9 1

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 1"];

In[4]:=

PD[K]

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

Out[4]=

X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

10 12 14 16 18 2 4 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[9]

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(2,{−1,−1,−1,−1,−1,−1,−1,−1,−1})

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]=

{ 2, 9, 2 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_1.

A1 Invariants.

Weight	Invariant
1	−q27 + q11 + q9 + q7
2	q72−q56−q54−q52 + q22 + q20 + q18 + q16 + q14
3	−q135 + q119 + q117 + q115−q85−q83−q81−q79−q77 + q33 + q31 + q29 + q27 + q25 + q23 + q21
4	q216−q200−q198−q196 + q166 + q164 + q162 + q160 + q158−q114−q112−q110−q108−q106−q104−q102 + q44 + q42 + q40 + q38 + q36 + q34 + q32 + q30 + q28
5	−q315 + q299 + q297 + q295−q265−q263−q261−q259−q257 + q213 + q211 + q209 + q207 + q205 + q203 + q201−q143−q141−q139−q137−q135−q133−q131−q129−q127 + q55 + q53 + q51 + q49 + q47 + q45 + q43 + q41 + q39 + q37 + q35
6	q432−q416−q414−q412 + q382 + q380 + q378 + q376 + q374−q330−q328−q326−q324−q322−q320−q318 + q260 + q258 + q256 + q254 + q252 + q250 + q248 + q246 + q244−q172−q170−q168−q166−q164−q162−q160−q158−q156−q154−q152 + q66 + q64 + q62 + q60 + q58 + q56 + q54 + q52 + q50 + q48 + q46 + q44 + q42
8	q720−q704−q702−q700 + q670 + q668 + q666 + q664 + q662−q618−q616−q614−q612−q610−q608−q606 + q548 + q546 + q544 + q542 + q540 + q538 + q536 + q534 + q532−q460−q458−q456−q454−q452−q450−q448−q446−q444−q442−q440 + q354 + q352 + q350 + q348 + q346 + q344 + q342 + q340 + q338 + q336 + q334 + q332 + q330−q230−q228−q226−q224−q222−q220−q218−q216−q214−q212−q210−q208−q206−q204−q202 + q88 + q86 + q84 + q82 + q80 + q78 + q76 + q74 + q72 + q70 + q68 + q66 + q64 + q62 + q60 + q58 + q56

A2 Invariants.

Weight	Invariant
1,0	−q38−q36−q34 + q22 + q20 + 2q18 + q16 + q14
1,1	q108−2q60−2q58−4q56−4q54−4q52−2q50−2q48 + q44 + 2q42 + 4q40 + 4q38 + 5q36 + 4q34 + 4q32 + 2q30 + q28
2,0	q94 + q92 + 2q90 + q88 + q86−q78−2q76−3q74−3q72−3q70−2q68−q66 + q44 + q42 + 2q40 + 2q38 + 3q36 + 2q34 + 2q32 + q30 + q28

A3 Invariants.

Weight	Invariant
0,1,0	q90−q60−2q58−3q56−3q54−3q52−2q50−q48 + q44 + q42 + 3q40 + 3q38 + 4q36 + 3q34 + 3q32 + q30 + q28
1,0,0	−q49−q47−2q45−q43−q41 + q33 + q31 + 2q29 + 2q27 + 2q25 + q23 + q21
1,0,1	q144 + q98 + q96 + 3q94 + 3q92 + 4q90 + 3q88 + 3q86 + q84−q82−4q80−8q78−10q76−14q74−14q72−14q70−10q68−7q66−2q64 + 3q62 + 7q60 + 10q58 + 11q56 + 12q54 + 11q52 + 10q50 + 7q48 + 5q46 + 2q44 + q42

A4 Invariants.

Weight	Invariant
0,1,0,0	q112 + q110 + q108 + q106 + q104−q82−2q80−4q78−5q76−7q74−7q72−7q70−5q68−3q66−q64 + 2q62 + 4q60 + 6q58 + 6q56 + 8q54 + 6q52 + 6q50 + 4q48 + 3q46 + q44 + q42
1,0,0,0	−q60−q58−2q56−2q54−2q52−q50−q48 + q44 + q42 + 2q40 + 2q38 + 3q36 + 2q34 + 2q32 + q30 + q28

B2 Invariants.

Weight	Invariant
0,1	−q90−q60−q56−q54−q52−q48 + q44 + q42 + q40 + q38 + 2q36 + q34 + q32 + q30 + q28
1,0	q144−q98−q96−q94−q92−2q90−q88−q86−q84−q82 + q66 + q62 + q60 + 2q58 + q56 + 2q54 + q52 + 2q50 + q48 + q46 + q42

D4 Invariants.

Weight	Invariant
0,1,0,0	q216−q172−q170−3q168−3q166−4q164−4q162−4q160−3q158 + 2q154 + 5q152 + 9q150 + 12q148 + 12q146 + 15q144 + 12q142 + 12q140 + 9q138 + 6q136 + 3q134 + 3q132−q126−3q124−6q122−10q120−16q118−22q116−28q114−33q112−36q110−37q108−33q106−27q104−18q102−8q100 + 4q98 + 12q96 + 22q94 + 26q92 + 29q90 + 29q88 + 28q86 + 22q84 + 20q82 + 14q80 + 10q78 + 6q76 + 4q74 + q72 + q70
1,0,0,0	q126−q82−q80−3q78−3q76−4q74−4q72−4q70−3q68−2q66 + q62 + 3q60 + 4q58 + 4q56 + 5q54 + 4q52 + 4q50 + 3q48 + 2q46 + q44 + q42

G2 Invariants.

