8 19

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["8 19"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,2-]

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(3,{1,1,1,2,1,1,1,2})

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]=

{ 3, 8, 3 }

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}, {2, 6}, {10, 3}, {9, 7}, {8, 2}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 8_19.

A1 Invariants.

Weight	Invariant
1	q−5 + q−7 + q−9 + q−11−q−15−q−17
2	q−10 + q−12 + q−14 + q−16 + q−18 + q−20 + q−22−q−28−q−30−q−32−q−34−q−36 + q−48
3	q−15 + q−17 + q−19 + q−21 + q−23 + q−25 + q−27 + q−29 + q−31 + q−33−q−41−q−43−q−45−q−47−q−49−q−51−q−53−q−55 + q−77 + q−79 + q−81 + q−83−q−87−q−89
4	q−20 + q−22 + q−24 + q−26 + q−28 + q−30 + q−32 + q−34 + q−36 + q−38 + q−40 + q−42 + q−44−q−54−q−56−q−58−q−60−q−62−q−64−q−66−q−68−q−70−q−72−q−74 + q−106 + q−108 + q−110 + q−112 + q−114 + q−116 + q−118−q−124−q−126−q−128−q−130−q−132 + q−144
5	q−25 + q−27 + q−29 + q−31 + q−33 + q−35 + q−37 + q−39 + q−41 + q−43 + q−45 + q−47 + q−49 + q−51 + q−53 + q−55−q−67−q−69−q−71−q−73−q−75−q−77−q−79−q−81−q−83−q−85−q−87−q−89−q−91−q−93 + q−135 + q−137 + q−139 + q−141 + q−143 + q−145 + q−147 + q−149 + q−151 + q−153−q−161−q−163−q−165−q−167−q−169−q−171−q−173−q−175 + q−197 + q−199 + q−201 + q−203−q−207−q−209
6	q−30 + q−32 + q−34 + q−36 + q−38 + q−40 + q−42 + q−44 + q−46 + q−48 + q−50 + q−52 + q−54 + q−56 + q−58 + q−60 + q−62 + q−64 + q−66−q−80−q−82−q−84−q−86−q−88−q−90−q−92−q−94−q−96−q−98−q−100−q−102−q−104−q−106−q−108−q−110−q−112 + q−164 + q−166 + q−168 + q−170 + q−172 + q−174 + q−176 + q−178 + q−180 + q−182 + q−184 + q−186 + q−188−q−198−q−200−q−202−q−204−q−206−q−208−q−210−q−212−q−214−q−216−q−218 + q−250 + q−252 + q−254 + q−256 + q−258 + q−260 + q−262−q−268−q−270−q−272−q−274−q−276 + q−288

A2 Invariants.

Weight	Invariant
1,0	q−10 + q−12 + 2q−14 + 2q−16 + 2q−18−q−22−2q−24−2q−26−q−28 + q−32
1,1	q−20 + 2q−22 + 4q−24 + 6q−26 + 7q−28 + 6q−30 + 4q−32 + 2q−34−q−36−4q−38−6q−40−8q−42−7q−44−6q−46−4q−48 + 2q−52 + 4q−54 + 4q−56 + 4q−58 + q−60−2q−64−2q−66−q−68 + 2q−72
2,0	q−20 + q−22 + 2q−24 + 2q−26 + 3q−28 + 3q−30 + 4q−32 + 3q−34 + 3q−36 + q−38−2q−42−3q−44−5q−46−5q−48−5q−50−4q−52−3q−54−2q−56 + q−58 + 2q−60 + 4q−62 + 4q−64 + 4q−66 + q−68−2q−72−2q−74−q−76 + q−80

A3 Invariants.

Weight	Invariant
0,1,0	q−20 + q−22 + 3q−24 + 4q−26 + 6q−28 + 5q−30 + 5q−32 + q−34−2q−36−6q−38−7q−40−7q−42−5q−44−q−46 + q−48 + 3q−50 + 2q−52 + 2q−54
1,0,0	q−15 + q−17 + 2q−19 + 3q−21 + 3q−23 + 2q−25 + q−27−q−29−3q−31−3q−33−3q−35−q−37 + q−41 + q−43
1,0,1	q−30 + 2q−32 + 5q−34 + 9q−36 + 14q−38 + 17q−40 + 19q−42 + 16q−44 + 9q−46−10q−50−19q−52−25q−54−27q−56−23q−58−16q−60−7q−62 + 3q−64 + 9q−66 + 15q−68 + 15q−70 + 11q−72 + 7q−74 + 2q−76−2q−78−4q−80−3q−82−3q−84−q−86−q−88 + 2q−96

A4 Invariants.

