 + 3q−36 + 2q−38 + q−40 + q−42−q−46 + q−52 + 2q−54 + 3q−56 + 2q−58 + q−60−4q−62−6q−64−10q−66−9q−68−5q−70−q−72 + 3q−74 + 4q−76 + 6q−78 + 5q−80 + 4q−82−2q−88−2q−90−q−92 + q−96

A3 Invariants.

Weight	Invariant
0,1,0	q−20 + q−24 + q−26 + q−30 + 2q−32 + 3q−34 + 5q−36 + 4q−38 + 4q−40 + 2q−42−3q−46−3q−48−5q−50−4q−52−4q−54−3q−56−q−58−q−60 + q−62 + q−64 + q−66 + q−68 + 2q−70
1,0,0	q−15 + q−19 + q−23 + 2q−27 + 3q−29 + 3q−31 + 3q−33−q−37−4q−39−3q−41−3q−43−q−45 + q−49 + q−51

A4 Invariants.

Weight	Invariant
0,1,0,0	q−30 + q−34 + q−36 + q−38 + 2q−42 + 2q−44 + 4q−46 + 5q−48 + 5q−50 + 7q−52 + 7q−54 + 5q−56 + 4q−58 + 4q−60 + 2q−62−2q−64−4q−66−6q−68−10q−70−14q−72−12q−74−12q−76−10q−78−2q−80 + 3q−82 + 6q−84 + 8q−86 + 10q−88 + 6q−90 + 2q−92−2q−98−2q−100
1,0,0,0	q−20 + q−24 + q−28 + q−32 + 2q−34 + 3q−36 + 4q−38 + 3q−40 + 3q−42−q−46−4q−48−4q−50−4q−52−3q−54−q−56 + q−60 + q−62 + q−64

B2 Invariants.

Weight	Invariant
0,1	q−20 + q−24−q−26 + 2q−28−q−30 + 2q−32 + q−34 + q−36 + 2q−38 + 2q−42−2q−44 + 3q−46−3q−48 + q−50−2q−52−q−56−q−58 + q−60−q−62 + q−64−q−66 + q−68−2q−70
1,0	q−30 + q−38 + q−40−q−44 + q−46 + 2q−48 + q−50−q−52 + 2q−54 + 3q−56 + 3q−58 + q−60 + q−62 + q−64 + 2q−66−q−70−q−72−q−76−2q−78−2q−80−q−82−2q−86−3q−88−2q−90−q−94−2q−96−q−98 + q−100 + q−102 + q−108 + q−110 + 2q−112

D4 Invariants.

Weight	Invariant
1,0,0,0	q−30 + q−34 + 2q−38−q−40 + 2q−42 + 3q−46 + 3q−48 + 3q−50 + 5q−52 + 5q−54 + 6q−56 + 3q−58 + 4q−60−2q−62−6q−66−4q−68−8q−70−5q−72−6q−74−3q−76−2q−78−q−80 + q−82 + 2q−86 + 2q−90 + q−94 + 2q−98

G2 Invariants.

Weight	Invariant
1,0	q−50 + q−54−q−56 + q−58 + 3q−64−2q−66 + 4q−68−q−70−q−72 + 3q−74−2q−76 + 2q−78−q−82 + 3q−84 + 3q−90−4q−92 + 4q−94−2q−98 + 5q−100−2q−102 + 4q−104 + q−106 + 3q−108 + 2q−110−2q−112 + q−114 + 3q−118−q−120−3q−122−q−124 + q−126−q−130−9q−132 + q−136−4q−138−7q−142 + q−144 + 3q−146−3q−148−2q−150−2q−152 + q−154 + 3q−156−q−158−q−160 + 2q−162 + 2q−166 + 2q−168−q−170 + q−172

.

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["10 142"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−3t2 + 2t−1 + 2t−1−3t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 9z4 + 8z2 + 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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

−2q10 + 2q9−2q8 + 3q7−2q6 + 2q5−q4 + q3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−6 + z6a−8 + 5z4a−6 + 5z4a−8−z4a−10 + 6z2a−6 + 7z2a−8−5z2a−10 + a−6 + 4a−8−5a−10 + a−12

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−8 + z8a−10 + z7a−7 + 2z7a−9 + z7a−11 + z6a−6−5z6a−8−6z6a−10−4z5a−7−9z5a−9−5z5a−11−5z4a−6 + 9z4a−8 + 15z4a−10 + z4a−12 + 3z3a−7 + 12z3a−9 + 9z3a−11 + 6z2a−6−10z2a−8−17z2a−10−z2a−12−6za−9−4za−11 + 2za−13−a−6 + 4a−8 + 5a−10 + a−12

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["10 142"];

