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.org/wiki/KnotTheory.

In[3]:=

K = Knot["6 3"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 2 12 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2112]

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,−1,2,−1,2,2})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 6_3.

A1 Invariants.

Weight	Invariant
1	−q7 + q5 + q + q−1 + q−5−q−7
2	q20−q18−2q16 + 2q14−2q10 + 2q8 + q6−q4 + q2 + 1 + q−2−q−4 + q−6 + 2q−8−2q−10 + 2q−14−2q−16−q−18 + q−20
3	−q39 + q37 + 2q35−3q31−2q29 + 4q27 + 2q25−4q23−4q21 + 3q19 + 4q17−2q15−4q13 + 3q11 + 4q9−2q5 + q + q−1−2q−5 + 4q−9 + 3q−11−4q−13−2q−15 + 4q−17 + 3q−19−4q−21−4q−23 + 2q−25 + 4q−27−2q−29−3q−31 + 2q−35 + q−37−q−39
4	q64−q62−2q60 + q56 + 5q54−4q50−4q48−2q46 + 10q44 + 5q42−4q40−9q38−8q36 + 9q34 + 9q32 + q30−9q28−11q26 + 7q24 + 10q22 + 4q20−6q18−9q16 + 3q14 + 6q12 + 4q10−2q8−5q6 + 2q2 + 3 + 2q−2−5q−6−2q−8 + 4q−10 + 6q−12 + 3q−14−9q−16−6q−18 + 4q−20 + 10q−22 + 7q−24−11q−26−9q−28 + q−30 + 9q−32 + 9q−34−8q−36−9q−38−4q−40 + 5q−42 + 10q−44−2q−46−4q−48−4q−50 + 5q−54 + q−56−2q−60−q−62 + q−64
5	−q95 + q93 + 2q91−q87−3q85−3q83 + 6q79 + 6q77 + q75−6q73−10q71−6q69 + 5q67 + 17q65 + 13q63−3q61−17q59−20q57−5q55 + 17q53 + 25q51 + 9q49−15q47−27q45−17q43 + 12q41 + 27q39 + 19q37−6q35−25q33−19q31 + 4q29 + 21q27 + 17q25−16q21−16q19 + 11q15 + 11q13 + 3q11−6q9−8q7−3q5 + 3q3 + 6q + 6q−1 + 3q−3−3q−5−8q−7−6q−9 + 3q−11 + 11q−13 + 11q−15−16q−19−16q−21 + 17q−25 + 21q−27 + 4q−29−19q−31−25q−33−6q−35 + 19q−37 + 27q−39 + 12q−41−17q−43−27q−45−15q−47 + 9q−49 + 25q−51 + 17q−53−5q−55−20q−57−17q−59−3q−61 + 13q−63 + 17q−65 + 5q−67−6q−69−10q−71−6q−73 + q−75 + 6q−77 + 6q−79−3q−83−3q−85−q−87 + 2q−91 + q−93−q−95
6	q132−q130−2q128 + q124 + 3q122 + q120 + 3q118−2q116−8q114−5q112−q110 + 6q108 + 8q106 + 14q104 + q102−14q100−19q98−15q96 + 13q92 + 38q90 + 25q88−4q86−29q84−41q82−28q80−q78 + 49q76 + 53q74 + 26q72−18q70−53q68−58q66−30q64 + 39q62 + 64q60 + 53q58 + 7q56−43q54−66q52−49q50 + 18q48 + 53q46 + 57q44 + 22q42−24q40−53q38−47q36 + 4q34 + 33q32 + 43q30 + 21q28−9q26−32q24−33q22−3q20 + 14q18 + 23q16 + 15q14 + q12−13q10−17q8−7q6 + 2q4 + 11q2 + 13 + 11q−2 + 2q−4−7q−6−17q−8−13q−10 + q−12 + 15q−14 + 23q−16 + 14q−18−3q−20−33q−22−32q−24−9q−26 + 21q−28 + 43q−30 + 33q−32 + 4q−34−47q−36−53q−38−24q−40 + 22q−42 + 57q−44 + 53q−46 + 18q−48−49q−50−66q−52−43q−54 + 7q−56 + 53q−58 + 64q−60 + 39q−62−30q−64−58q−66−53q−68−18q−70 + 26q−72 + 53q−74 + 49q−76−q−78−28q−80−41q−82−29q−84−4q−86 + 25q−88 + 38q−90 + 13q−92−15q−96−19q−98−14q−100 + q−102 + 14q−104 + 8q−106 + 6q−108−q−110−5q−112−8q−114−2q−116 + 3q−118 + q−120 + 3q−122 + q−124−2q−128−q−130 + q−132

