10 11

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["10 11"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,18,8,19 X9,20,10,1 X19,8,20,9 X17,10,18,11

In[5]:=

GaussCode[K]

Out[5]=

-1, 6, -4, 5, -3, 1, -7, 9, -8, 10, -2, 3, -5, 4, -6, 2, -10, 7, -9, 8

In[6]:=

DTCode[K]

Out[6]=

6 14 12 18 20 16 4 2 10 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[433]

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,−1,−2,1,3,−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[{12, 3}, {4, 2}, {3, 5}, {6, 4}, {5, 11}, {1, 6}, {10, 12}, {11, 7}, {2, 8}, {7, 9}, {8, 10}, {9, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_11.

A1 Invariants.

Weight	Invariant
1	q15−q13 + 2q11−2q9 + q7−q3 + q−2q−1 + 2q−3 + q−7
2	q42−q40 + 3q36−3q34−2q32 + 6q30−5q28−5q26 + 10q24−3q22−6q20 + 6q18 + q16−3q14−q12 + 5q10−6q6 + 6q4 + 4q2−8 + 3q−2 + 6q−4−7q−6−q−8 + 4q−10−3q−12−q−14 + 2q−16 + q−22
3	q81−q79 + q75 + q73−2q71−2q69 + 2q67 + 2q65−4q63−3q61 + 6q59 + 6q57−10q55−11q53 + 14q51 + 17q49−12q47−25q45 + 12q43 + 28q41−6q39−26q37−q35 + 22q33 + 7q31−14q29−12q27 + 5q25 + 15q23 + q21−18q19−9q17 + 20q15 + 13q13−19q11−16q9 + 16q7 + 22q5−13q3−24q + 6q−1 + 26q−3 + 2q−5−21q−7−9q−9 + 19q−11 + 12q−13−10q−15−12q−17 + 4q−19 + 10q−21−q−23−6q−25−2q−27 + 3q−29−q−33−q−35 + q−37 + q−45
4	q132−q130 + q126−q124 + 2q122−3q120−q118 + 2q116−3q114 + 6q112−3q110−q108 + 3q106−9q104 + 6q102−5q100 + 6q98 + 15q96−9q94−4q92−29q90 + 4q88 + 46q86 + 19q84−6q82−76q80−33q78 + 62q76 + 71q74 + 33q72−100q70−89q68 + 31q66 + 92q64 + 79q62−64q60−100q58−24q56 + 53q54 + 88q52 + 2q50−57q48−52q46−3q44 + 53q42 + 45q40−4q38−56q36−41q34 + 20q32 + 69q30 + 29q28−58q26−61q24 + q22 + 88q20 + 51q18−61q16−80q14−25q12 + 93q10 + 79q8−31q6−84q4−72q2 + 60 + 94q−2 + 25q−4−48q−6−98q−8−5q−10 + 60q−12 + 61q−14 + 16q−16−72q−18−44q−20 + 5q−22 + 44q−24 + 47q−26−18q−28−30q−30−24q−32 + 6q−34 + 31q−36 + 7q−38−3q−40−15q−42−8q−44 + 8q−46 + 3q−48 + 5q−50−3q−52−4q−54 + q−56−2q−58 + 2q−60−q−64 + q−66−q−68 + q−76
5	q195−q193 + q189−q187 + q183−2q181−2q179 + 2q177 + q175 + 4q171−q169−6q167−4q165−q163 + 3q161 + 10q159 + 10q157−q155−11q153−18q151−10q149 + 5q147 + 26q145 + 34q143 + 12q141−29q139−61q137−51q135 + 11q133 + 91q131 + 118q129 + 34q127−109q125−196q123−119q121 + 90q119 + 268q117 + 239q115−30q113−320q111−357q109−72q107 + 303q105 + 458q103 + 210q101−249q99−495q97−320q95 + 131q93 + 464q91 + 396q89 + 3q87−374q85−408q83−111q81 + 237q79 + 361q77 + 192q75−102q73−277q71−221q69−11q67 + 177q65 + 217q63 + 96q61−92q59−202q57−145q55 + 31q53 + 185q51 + 182q49 + 3q47−194q45−210q43−16q41 + 212q39 + 244q37 + 31q35−243q33−297q31−53q29 + 266q27 + 353q25 + 100q23−261q21−402q19−180q17 + 227q15 + 432q13 + 260q11−136q9−420q7−339q5 + 22q3 + 355q + 388q−1 + 111q−3−247q−5−380q−7−213q−9 + 97q−11 + 317q−13 + 283q−15 + 37q−17−208q−19−275q−21−146q−23 + 73q−25 + 227q−27 + 194q−29 + 29q−31−132q−33−183q−35−102q−37 + 38q−39 + 135q−41 + 118q−43 + 25q−45−66q−47−97q−49−60q−51 + 16q−53 + 63q−55 + 55q−57 + 16q−59−24q−61−41q−63−22q−65 + 4q−67 + 19q−69 + 20q−71 + 7q−73−9q−75−10q−77−6q−79−2q−81 + 6q−83 + 6q−85−q−89−q−91−4q−93 + 2q−97 + q−103−q−105−q−107 + q−115

