9 41

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 41"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X12,8,13,7 X14,5,15,6 X10,3,11,4 X2,11,3,12 X4,15,5,16 X8,17,9,18 X16,9,17,10 X18,14,1,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 12 16 2 18 4 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[20:20:20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_41.

A1 Invariants.

Weight	Invariant
1	q13−2q11 + 2q9−2q7 + q5 + 3q−1−2q−3 + 2q−5−q−7
2	q38−2q36−3q34 + 7q32 + q30−11q28 + 7q26 + 9q24−13q22 + 12q18−8q16−6q14 + 9q12−8q8 + 2q6 + 9q4−5q2−8 + 14q−2 + q−4−13q−6 + 9q−8 + 4q−10−7q−12 + 3q−14−2q−18 + q−20
3	q75−2q73−3q71 + 2q69 + 10q67 + 4q65−18q63−16q61 + 16q59 + 34q57−4q55−47q53−17q51 + 46q49 + 41q47−34q45−60q43 + 15q41 + 66q39 + 9q37−62q35−28q33 + 55q31 + 39q29−43q27−46q25 + 32q23 + 50q21−19q19−53q17 + 7q15 + 49q13 + 11q11−47q9−33q7 + 32q5 + 57q3−11q−66q−1−11q−3 + 66q−5 + 35q−7−56q−9−42q−11 + 34q−13 + 40q−15−15q−17−26q−19 + 5q−21 + 14q−23−4q−25−4q−27 + q−31−q−33 + 2q−37−q−39
4	q124−2q122−3q120 + 2q118 + 5q116 + 13q114−3q112−23q110−24q108−8q106 + 55q104 + 57q102 + 3q100−72q98−121q96−2q94 + 116q92 + 159q90 + 43q88−191q86−202q84−47q82 + 217q80 + 294q78 + 9q76−254q74−328q72−12q70 + 352q68 + 302q66−22q64−398q62−304q60 + 142q58 + 393q56 + 244q54−237q52−398q50−86q48 + 293q46 + 335q44−67q42−340q40−178q38 + 193q36 + 310q34 + 9q32−275q30−205q28 + 130q26 + 287q24 + 96q22−210q20−278q18−8q16 + 244q14 + 274q12−17q10−326q8−273q6 + 42q4 + 409q2 + 310−163q−2−441q−4−287q−6 + 269q−8 + 479q−10 + 133q−12−301q−14−416q−16−3q−18 + 316q−20 + 233q−22−46q−24−251q−26−98q−28 + 88q−30 + 116q−32 + 38q−34−76q−36−41q−38 + 9q−40 + 20q−42 + 20q−44−17q−46−6q−48 + 2q−50 + 6q−54−3q−56 + q−58−2q−62 + q−64
5	q185−2q183−3q181 + 2q179 + 5q177 + 8q175 + 6q173−8q171−31q169−30q167−2q165 + 44q163 + 84q161 + 72q159−17q157−143q155−190q153−106q151 + 98q149 + 309q147 + 346q145 + 105q143−294q141−567q139−486q137 + 2q135 + 617q133 + 899q131 + 537q129−314q127−1081q125−1177q123−371q121 + 857q119 + 1618q117 + 1232q115−157q113−1597q111−1983q109−858q107 + 1052q105 + 2327q103 + 1871q101−92q99−2114q97−2599q95−1026q93 + 1445q91 + 2851q89 + 1988q87−505q85−2620q83−2615q81−445q79 + 2075q77 + 2828q75 + 1183q73−1396q71−2689q69−1637q67 + 776q65 + 2358q63 + 1770q61−329q59−1961q57−1691q55 + 88q53 + 1632q51 + 1513q49−23q47−1421q45−1360q43 + 34q41 + 1345q39 + 1316q37 + 17q35−1317q33−1448q31−230q29 + 1238q27 + 1676q25 + 701q23−931q21−1927q19−1397q17 + 334q15 + 1957q13 + 2167q11 + 614q9−1637q7−2789q5−1726q3 + 888q + 2982q−1 + 2752q−3 + 198q−5−2647q−7−3387q−9−1306q−11 + 1844q−13 + 3402q−15 + 2149q−17−804q−19−2892q−21−2461q−23−109q−25 + 2034q−27 + 2259q−29 + 678q−31−1154q−33−1723q−35−843q−37 + 492q−39 + 1118q−41 + 702q−43−119q−45−603q−47−478q−49−30q−51 + 297q−53 + 265q−55 + 36q−57−118q−59−124q−61−33q−63 + 51q−65 + 55q−67 + 9q−69−23q−71−17q−73 + q−75 + 6q−77 + 8q−79 + 2q−81−6q−83−4q−85 + 3q−87−q−89 + 2q−93−q−95

A2 Invariants.

Weight	Invariant
1,0	q20 + q18−2q16−q12−2q10 + 2q8 + 2q4 + q2 + 2q−2−2q−4 + q−6 + q−8−q−10
1,1	q52−4q50 + 14q48−36q46 + 66q44−110q42 + 156q40−196q38 + 224q36−216q34 + 182q32−112q30 + 22q28 + 80q26−188q24 + 272q22−339q20 + 376q18−378q16 + 346q14−280q12 + 190q10−96q8−4q6 + 84q4−134q2 + 168−158q−2 + 149q−4−122q−6 + 98q−8−78q−10 + 56q−12−46q−14 + 38q−16−28q−18 + 18q−20−12q−22 + 8q−24−4q−26 + q−28
2,0	q52 + q50−q48−5q46−3q44 + 4q42 + 5q40−2q38−3q36 + 6q34 + 8q32−3q30−8q28 + 2q26 + 2q24−4q22−7q20 + 3q18 + 4q16−3q14 + q12−q8 + q6 + 6q4−3q2−3 + 7q−2 + 8q−4−5q−6−8q−8 + 7q−10 + 5q−12−5q−14−3q−16 + 2q−18 + 2q−20−2q−22−q−24 + q−26

A3 Invariants.