A2 Invariants.

Weight	Invariant
1,0	−q10 + 2q2 + 1 + 2q−2−q−10
1,1	q28−2q26 + 4q24−6q22 + 7q20−10q18 + 8q16−8q14 + 4q12−4q8 + 10q6−11q4 + 18q2−14 + 18q−2−11q−4 + 10q−6−4q−8 + 4q−12−8q−14 + 8q−16−10q−18 + 7q−20−6q−22 + 4q−24−2q−26 + q−28
2,0	q26−q22−q20−2q14 + q10 + 2q4 + 2q2 + 2 + 2q−2 + 2q−4 + q−10−2q−14−q−20−q−22 + q−26
3,0	−q48 + q44 + 2q42 + q40−2q38−q36 + 3q32−4q28−4q26−q24 + 2q22−q20−3q18 + 4q14 + 5q12 + 2q10 + 2q6 + q4−2 + q−4 + 2q−6 + 2q−10 + 5q−12 + 4q−14−3q−18−q−20 + 2q−22−q−24−4q−26−4q−28 + 3q−32−q−36−2q−38 + q−40 + 2q−42 + q−44−q−48

A3 Invariants.

Weight	Invariant
0,1,0	q22−q20 + q16−3q14−q12−2q8 + 3q4 + 3q2 + 4 + 3q−2 + 3q−4−2q−8−q−12−3q−14 + q−16−q−20 + q−22
1,0,0	−q13−q9 + 2q3 + 2q + 2q−1 + 2q−3−q−9−q−13
1,0,1	q36−2q34 + 3q32−q30−3q28 + 7q26−9q24 + 6q22−q20−9q18 + 10q16−15q14 + 5q12−q10−7q8 + 11q6 + 8q2 + 9 + 8q−2 + 11q−6−7q−8−q−10 + 5q−12−15q−14 + 10q−16−9q−18−q−20 + 6q−22−9q−24 + 7q−26−3q−28−q−30 + 3q−32−2q−34 + q−36

A4 Invariants.

Weight	Invariant
0,1,0,0	q28 + q22−3q18−3q16−2q14−3q12−3q10 + 4q6 + 3q4 + 6q2 + 8 + 6q−2 + 3q−4 + 4q−6−3q−10−3q−12−2q−14−3q−16−3q−18 + q−22 + q−28
1,0,0,0	−q16−q12−q10 + 2q4 + 2q2 + 3 + 2q−2 + 2q−4−q−10−q−12−q−16

B2 Invariants.

Weight	Invariant
0,1	−q22 + q20−2q18 + q16−q14 + q12 + 2q6−q4 + 3q2−2 + 3q−2−q−4 + 2q−6 + q−12−q−14 + q−16−2q−18 + q−20−q−22
1,0	q36−q32−q30 + q28 + q26−q24−2q22−q20 + q18−q14−q12 + q10 + q8 + q6 + 2q2 + 3 + 2q−2 + q−6 + q−8 + q−10−q−12−q−14 + q−18−q−20−2q−22−q−24 + q−26 + q−28−q−30−q−32 + q−36

D4 Invariants.

Weight	Invariant
1,0,0,0	q30−q28 + q26−q24 + q22−2q20−q18−2q16−q14−q12−2q10 + q8 + q6 + 5q4 + 2q2 + 6 + 2q−2 + 5q−4 + q−6 + q−8−2q−10−q−12−q−14−2q−16−q−18−2q−20 + q−22−q−24 + q−26−q−28 + q−30

G2 Invariants.