Weight	Invariant
1,0	q216−q172−q170−q164−q162−q160−q154−q152−q126−q120−q118−q116−q114−q108 + q100 + q98 + q94 + 2q92 + 2q90 + 2q88 + q86 + q84 + 2q82 + 2q80 + q78 + q74 + q72 + q70

.

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t4−t3 + t2−t + 1−t−1 + t−2−t−3 + t−4

In[5]:=

Conway[K][z]

Out[5]=

z8 + 7z6 + 15z4 + 10z2 + 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]=

{ 9, -8 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−4 + q−6−q−7 + q−8−q−9 + q−10−q−11 + q−12−q−13

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6a10−6z4a10−10z2a10−4a10 + z8a8 + 8z6a8 + 21z4a8 + 20z2a8 + 5a8

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

za17 + z2a16 + z3a15−za15 + z4a14−2z2a14 + z5a13−3z3a13 + za13 + z6a12−4z4a12 + 3z2a12 + z7a11−5z5a11 + 6z3a11−za11 + z8a10−7z6a10 + 16z4a10−14z2a10 + 4a10 + z7a9−6z5a9 + 10z3a9−4za9 + z8a8−8z6a8 + 21z4a8−20z2a8 + 5a8

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 1"];

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]=

{ t4−t3 + t2−t + 1−t−1 + t−2−t−3 + t−4, q−4 + q−6−q−7 + q−8−q−9 + q−10−q−11 + q−12−q−13 }

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−8 + q−11−q−13 + q−14−q−16 + q−17−q−19 + q−20−q−22 + q−23−q−25 + q−26−q−27−q−28 + q−29−q−31 + q−32−q−34 + q−35
3	q−12 + q−16−q−19 + q−20−q−23 + q−24−q−27 + q−28−q−31 + q−32−q−35 + q−36−q−39−q−43 + q−45−q−47 + q−49−q−51 + q−53−q−55 + q−57 + q−61−q−62 + q−65−q−66
4	q−16 + q−21−q−25 + q−26−q−30 + q−31−q−35 + q−36−q−40 + q−41−q−45 + q−46−q−50 + q−51−q−53−q−55 + q−56−q−58 + q−61−q−63 + q−66−q−68 + q−71−q−73 + q−76−q−78 + 2q−81−q−83 + q−86−q−88 + q−91−q−93 + q−96−q−98−q−100 + q−101−q−105 + q−106
5	q−20 + q−26−q−31 + q−32−q−37 + q−38−q−43 + q−44−q−49 + q−50−q−55 + q−56−q−61 + q−62−q−66−q−67 + q−68−q−72−q−73 + q−74 + q−75−q−78−q−79 + q−80 + q−81−q−84−q−85 + q−86 + q−87−q−90−q−91 + q−92 + q−93−q−96−q−97 + q−98 + q−99−q−102 + q−104 + q−105−q−108 + q−111−q−114 + q−117−q−120 + q−123−q−126 + q−129−q−131−q−132 + q−135 + q−136−q−137−q−138 + q−141 + q−142−q−143−q−144 + q−147 + q−148−q−149 + q−154−q−155
6	q−24 + q−31−q−37 + q−38−q−44 + q−45−q−51 + q−52−q−58 + q−59−q−65 + q−66−q−72 + q−73−2q−79 + q−80−2q−86 + q−87 + q−90−2q−93 + q−94 + q−97−2q−100 + q−101 + q−104−2q−107 + q−108 + q−111−2q−114 + q−115 + q−118−2q−121 + q−122 + 2q−125−2q−128 + q−129 + 2q−132−q−134−2q−135 + q−136 + 2q−139−q−141−2q−142 + q−143 + 2q−146−q−148−2q−149 + q−150 + 2q−153−q−155−2q−156 + q−157 + 2q−160−2q−162−2q−163 + q−164 + 2q−167−q−169−2q−170 + q−171 + 2q−174−q−176−2q−177 + q−178 + 2q−181−q−183−2q−184 + q−185 + 2q−188−2q−191 + q−192 + q−195−2q−198 + q−199 + q−202−2q−205 + q−206−q−212 + q−213
7	q−28 + q−36−q−43 + q−44−q−51 + q−52−q−59 + q−60−q−67 + q−68−q−75 + q−76−q−83 + q−84−q−91−q−99 + q−105−q−107 + q−113−q−115 + q−121−q−123 + q−129−q−131 + q−137−q−139 + q−145 + q−153−q−158 + q−161−q−166 + q−169−q−174 + q−177−q−182 + q−185−q−190−q−198 + q−202−q−206 + q−210−q−214 + q−218−q−222 + q−226 + q−234−q−237 + q−242−q−245 + q−250−q−253−q−261 + q−263−q−269 + q−271 + q−279−q−280