10 12

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 12"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X11,1,12,20 X19,13,20,12 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 2 20 18 6 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4312]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_12.

A1 Invariants.

Weight	Invariant
1	−q5 + q3−q + 3q−1−q−3 + q−5 + q−7−q−9 + 2q−11−2q−13 + q−15−q−17
2	q16−q14−q12 + 2q10−3q8−2q6 + 6q4−2q2−5 + 10q−2−9q−6 + 8q−8 + 3q−10−6q−12 + 2q−14 + 4q−16−6q−20 + 4q−22 + 5q−24−10q−26 + q−28 + 8q−30−8q−32−2q−34 + 7q−36−3q−38−2q−40 + 3q−42−q−44−q−46 + q−48
3	−q33 + q31 + q29−q25 + q23 + q21−3q19−2q17 + 4q15 + 3q13−8q11−7q9 + 7q7 + 15q5−6q3−19q−q−1 + 29q−3 + 9q−5−28q−7−18q−9 + 27q−11 + 27q−13−21q−15−26q−17 + 14q−19 + 25q−21−5q−23−21q−25−2q−27 + 16q−29 + 9q−31−11q−33−19q−35 + 6q−37 + 25q−39−32q−43−6q−45 + 31q−47 + 14q−49−29q−51−20q−53 + 21q−55 + 24q−57−13q−59−21q−61 + 3q−63 + 18q−65 + 2q−67−12q−69−3q−71 + 6q−73 + 3q−75−3q−77−q−79 + 2q−81−2q−85 + q−89 + q−91−q−93
4	q56−q54−q52−q48 + 3q46−q44 + q42 + 2q40−5q38 + 2q36−3q34 + 4q32 + 11q30−7q28−5q26−16q24 + 2q22 + 27q20 + 6q18−2q16−40q14−23q12 + 29q10 + 37q8 + 40q6−41q4−70q2−21 + 40q−2 + 113q−4 + 16q−6−89q−8−99q−10−19q−12 + 143q−14 + 97q−16−40q−18−132q−20−92q−22 + 103q−24 + 124q−26 + 21q−28−96q−30−107q−32 + 35q−34 + 88q−36 + 50q−38−39q−40−79q−42−15q−44 + 44q−46 + 58q−48 + 5q−50−51q−52−63q−54 + 10q−56 + 78q−58 + 51q−60−32q−62−116q−64−33q−66 + 90q−68 + 106q−70 + 12q−72−143q−74−87q−76 + 55q−78 + 127q−80 + 78q−82−102q−84−112q−86−23q−88 + 81q−90 + 113q−92−21q−94−71q−96−65q−98 + 3q−100 + 78q−102 + 27q−104−4q−106−44q−108−33q−110 + 22q−112 + 15q−114 + 21q−116−6q−118−18q−120−6q−124 + 10q−126 + 4q−128−2q−130 + 3q−132−7q−134 + 4q−142−q−144−q−148−q−150 + q−152
5	−q85 + q83 + q81 + q77−q75−3q73−q71 + q69 + 4q65 + 4q63−3q61−7q59−5q57−q55 + 8q53 + 16q51 + 8q49−13q47−22q45−17q43 + 7q41 + 34q39 + 39q37 + 4q35−45q33−61q31−32q29 + 33q27 + 87q25 + 76q23−6q21−94q19−130q17−67q15 + 68q13 + 171q11 + 160q9 + 26q7−174q5−268q3−154q + 106q−1 + 332q−3 + 336q−5 + 34q−7−350q−9−478q−11−219q−13 + 263q−15 + 583q−17 + 426q−19−124q−21−595q−23−574q−25−54q−27 + 524q−29 + 655q−31 + 223q−33−403q−35−650q−37−324q−39 + 247q−41 + 566q−43 + 376q−45−109q−47−447q−49−364q−51 + 8q−53 + 312q−55 + 316q−57 + 64q−59−200q−61−265q−63−107q−65 + 115q−67 + 220q−69 + 142q−71−54q−73−208q−75−196q−77 + 4q−79 + 224q−81 + 261q−83 + 60q−85−241q−87−366q−89−142q−91 + 255q−93 + 465q−95 + 258q−97−220q−99−553q−101−399q−103 + 145q−105 + 589q−107 + 528q−109−5q−111−554q−113−618q−115−158q−117 + 429q−119 + 647q−121 + 316q−123−255q−125−579q−127−419q−129 + 45q−131 + 440q−133 + 455q−135 + 116q−137−260q−139−392q−141−224q−143 + 79q−145 + 283q−147 + 250q−149 + 45q−151−148q−153−207q−155−111q−157 + 37q−159 + 135q−161 + 121q−163 + 25q−165−60q−167−89q−169−54q−171 + 8q−173 + 51q−175 + 50q−177 + 16q−179−18q−181−32q−183−22q−185−3q−187 + 16q−189 + 20q−191 + 7q−193−4q−195−9q−197−10q−199−4q−201 + 6q−203 + 6q−205 + 2q−207 + 2q−209−2q−211−4q−213−q−215 + q−217 + q−221 + q−223−q−225

A2 Invariants.