A2 Invariants.

Weight	Invariant
1,0	q22 + 2q16−q14−q8 + q6−2q4−1−q−2 + 2q−4 + q−6 + q−8 + q−10
1,1	q60−2q58 + 4q56−8q54 + 15q52−20q50 + 28q48−40q46 + 54q44−60q42 + 68q40−80q38 + 80q36−70q34 + 52q32−34q30−3q28 + 44q26−86q24 + 120q22−150q20 + 170q18−168q16 + 166q14−137q12 + 110q10−66q8 + 30q6 + 7q4−48q2 + 72−84q−2 + 84q−4−86q−6 + 72q−8−56q−10 + 40q−12−32q−14 + 22q−16−12q−18 + 10q−20−4q−22 + 4q−24 + q−28
2,0	q56 + q50 + 2q48−3q44 + 3q40−3q38−5q36 + q34 + 4q32−2q30−5q28 + 3q26 + 2q24−3q22 + q20 + 4q18 + 3q12−2q8 + 3q6 + 6q4−q2−3 + 4q−2 + q−4−5q−6−5q−8−q−10−3q−14−q−16 + 2q−18 + 3q−20 + q−22 + q−24 + q−26 + q−28

A3 Invariants.

Weight	Invariant
0,1,0	q48−q46 + 2q42−3q40 + 6q36−5q34−2q32 + 8q30−4q28−3q26 + 6q24−2q22−3q20 + q18 + 2q16−2q12 + 6q10 + 2q8−7q6 + 2q4 + 2q2−9 + q−2 + 2q−4−4q−6 + 3q−8 + 2q−10 + 2q−14 + 2q−16 + q−20
1,0,0	q29 + q25 + 2q21−q19 + q17−q15−q11−2q5−2q−q−3 + 2q−5 + q−7 + 2q−9 + q−11 + q−13

B2 Invariants.

Weight	Invariant
0,1	q48−q46 + 2q44−4q42 + 5q40−6q38 + 8q36−7q34 + 8q32−6q30 + 4q28 + q26−4q24 + 8q22−11q20 + 13q18−16q16 + 14q14−14q12 + 10q10−8q8 + 3q6−4q2 + 5−7q−2 + 8q−4−6q−6 + 7q−8−4q−10 + 4q−12−2q−14 + 2q−16 + q−20
1,0	q78−q74−q72 + q70 + 3q68−4q64−3q62 + 3q60 + 7q58−7q54−5q52 + 5q50 + 8q48−q46−8q44−2q42 + 6q40 + 4q38−5q36−5q34 + 3q32 + 5q30−q28−5q26 + q24 + 5q22 + 2q20−4q18−q16 + 5q14 + 4q12−5q10−7q8 + 2q6 + 8q4 + q2−8−7q−2 + 4q−4 + 7q−6−q−8−6q−10−3q−12 + 4q−14 + 3q−16−2q−20 + 2q−24 + 2q−26 + q−34

G2 Invariants.

