9 46

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["9 46"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_46.

A1 Invariants.

Weight	Invariant
1	q13−q7−q5 + q + 2q−1
2	q38−q34−q28−q26 + q22 + q20 + q18 + 2q16−2q8−2q6−q4 + q2 + 1 + q−2 + q−4 + q−6
3	q75−q71−q69 + q65−q61−q59 + q55 + 2q53 + q51 + q45 + q43−q41−3q39−q37−q33−2q31−q29 + q27 + q25 + q23 + 2q21 + 3q19 + q17 + q15−q9−q7−3q5−2q3 + q + 3q−1 + q−3−3q−5 + 2q−9 + 2q−11
4	q124−q120−q118−q116 + q114 + q112 + q110−2q106−q104 + q100 + 2q98 + 2q96 + q94−q92−2q90−q88 + q86 + q84 + q82−2q80−4q78−3q76 + 3q72 + q70−2q66−q64 + 3q62 + 4q60 + 3q58−2q54 + q52 + 2q50 + 2q48−q46−4q44−2q42−q40−q38−2q36−3q34−q32 + q30 + q28 + 3q22 + 3q20 + 3q18 + 2q16−2q12−q10 + q8 + 2q6 + q4−4q2−3−q−2 + 4q−4 + 4q−6−2q−8−3q−10−3q−12 + q−14 + 3q−16 + q−18 + q−20
5	q185−q181−q179−q177 + q173 + 2q171 + q169−q165−2q163−2q161 + 2q157 + 2q155 + 2q153 + 2q151−2q147−3q145−3q143−q141 + 2q139 + 3q137 + 2q135−3q131−5q129−4q127−q125 + 4q123 + 6q121 + 4q119 + q117−3q115−4q113−2q111 + 3q109 + 6q107 + 6q105 + 2q103−3q101−7q99−6q97 + 4q93 + 4q91 + q89−5q87−8q85−5q83 + q81 + 4q79 + 4q77−4q73−3q71 + q69 + 6q67 + 7q65 + 3q63−2q61−2q59 + q57 + 4q55 + 3q53−3q49−2q47−q45−3q41−4q39−3q37−q35 + q33 + q31−q27−q25−q23 + q21 + 3q19 + 5q17 + 5q15 + 3q13−4q11−6q9−5q7 + q5 + 7q3 + 10q + 5q−1−5q−3−10q−5−7q−7 + q−9 + 7q−11 + 7q−13 + q−15−5q−17−5q−19−2q−21 + 2q−25 + 2q−27 + 2q−29

A2 Invariants.

Weight	Invariant
1,0	q20 + q18−q12−q10−q8−q6 + q2 + 2 + 2q−2
1,1	q52 + 2q48−2q46 + 2q44−2q42−2q38−4q36−4q32 + 2q30 + q28 + 6q26 + 4q24 + 6q22 + 3q20 + 2q18−2q16−4q14−4q12−6q10−4q8−2q6−q4 + 2q2 + 4 + 4q−2 + 4q−4 + 2q−8
2,0	q52 + q50 + q48−q46−q44−q42−q40−q38−2q36−q34−q32 + q30 + 2q28 + 3q26 + 4q24 + 4q22 + 2q20−q16−2q14−3q12−3q10−3q8−2q6−q4 + 2 + 2q−2 + 4q−4 + 2q−6 + q−8

A3 Invariants.

Weight	Invariant
0,1,0	q40 + q36−q32−2q30−2q28 + q24 + 4q22 + 4q20 + 4q18−2q14−5q12−6q10−4q8−2q6 + 2q4 + 3q2 + 5 + 3q−2 + 2q−4
1,0,0	q27 + q25 + q23−q17−q15−q13−q11−q9−q7 + q3 + 2q + 2q−1 + 2q−3

A4 Invariants.

Weight	Invariant
0,1,0,0	q54 + q52 + q50 + q48 + q46−q44−2q42−3q40−4q38−4q36−2q34 + 2q32 + 4q30 + 7q28 + 9q26 + 8q24 + 4q22 + q20−4q18−8q16−10q14−9q12−7q10−4q8 + q6 + 4q4 + 6q2 + 6 + 6q−2 + 3q−4 + 2q−6
1,0,0,0	q34 + q32 + q30 + q28−q22−q20−q18−q16−q14−q12−q10−q8 + q4 + 2q2 + 2 + 2q−2 + 2q−4

B2 Invariants.