Weight	Invariant
0,1,0,0	q−30 + q−32 + 3q−34 + 5q−36 + 8q−38 + 9q−40 + 12q−42 + 10q−44 + 8q−46 + 2q−48−4q−50−11q−52−15q−54−17q−56−15q−58−10q−60−5q−62 + 2q−64 + 5q−66 + 8q−68 + 7q−70 + 6q−72 + 3q−74 + q−76−q−78−q−80−q−82−q−84
1,0,0,0	q−20 + q−22 + 2q−24 + 3q−26 + 4q−28 + 3q−30 + 3q−32 + q−34−q−36−3q−38−4q−40−4q−42−3q−44−q−46 + q−50 + q−52 + q−54

B2 Invariants.

Weight	Invariant
0,1	q−20 + q−22 + q−24 + 2q−26 + 2q−28 + q−30 + q−32 + q−34−q−40−q−42−q−44−q−46−q−48−q−50
1,0	q−30 + q−34 + q−36 + 2q−38 + q−40 + 3q−42 + 2q−44 + 3q−46 + 2q−48 + 2q−50 + q−52 + q−54−q−56−q−58−2q−60−3q−62−3q−64−3q−66−3q−68−3q−70−q−72−q−74 + 2q−80 + q−82 + q−84 + q−86 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q−30 + q−32 + 2q−34 + 4q−36 + 5q−38 + 6q−40 + 7q−42 + 6q−44 + 4q−46 + 2q−48−2q−50−5q−52−7q−54−8q−56−8q−58−6q−60−4q−62−q−64 + q−66 + 2q−68 + 3q−70 + 2q−72 + 2q−74 + q−76

G2 Invariants.

Weight	Invariant
1,0	q−50 + q−52 + q−54 + q−56 + q−58 + q−60 + 2q−62 + 2q−64 + q−66 + q−68 + 2q−70 + 2q−72 + 2q−74 + q−76 + q−80 + 2q−82−q−94−2q−96−q−98−q−100−2q−102−2q−104−2q−106−q−108−q−110−2q−112−2q−114−q−116−q−122−q−124 + q−126 + q−128 + q−136 + q−138 + q−144

.

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["8 19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−t2 + 1−t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + 5z4 + 5z2 + 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]=

{ 3, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + q5 + q3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−6 + 6z4a−6−z4a−8 + 10z2a−6−5z2a−8 + 5a−6−5a−8 + a−10

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a−6 + z6a−8 + z5a−7 + z5a−9−6z4a−6−6z4a−8−5z3a−7−5z3a−9 + 10z2a−6 + 10z2a−8 + 5za−7 + 5za−9−5a−6−5a−8−a−10

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["8 19"];

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−t2 + 1−t−2 + t−3, −q8 + q5 + 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	q23−q22 + q20−q19−q16−q13 + q12 + q9 + q6
3	−q43 + q41 + q40−q39 + q37−q35 + q33−q31 + q29−q27−q26 + q25−q23−q22 + q21−q19 + q17 + q13 + q9
4	q70−q69 + q65−2q64 + q60−q59 + q57 + q55−q54 + q52−q49 + q47−q44 + q42−q39 + q37−q35−q34 + q32−q30−q29 + q27−q25 + q22 + q17 + q12
5	−q102 + q100 + q99−q96−q95 + q94 + q93−q90−q89 + q88 + q87−q85−q84−q83 + q82 + q81−q79−q78 + q76 + q75 + q74−q73−q72 + q70 + q69 + q68−q67−q66 + q63 + q62−q61−q60 + q57 + q56−q55−q54 + q51 + q50−q49−q48 + q45−q43−q42 + q39−q37−q36 + q33−q31 + q27 + q21 + q15
6	q141−q140−q135 + q134−q133 + q130 + q127−q126 + q123−q121 + q120−q119 + q116−q114 + q113−q112 + q109−q107−q105 + q102−q100−q98 + 2q95−q93 + 2q88−q86 + 2q81−q79−q78 + 2q74−q72−q71 + 2q67−q65−q64 + 2q60−q58−q57 + q53−q51−q50 + q46−q44−q43 + q39−q37 + q32 + q25 + q18
7	−q185 + q183 + q182−q178−q177 + q175 + q174−q170−q169−q168 + q167 + q166−q162−q161 + q159 + q158 + q157−q154−q153 + q151 + q150 + q149−q147−q146−q145 + q143 + q142 + q141−q139−q138−q137 + q135 + q134 + q133−q131−q130−q129 + q126 + q125−q123−q122−q121 + q118 + q117−q115−q114 + q110 + q109 + q108−q107−q106 + q102 + q101 + q100−q99−q98 + q94 + q93−q91−q90 + q86 + q85−q83−q82 + q78 + q77−q75−q74 + q70 + q69−q67−q66 + q61−q59−q58 + q53−q51−q50 + q45−q43 + q37 + q29 + q21