9 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.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 2"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 18 16 14 2 10 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[72]

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(5,{−1,−1,−1,−2,1,−2,−3,2,−3,−4,3,−4})

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]=

{ 5, 12, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_2.

A1 Invariants.

Weight	Invariant
1	−q21 + q15 + q5 + q
2	q60−q56−q50 + q46−q30−q28 + q18 + q16 + q14 + q8 + q2
3	−q117 + q113 + q111−q107 + q103−q99−q97 + q93 + q73 + q71−q67−q61−q59−q53 + q49 + q47−q43−q41−q39 + q37−q33−q31 + q29 + 2q27−q23 + q21 + 2q19 + q17 + q11 + q3
4	q192−q188−q186−q184 + q182 + q180 + q178−2q174 + q170 + q168 + q166−q164−q162−q160 + q156−q134−q132 + q128 + 2q126−q122 + q118 + 2q116−2q112−q110 + q106−2q102−q100 + q96 + q94 + q90 + q88 + q86−q82−q80 + q76−q72−q70−q68−q66 + q62−q60−2q58−2q56 + 2q52 + q50−2q46 + q44 + 2q42 + q40−q38−2q36 + q34 + 2q32 + q30−q26 + q24 + q22 + q20 + q18 + q14 + q4
5	−q285 + q281 + q279 + q277−q273−2q271−q269 + q265 + 2q263 + q261−q259−2q257−q255 + q251 + 2q249 + q247−q243−q241−q239 + q235 + q213 + q211−q207−2q205−2q203 + 2q199 + 2q197−2q193−2q191−q189 + 2q187 + 4q185 + 2q183−q181−2q179−2q177 + 2q173 + 2q171−2q167−2q165−q163 + q159 + q157−q155−q153−q151 + q147 + 2q145 + q143−q139−q137 + q135 + 2q133 + q131−q127 + q123−q119−2q117−q115 + q113 + 2q111 + q109−q105−q103−2q101 + q97 + 2q95 + 2q93 + q91−2q89−3q87−2q85 + q81 + q79−q77−2q75−3q73−q71 + q69 + q67−q63 + q57−q53 + 2q49 + 3q47 + q45−q43−q41−q39 + q37 + 2q35 + q33 + q23 + q21 + q19 + q17 + q5

A2 Invariants.

Weight	Invariant
1,0	−q32−q30 + q24 + q22 + q8 + q6 + q2
1,1	q84 + 2q80−2q78 + 2q76−4q74 + 2q72−4q70 + 4q62−q60 + 4q58−4q56 + 2q54−4q52 + 2q50−2q48 + 2q46−2q42−2q40−2q38−2q36 + 2q30 + 2q28 + 2q26 + 2q24 + q20 + 2q16 + 2q12 + 2q8 + q4
2,0	q82 + q80 + q78−q76−q74−q72−q70−q68−q66 + q64 + q62 + q60−q44−2q42−2q40−q38 + q32 + q30 + q28 + 2q26 + q24 + q20 + q18 + q16 + q12 + q10 + q4

A3 Invariants.

Weight	Invariant
0,1,0	q66 + q62−q58−q56−q54−q52−q50−q46−q42 + q38 + q36 + q34 + q32 + q30 + q16 + q12 + 2q10 + q8 + q4
1,0,0	−q43−q41−q39 + q33 + q31 + q29 + q11 + q9 + q7 + q3

B2 Invariants.

Weight	Invariant
0,1	−q66−q62−q58 + q56−q54 + q52 + q50 + q46 + q42−2q40 + q38−q36 + q34−q32 + q30 + q16 + q12 + q8 + q4
1,0	q108 + q100−q96−q94−q88−q86 + q82−q76−q68−q66 + q56 + q54 + q48 + q46 + q26 + q18 + q16 + q14 + q6

G2 Invariants.

Weight	Invariant
1,0	q156 + q152−q150 + q142−2q140 + q138−q136−q134−2q130−q128−q126−q124−q118 + q112−q108 + q106 + q104 + 2q102 + q98 + q94 + q92−2q90 + q88 + q86 + q76−q72 + q66−q62−q52 + q48 + q38 + q34 + q28 + q24 + q20 + q14 + q10

.

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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

4t−7 + 4t−1

In[5]:=

Conway[K][z]

Out[5]=

4z2 + 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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−1−q−2 + 2q−3−2q−4 + 2q−5−2q−6 + 2q−7−q−8 + q−9−q−10

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−a10 + z2a8 + a8 + z2a6 + z2a4 + z2a2 + a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z7a11−6z5a11 + 10z3a11−4za11 + z8a10−6z6a10 + 11z4a10−7z2a10 + a10 + 2z7a9−10z5a9 + 13z3a9−4za9 + z8a8−5z6a8 + 8z4a8−6z2a8 + a8 + z7a7−3z5a7 + z3a7 + z6a6−2z4a6 + z5a5−z3a5 + z4a4 + z3a3 + z2a2−a2

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 2"];

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]=

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

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]=

{7_4,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n13,}

n	Jn
2	q−2−q−3 + 2q−5−2q−6 + 3q−8−2q−9 + 2q−11−2q−12 + 2q−14−3q−15 + 3q−17−3q−18 + 3q−20−3q−21 + 3q−23−2q−24−q−25 + 2q−26−q−27−q−28 + q−29
3	q−3−q−4 + 2q−7−2q−8 + q−10 + 3q−11−3q−12−2q−13 + 2q−14 + 5q−15−4q−16−4q−17 + 2q−18 + 6q−19−3q−20−6q−21 + 2q−22 + 6q−23−2q−24−5q−25 + 2q−26 + 5q−27−3q−28−4q−29 + 2q−30 + 4q−31−3q−32−3q−33 + 2q−34 + 3q−35−2q−36−2q−37 + 2q−38 + 2q−39−2q−40−2q−41 + 2q−42 + 2q−43−2q−44−2q−45 + 2q−46 + 2q−47−q−48−3q−49 + q−50 + 2q−51−2q−53 + q−55 + q−56−q−57
4	q−4−q−5 + 2q−9−2q−10 + q−11 + 2q−14−4q−15 + 2q−16 + q−17 + q−18 + q−19−7q−20 + 3q−21 + 3q−22 + 2q−23−10q−25 + 5q−26 + 4q−27 + 3q−28−2q−29−12q−30 + 5q−31 + 5q−32 + 5q−33−3q−34−13q−35 + 5q−36 + 5q−37 + 5q−38−2q−39−12q−40 + 4q−41 + 4q−42 + 5q−43−q−44−11q−45 + 4q−46 + 4q−47 + 4q−48−11q−50 + 3q−51 + 3q−52 + 3q−53 + 2q−54−10q−55 + 2q−56 + 2q−57 + 2q−58 + 4q−59−8q−60 + q−61 + q−62 + q−63 + 5q−64−6q−65 + 5q−69−5q−70 + 5q−74−5q−75 + 5q−79−4q−80−q−81−q−82 + 5q−84−2q−85−q−86−q−87−q−88 + 3q−89−q−92−q−93 + q−94