Weight	Invariant
0,1	q40 + q36 + q32−q24−2q20−2q16−q12 + 2q4 + q2 + 1 + q−2 + 2q−4
1,0	q66 + q58−q54−q52−q48−2q46−q44 + q40 + q38 + 2q36 + 2q34 + 3q32 + 2q30 + 2q28−q22−2q20−4q18−3q16−2q14−2q12−2q10−q8 + 2q6 + q4 + 2q2 + 2 + 3q−2 + q−4 + 2q−6

D4 Invariants.

Weight	Invariant
1,0,0,0	q54 + q50 + q46−q44−q42−q40−q38 + 2q32 + 2q30 + 4q28 + 2q26 + 2q24−q22−q20−4q18−4q16−5q14−4q12−2q10−q8 + 2q6 + 2q4 + 4q2 + 4 + 3q−2 + 2q−4 + 2q−6

G2 Invariants.

Weight	Invariant
1,0	q94 + q90−q88−q82 + 2q80 + 2q70 + q68−3q66 + q62 + 2q60 + 2q58−4q56 + 3q52 + 2q50−2q48−5q46 + 3q42 + 2q40−4q38−3q36 + 3q32−5q28−4q26 + 2q24 + 2q22−2q20−q18−3q16 + 3q14 + 2q12−q10 + 2q4 + 3q2 + 1 + q−2 + q−4 + 3q−8 + q−10−q−12 + q−14

.

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["9 46"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t + 5−2t−1

In[5]:=

Conway[K][z]

Out[5]=

1−2z2

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]=

{3,t + 1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

2−q−1 + q−2−2q−3 + q−4−q−5 + q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a6−z2a4−a4−z2a2−a2 + 2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a5z7 + a3z7 + a6z6 + 2a4z6 + a2z6−5a5z5−5a3z5−5a6z4−9a4z4−4a2z4 + 7a5z3 + 8a3z3 + az3 + 6a6z2 + 9a4z2 + 3a2z2−4a5z−6a3z−2az−a6−a4 + a2 + 2

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["9 46"];

