9 4

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 4"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3,12,4,13 X7,18,8,1 X9,16,10,17 X15,10,16,11 X17,8,18,9 X5,14,6,15 X11,2,12,3 X13,4,14,5

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 12 14 18 16 2 4 10 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[54]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_4.

A1 Invariants.

Weight	Invariant
1	−q23−q19 + q17 + q13 + q11−q9 + q7 + q3
2	q64 + q58−q56−2q54 + q52−2q48 + q46 + q44−2q42 + q38−q36 + q30−2q28 + 3q24−q22 + 2q18 + q12 + q6
3	−q123 + q113 + q111−q107 + q105 + 2q103−3q99−2q97 + 2q95 + 2q93−3q89 + 3q85 + 2q83−3q81−2q79 + 2q77 + 3q75−2q73−2q71−2q65−q63−q61 + 2q57−2q55−2q53 + q51 + 4q49−4q45 + 3q41 + 2q39−q37−q35 + q31 + q29 + q21 + q19 + q17 + q9
4	q200−q192−q188 + q184−q182−2q178−q176 + 3q174 + 2q172 + 3q170−2q168−4q166−q164 + q162 + 6q160 + q158−3q156−4q154−5q152 + 3q150 + 4q148 + 4q146−8q142−3q140 + 2q138 + 7q136 + 6q134−5q132−6q130−2q128 + 7q126 + 8q124−2q122−5q120−3q118 + 3q116 + 4q114−q112−2q110−2q108−2q102−q100−q98−q96 + q94 + 4q92−q90−3q88−4q86−q84 + 8q82 + 2q80−3q78−9q76−5q74 + 9q72 + 6q70 + 2q68−6q66−8q64 + q62 + 4q60 + 6q58 + q56−5q54−2q52−q50 + 4q48 + 4q46−q42−3q40 + q38 + 2q36 + q34 + q32−2q30 + q26 + q24 + 2q22 + q12
5	−q295 + q287 + q285−q277 + 2q273 + q271−q267−3q265−3q263 + 3q259 + 3q257 + 2q255−2q253−5q251−5q249 + 5q245 + 8q243 + 5q241−q239−6q237−8q235−4q233 + 3q231 + 7q229 + 8q227 + 3q225−3q223−10q221−11q219−3q217 + 6q215 + 13q213 + 11q211−q209−15q207−16q205−6q203 + 8q201 + 19q199 + 14q197−4q195−18q193−16q191−q189 + 16q187 + 19q185 + 4q183−11q181−16q179−6q177 + 10q175 + 12q173 + 5q171−3q169−8q167−4q165 + 3q163 + 5q161 + q159−q157−q155−q153 + q151 + q149−q147−q143−q141−q139 + 2q137 + 6q135 + 2q133−2q131−7q129−11q127−2q125 + 11q123 + 14q121 + 3q119−12q117−22q115−11q113 + 10q111 + 25q109 + 16q107−7q105−21q103−20q101−3q99 + 16q97 + 20q95 + 8q93−8q91−16q89−12q87 + 10q83 + 11q81 + 4q79−4q77−7q75−5q73 + 4q69 + 5q67 + 2q65−q63−3q61−2q59 + q57 + 2q55 + 3q53 + q51−2q49−q47 + q43 + 2q41 + 2q39−q37−q35 + q29 + 2q27 + q25 + q15

A2 Invariants.

Weight	Invariant
1,0	−q34−q32−q30−q28 + q26 + q24 + q22 + q20 + q16 + q10 + q6
1,1	q92 + 2q88−2q86 + 4q84−4q82 + 4q80−8q78 + 5q76−8q74 + 8q72−6q70 + 3q68 + 2q66−2q64 + 8q62−12q60 + 12q58−14q56 + 10q54−15q52 + 10q50−10q48 + 6q46−3q44−2q42 + 2q40−4q38 + 7q36−4q34 + 6q32 + 7q28−2q26 + 4q24−2q22 + 4q20−2q18 + 2q16 + q12
2,0	q86 + q84 + 2q82 + q80 + q78−q76−2q74−2q72−2q70−2q68−2q66 + q62 + q60−q58−q52−q50−q44−q42 + q40 + q36 + 3q34 + 2q32 + q30 + 2q28 + 2q26−q22 + q20 + q18 + q12

A3 Invariants.

Weight	Invariant
0,1,0	q74 + q70 + q68−3q60−q58−q56−4q54−q52−q48−q46 + q44 + q40 + q38 + 3q36 + q34 + 3q30−q26 + 2q24 + q22 + q18 + q16 + q12
1,0,0	−q45−q43−2q41−q39−q37 + q35 + q33 + 2q31 + q29 + q27 + q21 + q17 + q13 + q9

A4 Invariants.

Weight	Invariant
0,1,0,0	q96 + q94 + q92 + 2q90 + 3q88 + q86 + q84 + q82−q80−5q78−6q76−5q74−6q72−6q70−2q68 + 2q62 + 2q60 + q58 + 2q54 + 2q52 + 2q50 + 2q48 + 4q46 + 2q44 + q40 + q38 + q36 + q32 + 2q30 + q28 + q26 + q24 + q22 + q18
1,0,0,0	−q56−q54−2q52−2q50−q48−q46 + q44 + q42 + 2q40 + 2q38 + q36 + q34 + q26 + q22 + q20 + q16 + q12

B2 Invariants.

