10 128

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["10 128"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 -14 2 -16 -18 -20 -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]=

[32,3,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_128.

A1 Invariants.

Weight	Invariant
1	q−5 + q−9 + q−11 + q−13−q−17−q−21
2	q−10 + 2q−16 + q−18 + q−22 + q−24 + q−26−q−28 + q−32−q−34−q−36−2q−40−q−42 + q−52−q−56 + q−60
3	q−15 + q−21 + 2q−23 + q−25−q−27 + 2q−31 + 2q−33 + q−35−q−41 + q−43 + 2q−45−3q−49−2q−51 + q−53 + q−55−2q−57−3q−59 + q−61 + 2q−63−q−65−3q−67−q−73−q−75 + q−77 + q−81 + q−83−q−85 + 3q−89 + 2q−91−3q−93−3q−95 + 3q−97 + 3q−99−2q−101−3q−103 + 3q−107 + q−109−q−111−q−113

A2 Invariants.

Weight	Invariant
1,0	q−10 + q−14 + q−16 + 2q−18 + q−20 + q−22 + q−24−q−26−q−28−2q−30−q−32−q−34 + q−38
1,1	q−20 + 2q−24 + 6q−28 + 8q−32 + 2q−34 + 7q−36 + 2q−38 + 2q−42−5q−44−8q−48−2q−50−9q−52−2q−54−6q−56 + 2q−58−2q−60 + 4q−62 + 2q−64 + 2q−66 + q−68 + 2q−70−2q−78 + 2q−80−4q−82 + 3q−84−2q−86 + 2q−88
2,0	q−20 + q−26 + 2q−28 + 2q−30 + 2q−32 + 2q−34 + 3q−36 + 2q−38 + 2q−40 + 2q−42 + 2q−44−q−50−2q−52−4q−54−5q−56−4q−58−3q−60−3q−62 + q−70 + q−72 + q−74 + 2q−78 + 2q−80 + 2q−82 + q−86−q−88−2q−90−q−92 + q−96

A3 Invariants.

Weight	Invariant
0,1,0	q−20 + q−24 + 2q−26 + 2q−28 + 2q−30 + 3q−32 + 3q−34 + 3q−36 + 2q−38 + 3q−40−q−44−3q−46−4q−48−6q−50−5q−52−3q−54−2q−56 + q−58 + 2q−60 + 3q−62 + q−64 + q−66
1,0,0	q−15 + q−19 + q−21 + 2q−23 + q−25 + 2q−27 + 2q−29 + q−31 + q−33−q−35−q−37−3q−39−2q−41−2q−43−q−45 + q−49 + q−51

A4 Invariants.

Weight	Invariant
0,1,0,0	q−30 + q−34 + 2q−36 + 3q−38 + 2q−40 + 4q−42 + 4q−44 + 5q−46 + 4q−48 + 4q−50 + 5q−52 + 5q−54 + 3q−56 + 3q−58 + 2q−60−2q−62−6q−64−10q−66−12q−68−14q−70−12q−72−7q−74−3q−76 + q−78 + 7q−80 + 8q−82 + 6q−84 + 5q−86 + 4q−88 + q−90−q−92−q−94−q−98−q−100
1,0,0,0	q−20 + q−24 + q−26 + 2q−28 + q−30 + 2q−32 + 2q−34 + 2q−36 + 2q−38 + q−40 + q−42−q−44−q−46−3q−48−3q−50−3q−52−2q−54−q−56 + q−60 + q−62 + q−64

B2 Invariants.

Weight	Invariant
0,1	q−20 + q−24 + 2q−28 + q−32 + q−34 + q−36 + 2q−38−q−40 + 2q−42−q−44 + q−46−2q−48−q−52−q−54−q−58−q−62 + q−64−q−66
1,0	q−30 + q−38 + q−40 + q−42 + 2q−46 + 2q−48 + q−50 + 2q−54 + 2q−56 + 2q−58 + q−62 + q−64 + q−66−q−68−q−70−q−72−q−74−2q−76−3q−78−2q−80−2q−82−q−84−2q−86−2q−88−q−90 + q−92 + q−98 + 2q−100 + q−102 + q−108

D4 Invariants.

Weight	Invariant
1,0,0,0	q−30 + q−34 + q−36 + 3q−38 + q−40 + 3q−42 + 2q−44 + 3q−46 + 3q−48 + 2q−50 + 3q−52 + 2q−54 + 4q−56 + q−58 + 2q−60−2q−62−q−64−5q−66−5q−68−7q−70−6q−72−5q−74−3q−76−q−78 + 3q−82 + 2q−84 + 3q−86 + q−88 + 2q−90

G2 Invariants.