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]=

{ −2t + 5−2t−1, 2−q−1 + q−2−2q−3 + q−4−q−5 + q−6 }

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]=

{6_1, K11n67, K11n97, K11n139,}

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	q2 + q−1−2q−3 + 2q−9−q−10 + 2q−12−2q−13−q−14 + 2q−15−q−16−q−17 + q−18
3	2q4−2q2−3q + 6 + 2q−1−4q−2−6q−3 + 5q−4 + 4q−5−4q−6−5q−7 + 5q−8 + 5q−9−4q−10−3q−11 + 4q−12 + 4q−13−4q−14−3q−15 + 2q−16 + 3q−17−3q−18−2q−19 + q−20 + q−21−q−22 + q−24−q−26 + q−27 + 2q−28−q−29−2q−30 + 2q−32−q−34−q−35 + q−36
4	q8 + 2q6−2q5−4q4 + q3 + q2 + 8q−2−9q−1−q−2 + 13q−4−q−5−10q−6−3q−7−q−8 + 15q−9 + q−10−9q−11−3q−12−q−13 + 12q−14 + q−15−8q−16−3q−17−3q−18 + 10q−19 + 2q−20−7q−21−3q−22−4q−23 + 8q−24 + 5q−25−4q−26−3q−27−5q−28 + 5q−29 + 7q−30−q−31−2q−32−6q−33 + q−34 + 6q−35 + q−36−q−37−4q−38−2q−39 + 3q−40 + q−42−q−43−2q−44 + 3q−45−2q−46 + 4q−50−2q−51−q−52−q−53−q−54 + 3q−55−q−58−q−59 + q−60
5	2q12−2q9−2q8−3q7 + 2q6 + 6q5 + 6q4−2q3−8q2−11q−1 + 11q−1 + 16q−2 + 3q−3−11q−4−17q−5−7q−6 + 10q−7 + 18q−8 + 10q−9−9q−10−17q−11−9q−12 + 8q−13 + 16q−14 + 10q−15−9q−16−16q−17−8q−18 + 8q−19 + 14q−20 + 8q−21−10q−22−15q−23−5q−24 + 7q−25 + 13q−26 + 7q−27−7q−28−12q−29−4q−30 + 4q−31 + 10q−32 + 7q−33−2q−34−8q−35−5q−36−q−37 + 6q−38 + 6q−39 + 2q−40−4q−41−5q−42−4q−43 + 3q−45 + 5q−46 + 2q−47−2q−48−4q−49−4q−50−3q−51 + 4q−52 + 6q−53 + 3q−54−4q−56−6q−57−q−58 + 4q−59 + 4q−60 + 4q−61−q−62−4q−63−3q−64−q−65 + q−66 + 3q−67 + q−68−q−69−q−70−q−74−q−75 + 2q−77 + 2q−78−q−80−q−81−2q−82 + 2q−84 + q−85−q−88−q−89 + q−90
6	q18 + 2q16−2q14−4q13−4q12−q11 + 3q10 + 12q9 + 7q8 + 2q7−10q6−15q5−17q4−2q3 + 24q2 + 22q + 19−7q−1−23q−2−37q−3−16q−4 + 24q−5 + 30q−6 + 32q−7 + 2q−8−19q−9−44q−10−25q−11 + 18q−12 + 27q−13 + 35q−14 + 6q−15−14q−16−42q−17−27q−18 + 16q−19 + 25q−20 + 33q−21 + 4q−22−13q−23−41q−24−24q−25 + 17q−26 + 25q−27 + 32q−28 + 3q−29−13q−30−39q−31−20q−32 + 15q−33 + 22q−34 + 30q−35 + 5q−36−10q−37−36q−38−18q−39 + 10q−40 + 15q−41 + 27q−42 + 8q−43−4q−44−32q−45−17q−46 + 2q−47 + 8q−48 + 23q−49 + 11q−50 + 5q−51−24q−52−15q−53−6q−54−q−55 + 17q−56 + 13q−57 + 14q−58−13q−59−9q−60−10q−61−10q−62 + 8q−63 + 9q−64 + 16q−65−3q−66 + q−67−7q−68−12q−69−2q−70 + 9q−72 + 9q−74 + q−75−6q−76−2q−77−6q−78−2q−79−5q−80 + 7q−81 + 5q−82 + 2q−83 + 5q−84−2q−85−4q−86−8q−87−q−88 + 2q−90 + 8q−91 + 3q−92 + q−93−3q−94−3q−95−4q−96−3q−97 + 5q−98 + 2q−100 + q−101−q−103−2q−104 + 4q−105−3q−106−q−107−q−108 + 5q−112−q−113−q−115−q−116−2q−117−q−118 + 3q−119 + q−121−q−124−q−125 + q−126
7	2q24−4q19−4q18−5q17 + 2q16 + 6q15 + 8q14 + 12q13 + 8q12−4q11−20q10−29q9−15q8 + 6q7 + 19q6 + 37q5 + 40q4 + 11q3−27q2−58q−47−17q−1 + 14q−2 + 56q−3 + 69q−4 + 38q−5−12q−6−63q−7−71q−8−40q−9−4q−10 + 53q−11 + 76q−12 + 52q−13 + 5q−14−53q−15−69q−16−47q−17−15q−18 + 46q−19 + 70q−20 + 49q−21 + 10q−22−47q−23−65q−24−44q−25−12q−26 + 46q−27 + 67q−28 + 44q−29 + 9q−30−48q−31−65q−32−43q−33−6q−34 + 48q−35 + 66q−36 + 40q−37 + 6q−38−46q−39−63q−40−41q−41−4q−42 + 44q−43 + 60q−44 + 35q−45 + 7q−46−38q−47−57q−48−39q−49−6q−50 + 35q−51 + 49q−52 + 33q−53 + 13q−54−25q−55−48q−56−37q−57−13q−58 + 21q−59 + 39q−60 + 32q−61 + 22q−62−8q−63−35q−64−34q−65−24q−66 + 4q−67 + 25q−68 + 29q−69 + 28q−70 + 10q−71−17q−72−26q−73−29q−74−14q−75 + 5q−76 + 16q−77 + 28q−78 + 22q−79 + 3q−80−8q−81−22q−82−24q−83−14q−84−2q−85 + 14q−86 + 22q−87 + 17q−88 + 12q−89−4q−90−17q−91−18q−92−16q−93−5q−94 + 7q−95 + 13q−96 + 18q−97 + 13q−98−5q−100−12q−101−14q−102−7q−103−2q−104 + 6q−105 + 10q−106 + 7q−107 + 8q−108 + 3q−109−5q−110−5q−111−6q−112−7q−113−3q−114−2q−115 + 4q−116 + 7q−117 + 4q−118 + 4q−119 + 4q−120−2q−121−4q−122−8q−123−6q−124−q−126 + 3q−127 + 8q−128 + 5q−129 + 3q−130−q−131−5q−132−2q−133−4q−134−3q−135 + 2q−136 + q−137 + 3q−138 + 2q−139−2q−140 + 2q−141−q−143 + 2q−144−2q−145−q−146−3q−148 + q−151 + 4q−152 + q−155−2q−156−q−157−2q−158−q−159 + 2q−160 + q−161 + q−163−q−166−q−167 + q−168