Weight	Invariant
1,0	q52−q50 + 2q48−2q46−q44 + q42−3q40 + 4q38−4q36 + q34−3q30 + 3q28−3q26 + q24 + q22−2q20 + q18 + q16−q14 + 4q12−3q10 + 3q8 + q6−q4 + 6q2−5 + 6q−2−q−4 + q−6 + 3q−8−3q−10 + 4q−12−q−14 + q−16 + q−18−2q−20 + q−22 + q−24−3q−26 + 3q−28−3q−30 + q−34−4q−36 + 4q−38−3q−40 + q−42−q−44−2q−46 + 2q−48−q−50 + q−52

.

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["6 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−3t + 5−3t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

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]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q3 + 2q2−2q + 3−2q−1 + 2q−2−q−3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z4−a2z2−z2a−2 + 3z2−a2−a−2 + 3

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

az5 + z5a−1 + 2a2z4 + 2z4a−2 + 4z4 + a3z3 + az3 + z3a−1 + z3a−3−3a2z2−3z2a−2−6z2−a3z−2az−2za−1−za−3 + a2 + a−2 + 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["6 3"];

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]=

{ t2−3t + 5−3t−1 + t−2, −q3 + 2q2−2q + 3−2q−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]=

{K11n12,}

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	q9−2q8−q7 + 5q6−4q5−3q4 + 9q3−5q2−5q + 11−5q−1−5q−2 + 9q−3−3q−4−4q−5 + 5q−6−q−7−2q−8 + q−9
3	−q18 + 2q17 + q16−2q15−4q14 + 3q13 + 7q12−4q11−10q10 + 3q9 + 14q8−3q7−16q6 + q5 + 21q4−2q3−20q2−q + 23−q−1−20q−2−2q−3 + 21q−4 + q−5−16q−6−3q−7 + 14q−8 + 3q−9−10q−10−4q−11 + 7q−12 + 3q−13−4q−14−2q−15 + q−16 + 2q−17−q−18
4	q30−2q29−q28 + 2q27 + q26 + 5q25−7q24−5q23 + 2q22 + 3q21 + 17q20−12q19−14q18−3q17 + 4q16 + 34q15−12q14−22q13−13q12 + 2q11 + 52q10−9q9−28q8−23q7−q6 + 64q5−6q4−30q3−29q2−4q + 69−4q−1−29q−2−30q−3−6q−4 + 64q−5−q−6−23q−7−28q−8−9q−9 + 52q−10 + 2q−11−13q−12−22q−13−12q−14 + 34q−15 + 4q−16−3q−17−14q−18−12q−19 + 17q−20 + 3q−21 + 2q−22−5q−23−7q−24 + 5q−25 + q−26 + 2q−27−q−28−2q−29 + q−30
5	−q45 + 2q44 + q43−2q42−q41−2q40−q39 + 5q38 + 7q37−2q36−6q35−9q34−5q33 + 9q32 + 18q31 + 10q30−10q29−25q28−19q27 + 6q26 + 33q25 + 32q24−2q23−41q22−43q21−6q20 + 43q19 + 61q18 + 13q17−49q16−68q15−25q14 + 49q13 + 84q12 + 30q11−53q10−85q9−41q8 + 49q7 + 100q6 + 41q5−53q4−93q3−50q2 + 47q + 105 + 47q−1−50q−2−93q−3−53q−4 + 41q−5 + 100q−6 + 49q−7−41q−8−85q−9−53q−10 + 30q−11 + 84q−12 + 49q−13−25q−14−68q−15−49q−16 + 13q−17 + 61q−18 + 43q−19−6q−20−43q−21−41q−22−2q−23 + 32q−24 + 33q−25 + 6q−26−19q−27−25q−28−10q−29 + 10q−30 + 18q−31 + 9q−32−5q−33−9q−34−6q−35−2q−36 + 7q−37 + 5q−38−q−39−2q−40−q−41−2q−42 + q−43 + 2q−44−q−45