Weight	Invariant
1,0	q−50 + q−54 + q−58 + 3q−64−q−66 + 2q−68 + 2q−74 + q−78 + q−80 + q−84 + 2q−86−q−88 + 3q−90−2q−92 + 2q−94 + 2q−96−q−98 + 3q−100−q−102 + 2q−104−q−114−q−116−q−120−3q−124−q−126−3q−128 + q−130−3q−132−2q−134−3q−138 + 2q−140−2q−142−q−144−q−148 + q−152−q−154 + 2q−156 + q−158−q−160 + 2q−162−q−164 + q−166 + q−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["10 128"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−3t2 + t + 1 + t−1−3t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 9z4 + 7z2 + 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]=

{ 11, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q10 + q9−2q8 + 2q7−q6 + 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 + 6z2a−8−5z2a−10 + 2a−6 + 2a−8−4a−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−7−10z5a−9−6z5a−11−5z4a−6 + 7z4a−8 + 12z4a−10 + 2z3a−7 + 13z3a−9 + 11z3a−11 + 6z2a−6−5z2a−8−11z2a−10 + za−7−5za−9−6za−11−2a−6 + 2a−8 + 4a−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["10 128"];

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 + t + 1 + t−1−3t−2 + 2t−3, −q10 + q9−2q8 + 2q7−q6 + 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−q28−q27 + 2q26−2q24 + 2q23−2q21 + q20−q19−q16 + 2q15−q14−2q13 + 4q12−q11−2q10 + 3q9−q7 + q6
3	−q55 + 2q53 + 2q52−4q51−3q50 + 3q49 + 7q48−4q47−9q46 + 3q45 + 12q44−3q43−12q42 + 2q41 + 14q40−3q39−13q38 + 3q37 + 12q36−3q35−12q34 + 3q33 + 9q32−q31−9q30 + 2q29 + 5q28−6q26 + 2q25 + 2q24−q23−3q22 + 4q21 + q20−3q19−2q18 + 4q17 + 2q16−2q15−2q14 + 2q13 + q12−q10 + q9
4	q90−q89−q87−q86 + 3q85−q84 + 4q83−2q82−6q81 + q80−3q79 + 12q78 + 3q77−8q76−6q75−11q74 + 19q73 + 10q72−6q71−9q70−19q69 + 20q68 + 13q67−3q66−8q65−23q64 + 20q63 + 14q62−2q61−7q60−22q59 + 16q58 + 15q57−q56−6q55−21q54 + 9q53 + 15q52 + 3q51−3q50−20q49 + q48 + 13q47 + 8q46 + 2q45−17q44−7q43 + 8q42 + 8q41 + 7q40−10q39−10q38 + 4q37 + 4q36 + 6q35−4q34−7q33 + 5q32 + 2q30−3q29−5q28 + 7q27 + q26 + q25−3q24−5q23 + 5q22 + q21 + 2q20−q19−3q18 + 2q17 + q15−q13 + q12
5	−q132 + q130 + 2q129 + q128−4q126−5q125−q124 + 4q123 + 7q122 + 7q121−10q119−13q118−6q117 + 8q116 + 18q115 + 16q114−2q113−22q112−25q111−7q110 + 22q109 + 33q108 + 15q107−19q106−38q105−22q104 + 17q103 + 41q102 + 24q101−15q100−40q99−27q98 + 14q97 + 41q96 + 25q95−14q94−40q93−25q92 + 14q91 + 40q90 + 24q89−14q88−40q87−22q86 + 14q85 + 36q84 + 23q83−11q82−35q81−19q80 + 9q79 + 27q78 + 21q77−4q76−24q75−17q74 + 14q72 + 16q71 + 5q70−8q69−10q68−8q67−q66 + 6q65 + 6q64 + 8q63 + 2q62−6q61−10q60−8q59−q58 + 11q57 + 13q56 + 5q55−7q54−14q53−11q52 + 4q51 + 12q50 + 11q49−8q47−11q46−q45 + 5q44 + 6q43 + q42−3q41−5q40 + q39 + 4q38 + 3q37−2q36−4q35−4q34 + 2q33 + 4q32 + 4q31−3q29−4q28 + q27 + q26 + 2q25 + 2q24−q23−2q22 + q21 + q18−q16 + q15
6	q183−q182−q179−q178−q177 + 3q176 + q175 + 4q174 + 2q173−q172−7q171−8q170−4q169−3q168 + 13q167 + 15q166 + 15q165−2q164−13q163−26q162−30q161 + q160 + 21q159 + 46q158 + 31q157 + 13q156−38q155−68q154−40q153−5q152 + 60q151 + 69q150 + 64q149−22q148−85q147−79q146−44q145 + 50q144 + 84q143 + 103q142−q141−81q140−94q139−66q138 + 37q137 + 83q136 + 118q135 + 6q134−75q133−95q132−72q131 + 33q130 + 81q129 + 123q128 + 4q127−74q126−95q125−74q124 + 33q123 + 82q122 + 122q121 + 3q120−69q119−94q118−74q117 + 30q116 + 78q115 + 117q114 + 6q113−58q112−88q111−74q110 + 20q109 + 64q108 + 109q107 + 16q106−34q105−75q104−76q103−q102 + 41q101 + 94q100 + 28q99−q98−50q97−71q96−25q95 + 11q94 + 67q93 + 29q92 + 30q91−17q90−50q89−33q88−15q87 + 30q86 + 11q85 + 38q84 + 8q83−18q82−16q81−16q80 + 4q79−14q78 + 20q77 + 6q76−2q75 + 7q74 + 4q73 + 7q72−19q71−12q69−12q68 + 11q67 + 15q66 + 18q65−3q64 + q63−16q62−22q61 + 2q60 + 6q59 + 14q58 + 5q57 + 8q56−7q55−16q54 + 2q53−q52 + 6q51 + q50 + 5q49−5q48−10q47 + 6q46 + 6q44 + q43 + 3q42−6q41−9q40 + 4q39−q38 + 4q37 + 3q36 + 4q35−3q34−5q33 + 2q32−2q31 + q30 + q29 + 3q28−q27−2q26 + 2q25−q24 + q21−q19 + q18