Weight

Invariant

1,0

q114−q112 + 2q110−3q108 + 2q106−q104−2q102 + 6q100−8q98 + 10q96−9q94 + 5q92 + 2q90−11q88 + 19q86−21q84 + 18q82−12q80−q78 + 15q76−23q74 + 29q72−21q70 + 9q68 + 5q66−16q64 + 19q62−13q60 + 2q58 + 14q56−19q54 + 16q52 + q50−19q48 + 33q46−37q44 + 25q42−7q40−18q38 + 36q36−42q34 + 36q32−20q30−5q28 + 19q26−29q24 + 25q22−17q20 + q18 + 13q16−18q14 + 15q12−q10−15q8 + 24q6−24q4 + 10q2 + 5−21q−2 + 31q−4−28q−6 + 19q−8−3q−10−14q−12 + 20q−14−19q−16 + 16q−18−8q−20 + q−22 + 5q−24−6q−26 + 9q−28−6q−30 + 4q−32 + 2q−38−2q−40 + 2q−42 + q−46

.

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["10 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−4t2 + 11t−13 + 11t−1−4t−2

In[5]:=

Conway[K][z]

Out[5]=

−4z4−5z2 + 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]=

{ 43, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a6 + a6−z4a4−z2a4−2z4a2−4z2a2−a2−z4−2z2−1 + z2a−2 + 2a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a3z9 + az9 + 2a4z8 + 3a2z8 + z8 + 3a5z7−2a3z7−4az7 + z7a−1 + 3a6z6−4a4z6−10a2z6 + z6a−2−2z6 + 2a7z5−7a5z5 + 5a3z5 + 11az5−3z5a−1 + a8z4−6a6z4 + 5a4z4 + 16a2z4−5z4a−2−z4−3a7z3 + 9a5z3−5a3z3−16az3 + z3a−1−2a8z2 + 5a6z2−12a2z2 + 7z2a−2 + 2z2−2a5z + 2a3z + 5az + za−1−a6 + a2−2a−2−1

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.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 11"];

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]=

{ −4t2 + 11t−13 + 11t−1−4t−2, q3−q2 + 3q−5 + 6q−1−7q−2 + 7q−3−6q−4 + 4q−5−2q−6 + q−7 }

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	q10−q9 + 3q7−4q6−2q5 + 10q4−9q3−8q2 + 23q−12−19q−1 + 35q−2−10q−3−31q−4 + 41q−5−5q−6−37q−7 + 39q−8−q−9−32q−10 + 27q−11 + 2q−12−19q−13 + 12q−14 + 2q−15−8q−16 + 4q−17 + q−18−2q−19 + q−20
3	q21−q20 + 2q17−3q16 + q14 + 5q13−8q12−4q11 + 6q10 + 16q9−14q8−20q7 + 8q6 + 38q5−7q4−48q3−4q2 + 61q + 17−68q−1−34q−2 + 72q−3 + 52q−4−74q−5−66q−6 + 69q−7 + 84q−8−67q−9−95q−10 + 60q−11 + 103q−12−53q−13−105q−14 + 43q−15 + 101q−16−32q−17−90q−18 + 20q−19 + 76q−20−12q−21−56q−22 + 4q−23 + 39q−24 + q−25−27q−26 + q−27 + 14q−28 + 2q−29−11q−30 + q−31 + 5q−32 + q−33−5q−34 + q−35 + q−36 + q−37−2q−38 + q−39
4	q36−q35−q32 + 3q31−3q30 + q29 + 2q28−5q27 + 6q26−8q25 + 2q24 + 10q23−7q22 + 11q21−24q20−5q19 + 22q18 + 3q17 + 35q16−49q15−35q14 + 16q13 + 15q12 + 100q11−52q10−74q9−33q8−13q7 + 188q6−7q5−75q4−98q3−106q2 + 238q + 66−6q−1−132q−2−238q−3 + 226q−4 + 119q−5 + 104q−6−118q−7−356q−8 + 171q−9 + 138q−10 + 216q−11−81q−12−443q−13 + 109q−14 + 141q−15 + 303q−16−41q−17−492q−18 + 48q−19 + 126q−20 + 355q−21 + 8q−22−484q−23−8q−24 + 77q−25 + 350q−26 + 67q−27−398q−28−43q−29 + 274q−31 + 103q−32−255q−33−30q−34−61q−35 + 154q−36 + 92q−37−122q−38 + 8q−39−70q−40 + 59q−41 + 49q−42−52q−43 + 33q−44−43q−45 + 17q−46 + 16q−47−27q−48 + 28q−49−19q−50 + 8q−51 + 5q−52−16q−53 + 13q−54−7q−55 + 4q−56 + 3q−57−7q−58 + 4q−59−2q−60 + q−61 + q−62−2q−63 + q−64