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]=

{ 2t3−3t2 + 2t−1 + 2t−1−3t−2 + 2t−3, −2q10 + 2q9−2q8 + 3q7−2q6 + 2q5−q4 + q3 }

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	q29−q26 + q25−q24−3q23 + 3q22−3q20 + 2q19 + 3q18−4q17 + 4q15−4q14−q13 + 5q12−2q11−2q10 + 3q9−q7 + q6
3	−2q55 + 2q53 + 4q52−5q51−3q50 + 2q49 + 10q48−2q47−10q46−2q45 + 12q44 + 3q43−13q42−4q41 + 11q40 + 5q39−12q38−3q37 + 9q36 + 4q35−9q34−q33 + 4q32 + 3q31−4q30−q29−q28 + 2q27 + q26 + 2q25−4q24−2q23 + 2q22 + 5q21−2q20−4q19−q18 + 5q17 + q16−2q15−2q14 + 2q13 + q12−q10 + q9
4	q90 + 2q88−2q87−4q86−q85−q84 + 10q83 + 5q82−9q81−11q80−11q79 + 19q78 + 23q77−4q76−20q75−30q74 + 19q73 + 36q72 + 8q71−22q70−42q69 + 12q68 + 39q67 + 15q66−19q65−45q64 + 9q63 + 37q62 + 13q61−14q60−41q59 + 6q58 + 34q57 + 12q56−9q55−36q54−q53 + 29q52 + 11q51−q50−28q49−10q48 + 18q47 + 9q46 + 9q45−14q44−13q43 + 6q42 + q41 + 12q40−q39−8q38 + 2q37−8q36 + 5q35 + 3q34 + 7q32−9q31−q30−2q29−q28 + 10q27−2q26−4q24−4q23 + 6q22 + 2q20−q19−3q18 + 2q17 + q15−q13 + q12
5	−2q132 + 2q129 + 2q128 + 4q127−q126−3q125−7q124−5q123 + 2q122 + 10q121 + 14q120 + 10q119−13q118−27q117−23q116 + 37q114 + 48q113 + 11q112−45q111−63q110−33q109 + 41q108 + 83q107 + 51q106−37q105−89q104−64q103 + 26q102 + 92q101 + 74q100−19q99−90q98−76q97 + 15q96 + 86q95 + 75q94−12q93−85q92−71q91 + 11q90 + 81q89 + 71q88−12q87−79q86−65q85 + 8q84 + 72q83 + 66q82−2q81−66q80−63q79−5q78 + 53q77 + 63q76 + 17q75−44q74−59q73−24q72 + 24q71 + 53q70 + 36q69−13q68−42q67−36q66−6q65 + 27q64 + 38q63 + 16q62−13q61−27q60−24q59−3q58 + 17q57 + 20q56 + 14q55−4q54−15q53−14q52−5q51 + 2q50 + 12q49 + 9q48 + 3q47−3q46−5q45−8q44−3q43 + q42 + 5q41 + 4q40 + 5q39−q38−3q37−5q36−4q35 + 5q33 + 3q32 + 3q31−q30−4q29−3q28 + 2q27 + 2q25 + 2q24−q23−2q22 + q21 + q18−q16 + q15
6	q183 + 2q181−2q179−4q178−4q177−3q176 + q175 + 10q174 + 9q173 + 8q172−q171−7q170−25q169−23q168−q167 + 14q166 + 37q165 + 38q164 + 33q163−28q162−65q161−63q160−45q159 + 27q158 + 88q157 + 136q156 + 42q155−58q154−131q153−156q152−60q151 + 83q150 + 227q149 + 151q148 + 16q147−138q146−234q145−161q144 + 27q143 + 247q142 + 211q141 + 87q140−101q139−245q138−204q137−20q136 + 228q135 + 215q134 + 111q133−76q132−230q131−205q130−29q129 + 216q128 + 203q127 + 106q126−73q125−220q124−197q123−25q122 + 211q121 + 192q120 + 103q119−68q118−208q117−190q116−33q115 + 193q114 + 178q113 + 110q112−43q111−180q110−182q109−60q108 + 145q107 + 154q106 + 126q105 + 5q104−129q103−167q102−100q101 + 70q100 + 110q99 + 135q98 + 65q97−52q96−127q95−126q94−12q93 + 37q92 + 110q91 + 102q90 + 30q89−50q88−104q87−62q86−41q85 + 38q84 + 80q83 + 69q82 + 28q81−32q80−43q79−68q78−29q77 + 12q76 + 37q75 + 49q74 + 23q73 + 14q72−29q71−33q70−28q69−16q68 + 14q67 + 14q66 + 32q65 + 13q64 + 4q63−11q62−22q61−8q60−17q59 + 7q58 + 8q57 + 15q56 + 8q55 + 6q53−16q52−4q51−7q50 + q49−q48 + 2q47 + 14q46−3q45 + 4q44−3q43−q42−8q41−5q40 + 7q39−2q38 + 5q37 + 2q36 + 3q35−4q34−4q33 + 3q32−3q31 + q30 + q29 + 3q28−q27−2q26 + 2q25−q24 + q21−q19 + q18