6	q63−2q62−q61 + 2q60 + q59 + 2q58−2q57 + 3q56−7q55−7q54 + 5q53 + 5q52 + 9q51 + 9q49−20q48−22q47 + 9q45 + 24q44 + 13q43 + 34q42−33q41−51q40−25q39−3q38 + 37q37 + 40q36 + 84q35−29q34−78q33−69q32−38q31 + 32q30 + 68q29 + 153q28−4q27−89q26−115q25−88q24 + 9q23 + 85q22 + 220q21 + 31q20−85q19−150q18−134q17−20q16 + 91q15 + 271q14 + 60q13−75q12−172q11−164q10−43q9 + 90q8 + 301q7 + 77q6−66q5−180q4−178q3−57q2 + 86q + 311 + 86q−1−57q−2−178q−3−180q−4−66q−5 + 77q−6 + 301q−7 + 90q−8−43q−9−164q−10−172q−11−75q−12 + 60q−13 + 271q−14 + 91q−15−20q−16−134q−17−150q−18−85q−19 + 31q−20 + 220q−21 + 85q−22 + 9q−23−88q−24−115q−25−89q−26−4q−27 + 153q−28 + 68q−29 + 32q−30−38q−31−69q−32−78q−33−29q−34 + 84q−35 + 40q−36 + 37q−37−3q−38−25q−39−51q−40−33q−41 + 34q−42 + 13q−43 + 24q−44 + 9q−45−22q−47−20q−48 + 9q−49 + 9q−51 + 5q−52 + 5q−53−7q−54−7q−55 + 3q−56−2q−57 + 2q−58 + q−59 + 2q−60−q−61−2q−62 + q−63
7	−q84 + 2q83 + q82−2q81−q80−2q79 + 2q78−q76 + 7q75 + 4q74−4q73−5q72−11q71−q70 + 3q69−q68 + 21q67 + 16q66 + q65−9q64−34q63−21q62−8q61−4q60 + 44q59 + 49q58 + 30q57 + 12q56−59q55−68q54−55q53−41q52 + 58q51 + 94q50 + 98q49 + 78q48−50q47−118q46−138q45−128q44 + 28q43 + 126q42 + 181q41 + 195q40 + 7q39−135q38−224q37−250q36−50q35 + 119q34 + 256q33 + 322q32 + 101q31−115q30−281q29−371q28−149q27 + 86q26 + 299q25 + 433q24 + 191q23−75q22−312q21−460q20−232q19 + 45q18 + 319q17 + 506q16 + 260q15−44q14−321q13−512q12−284q11 + 14q10 + 322q9 + 547q8 + 298q7−23q6−320q5−534q4−309q3−7q2 + 316q + 559 + 316q−1−7q−2−309q−3−534q−4−320q−5−23q−6 + 298q−7 + 547q−8 + 322q−9 + 14q−10−284q−11−512q−12−321q−13−44q−14 + 260q−15 + 506q−16 + 319q−17 + 45q−18−232q−19−460q−20−312q−21−75q−22 + 191q−23 + 433q−24 + 299q−25 + 86q−26−149q−27−371q−28−281q−29−115q−30 + 101q−31 + 322q−32 + 256q−33 + 119q−34−50q−35−250q−36−224q−37−135q−38 + 7q−39 + 195q−40 + 181q−41 + 126q−42 + 28q−43−128q−44−138q−45−118q−46−50q−47 + 78q−48 + 98q−49 + 94q−50 + 58q−51−41q−52−55q−53−68q−54−59q−55 + 12q−56 + 30q−57 + 49q−58 + 44q−59−4q−60−8q−61−21q−62−34q−63−9q−64 + q−65 + 16q−66 + 21q−67−q−68 + 3q−69−q−70−11q−71−5q−72−4q−73 + 4q−74 + 7q−75−q−76 + 2q−78−2q−79−q−80−2q−81 + q−82 + 2q−83−q−84