Weight	Invariant
0,1,0	q40−2q38 + 2q36 + 2q34−6q32 + 6q30−8q26 + 8q24−q22−8q20 + 7q18 + 2q16−7q14 + 2q12 + 2q10−q8−3q6 + 2q4 + 9q2−5 + q−2 + 11q−4−9q−6−2q−8 + 8q−10−5q−12−3q−14 + 5q−16−q−18−2q−20 + q−22
1,0,0	q27 + q25 + q23−2q21−3q17−q15−2q13 + 2q11 + q9 + 2q7 + 2q5 + q3 + q−q−1 + 2q−3−2q−5 + q−7 + q−11−q−13

B2 Invariants.

Weight	Invariant
0,1	q40−2q38 + 6q36−8q34 + 10q32−12q30 + 12q28−12q26 + 8q24−5q22−2q20 + 7q18−14q16 + 19q14−22q12 + 24q10−21q8 + 19q6−12q4 + 9q2−1−3q−2 + 9q−4−11q−6 + 12q−8−12q−10 + 9q−12−7q−14 + 5q−16−3q−18 + 2q−20−q−22
1,0	q66−2q62−2q60 + 4q58 + 5q56−4q54−8q52 + 11q48 + 5q46−11q44−10q42 + 7q40 + 13q38−13q34−5q32 + 9q30 + 8q28−5q26−8q24 + 2q22 + 7q20−q18−9q16−q14 + 9q12 + 3q10−9q8−5q6 + 8q4 + 11q2−4−10q−2 + 2q−4 + 14q−6 + 3q−8−10q−10−8q−12 + 5q−14 + 10q−16−7q−20−5q−22 + 2q−24 + 5q−26 + q−28−2q−30−2q−32 + q−36

G2 Invariants.

Weight

Invariant

1,0

q94−2q92 + 6q90−10q88 + 11q86−7q84−7q82 + 27q80−39q78 + 44q76−28q74−3q72 + 40q70−66q68 + 70q66−45q64 + q62 + 37q60−64q58 + 54q56−25q54−16q52 + 42q50−49q48 + 27q46 + 7q44−43q42 + 63q40−60q38 + 37q36 + 6q34−46q32 + 79q30−82q28 + 64q26−19q24−28q22 + 65q20−77q18 + 58q16−17q14−25q12 + 51q10−48q8 + 18q6 + 19q4−46q2 + 51−30q−2−3q−4 + 35q−6−51q−8 + 53q−10−32q−12 + 8q−14 + 16q−16−32q−18 + 33q−20−27q−22 + 18q−24−7q−26−2q−28 + 9q−30−14q−32 + 13q−34−10q−36 + 7q−38−2q−40−q−42 + 2q−44−4q−46 + 3q−48−2q−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["9 41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

3t2−12t + 19−12t−1 + 3t−2

In[5]:=

Conway[K][z]

Out[5]=

3z4 + 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]=

{7,t + 1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q3 + 3q2−5q + 8−8q−1 + 8q−2−7q−3 + 5q−4−3q−5 + q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a6−3z2a4−3a4 + 2z4a2 + 4z2a2 + 3a2 + z4−z2a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2a4z8 + 2a2z8 + 3a5z7 + 9a3z7 + 6az7 + a6z6−a4z6 + 5a2z6 + 7z6−10a5z5−26a3z5−11az5 + 5z5a−1−3a6z4−12a4z4−23a2z4 + 3z4a−2−11z4 + 9a5z3 + 19a3z3 + 6az3−3z3a−1 + z3a−3 + 3a6z2 + 13a4z2 + 17a2z2−z2a−2 + 6z2−2a5z−4a3z−2az−a6−3a4−3a2

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 41"];

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]=

{ 3t2−12t + 19−12t−1 + 3t−2, −q3 + 3q2−5q + 8−8q−1 + 8q−2−7q−3 + 5q−4−3q−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]=

{K11n83,}

In[6]:=

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

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

Out[6]=

{K11n4, K11n21,}

n	Jn
2	q9−3q8 + 2q7 + 4q6−13q5 + 13q4 + 9q3−35q2 + 27q + 22−57q−1 + 30q−2 + 36q−3−64q−4 + 20q−5 + 44q−6−55q−7 + 5q−8 + 42q−9−35q−10−7q−11 + 29q−12−13q−13−9q−14 + 11q−15−q−16−3q−17 + q−18
3	−q18 + 3q17−2q16−q15 + q14 + 2q13−6q12−q11 + 19q10−7q9−37q8 + 10q7 + 74q6−13q5−113q4−4q3 + 165q2 + 18q−190−59q−1 + 220q−2 + 86q−3−215q−4−124q−5 + 206q−6 + 144q−7−177q−8−166q−9 + 146q−10 + 178q−11−108q−12−184q−13 + 68q−14 + 181q−15−26q−16−168q−17−15q−18 + 147q−19 + 45q−20−111q−21−66q−22 + 72q−23 + 71q−24−36q−25−61q−26 + 9q−27 + 41q−28 + 7q−29−23q−30−9q−31 + 9q−32 + 5q−33−q−34−3q−35 + q−36