Weight	Invariant
0,1	−q74−q70 + q68−2q66 + 2q64−2q62 + q60−q58 + q56−q52 + 2q50−3q48 + 3q46−3q44 + 4q42−3q40 + 3q38−q36 + q34−q30 + 2q28−q26 + 2q24−q22 + 2q20−q18 + q16 + q12
1,0	q120 + q112 + q110−q106 + q102 + q100−2q98−3q96−q94 + q92−3q88−2q86 + q82−q78 + q74−q70−q68 + q66 + q64−q60 + q58 + 2q56 + q54−q52 + 3q48 + 2q46−q44−q42 + 2q38 + q36−q32 + q28 + q26 + q18

D4 Invariants.

Weight	Invariant
1,0,0,0	q102 + q98 + 2q94−q92 + q90−2q88 + q86−2q84−2q80−q78−2q76−3q74−q72−3q70−4q66 + 2q64−2q62 + 4q60−q58 + 4q56 + 4q52 + q50 + 2q48 + 2q42−q40 + q38−q36 + 2q34 + 2q30 + 2q26 + q22 + q18

G2 Invariants.

Weight	Invariant
1,0	q176 + q172−q170 + q168−q164 + 2q162−2q160 + 2q158−2q156 + q152−2q150 + 3q148−5q146 + 2q144−q142−3q140 + 2q138−5q136 + q134 + q132−3q130−3q126−q124 + 3q122−5q120 + 2q118−q116−q114 + 5q112−3q110 + 4q108−q106 + 3q104 + 2q102−3q100 + 6q98−3q96 + 4q94 + q92−2q90 + 3q88−q86 + q82−3q80 + q78−2q74 + 4q72−3q70 + 2q68−q64 + q62−2q60 + 3q58−q56 + q54 + 2q48−q46 + 2q44−q42 + q40 + q38−q36 + q34 + q30

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

3t2−5t + 5−5t−1 + 3t−2

In[5]:=

Conway[K][z]

Out[5]=

3z4 + 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]=

{ 21, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−2−q−3 + 2q−4−3q−5 + 4q−6−3q−7 + 3q−8−2q−9 + q−10−q−11

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a10−2a10 + z4a8 + 3z2a8 + 2a8 + z4a6 + 2z2a6 + z4a4 + 3z2a4 + a4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z5a13−4z3a13 + 3za13 + z6a12−3z4a12 + z2a12 + z7a11−3z5a11 + 2z3a11−za11 + z8a10−5z6a10 + 11z4a10−10z2a10 + 2a10 + 2z7a9−8z5a9 + 12z3a9−4za9 + z8a8−5z6a8 + 11z4a8−7z2a8 + 2a8 + z7a7−3z5a7 + 4z3a7 + z6a6−2z4a6 + z2a6 + z5a5−2z3a5 + z4a4−3z2a4 + a4

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 4"];

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]=

{ 3t2−5t + 5−5t−1 + 3t−2, q−2−q−3 + 2q−4−3q−5 + 4q−6−3q−7 + 3q−8−2q−9 + q−10−q−11 }

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	q−4−q−5 + 2q−7−2q−8 + 4q−10−4q−11−q−12 + 8q−13−7q−14−3q−15 + 11q−16−8q−17−3q−18 + 10q−19−6q−20−4q−21 + 8q−22−3q−23−4q−24 + 5q−25−q−26−3q−27 + 2q−28−q−30 + q−31
3	q−6−q−7 + 2q−10−q−11−q−13 + 2q−14−q−15 + q−16−q−17 + q−18−2q−19 + q−20 + 2q−21 + 2q−22−5q−23−3q−24 + 6q−25 + 6q−26−8q−27−6q−28 + 6q−29 + 10q−30−10q−31−7q−32 + 6q−33 + 9q−34−8q−35−7q−36 + 4q−37 + 9q−38−3q−39−8q−40 + 8q−42 + 2q−43−7q−44−3q−45 + 5q−46 + 5q−47−5q−48−3q−49 + q−50 + 4q−51−2q−52−q−53 + 2q−55−q−56 + q−59−q−60
4	q−8−q−9 + 3q−13−2q−14−q−16−2q−17 + 6q−18−2q−19 + q−20−2q−21−6q−22 + 8q−23−q−24 + 5q−25−2q−26−11q−27 + 7q−28−4q−29 + 11q−30 + 3q−31−13q−32 + 4q−33−13q−34 + 13q−35 + 11q−36−9q−37 + 7q−38−27q−39 + 9q−40 + 17q−41−4q−42 + 13q−43−36q−44 + 6q−45 + 18q−46−2q−47 + 18q−48−39q−49 + 4q−50 + 18q−51−2q−52 + 17q−53−37q−54 + 4q−55 + 16q−56−2q−57 + 18q−58−32q−59 + 3q−60 + 10q−61−4q−62 + 21q−63−22q−64 + 2q−65 + q−66−8q−67 + 22q−68−11q−69 + 3q−70−4q−71−13q−72 + 17q−73−3q−74 + 7q−75−4q−76−14q−77 + 9q−78−2q−79 + 8q−80−9q−82 + 4q−83−4q−84 + 5q−85 + 2q−86−4q−87 + 3q−88−3q−89 + q−90 + q−91−2q−92 + 2q−93−q−94−q−97 + q−98