Weight	Invariant
1,0	−q6 + 2q−2−q−4 + 2q−6 + q−8 + q−10 + 2q−12−q−14 + q−16−q−18−q−20−q−24
1,1	q20−2q18 + 4q16−8q14 + 15q12−22q10 + 28q8−42q6 + 55q4−70q2 + 78−96q−2 + 105q−4−98q−6 + 90q−8−56q−10 + 29q−12 + 30q−14−76q−16 + 128q−18−171q−20 + 202q−22−216q−24 + 214q−26−199q−28 + 162q−30−118q−32 + 64q−34−19q−36−32q−38 + 70q−40−92q−42 + 100q−44−102q−46 + 96q−48−80q−50 + 66q−52−54q−54 + 40q−56−28q−58 + 19q−60−12q−62 + 6q−64−2q−66 + q−68
2,0	q18−q14 + q10−2q8−4q6 + q4 + 3q2−2−2q−2 + 5q−4 + 2q−6−3q−8 + q−10 + 4q−12 + 5q−18 + 4q−20−2q−22 + 4q−24 + 4q−26−3q−28−3q−30 + 2q−32−q−34−6q−36−3q−38 + q−40−q−42−3q−44 + q−46 + 2q−48 + q−62

A3 Invariants.

Weight	Invariant
0,1,0	q14−q12 + q8−3q6 + 2q2−6 + 7q−4−7q−6 + 2q−8 + 10q−10−2q−12 + q−14 + 7q−16 + q−18−q−20 + q−22 + 4q−24−2q−26−6q−28 + 5q−30−q−32−10q−34 + 5q−36 + q−38−7q−40 + 3q−42 + q−44−3q−46 + 2q−48 + q−50−q−52 + q−54
1,0,0	−q7−q3 + q−q−1 + 2q−3−q−5 + 2q−7 + q−9 + 2q−11 + 2q−13 + q−15 + 2q−17−q−19 + q−21−2q−23−2q−27−q−31

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q6−q4−q2−3−3q−2−q−4 + q−6−5q−8 + q−10 + 8q−12 + 4q−14 + 8q−18 + 9q−20 + q−24 + 7q−26 + 4q−28−3q−30 + 4q−32 + 5q−34−6q−36−3q−38 + 2q−40−6q−42−10q−44−q−46−6q−50−4q−52 + 3q−54 + 2q−56−2q−58 + 2q−60 + 3q−62 + q−68
1,0,0,0	−q8−q4−q−2 + 2q−4−q−6 + 2q−8 + q−10 + 2q−12 + 2q−14 + 2q−16 + 2q−18 + q−20 + 2q−22−q−24 + q−26−2q−28−q−30−q−32−2q−34−q−38

B2 Invariants.

Weight	Invariant
0,1	−q14 + q12−2q10 + 3q8−5q6 + 6q4−8q2 + 8−8q−2 + 9q−4−5q−6 + 4q−8 + 2q−10−4q−12 + 11q−14−13q−16 + 17q−18−17q−20 + 17q−22−16q−24 + 12q−26−8q−28 + 3q−30 + q−32−4q−34 + 7q−36−9q−38 + 9q−40−9q−42 + 7q−44−5q−46 + 4q−48−3q−50 + q−52−q−54
1,0	q24−q20−q18 + q16 + 2q14−q12−4q10−2q8 + 3q6 + 5q4−2q2−8−3q−2 + 7q−4 + 9q−6−4q−8−9q−10−q−12 + 10q−14 + 6q−16−4q−18−5q−20 + 5q−22 + 7q−24−5q−28 + q−30 + 6q−32 + 2q−34−5q−36−3q−38 + 5q−40 + 4q−42−5q−44−7q−46 + 3q−48 + 8q−50−q−52−10q−54−6q−56 + 7q−58 + 8q−60−3q−62−9q−64−3q−66 + 6q−68 + 4q−70−2q−72−4q−74−q−76 + 3q−78 + 2q−80−q−82−q−84 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−q16 + q14−2q12 + 3q10−4q8 + 3q6−6q4 + 5q2−8 + 5q−2−7q−4 + 7q−6−5q−8 + 4q−10 + q−12 + 4q−14 + 6q−16−4q−18 + 10q−20−7q−22 + 15q−24−11q−26 + 14q−28−12q−30 + 15q−32−10q−34 + 9q−36−10q−38 + 3q−40−3q−42−2q−44−q−46−6q−48 + 5q−50−7q−52 + 6q−54−8q−56 + 7q−58−6q−60 + 4q−62−4q−64 + 4q−66−2q−68 + 2q−70−q−72 + q−74

G2 Invariants.

Weight

Invariant

1,0

q32−q30 + 2q28−3q26 + 2q24−2q22−2q20 + 6q18−9q16 + 9q14−10q12 + 6q10−10q6 + 17q4−23q2 + 23−15q−2 + q−4 + 14q−6−26q−8 + 38q−10−30q−12 + 14q−14 + 6q−16−23q−18 + 30q−20−21q−22 + 8q−24 + 12q−26−20q−28 + 21q−30−6q−32−14q−34 + 32q−36−39q−38 + 30q−40−9q−42−16q−44 + 40q−46−49q−48 + 48q−50−29q−52 + 3q−54 + 23q−56−40q−58 + 42q−60−27q−62 + 9q−64 + 13q−66−23q−68 + 22q−70−8q−72−10q−74 + 24q−76−29q−78 + 14q−80 + 4q−82−23q−84 + 34q−86−34q−88 + 22q−90−7q−92−13q−94 + 22q−96−28q−98 + 23q−100−14q−102 + 4q−104 + 4q−106−11q−108 + 13q−110−11q−112 + 8q−114−3q−116−q−118 + 3q−120−4q−122 + 3q−124−q−126 + q−128

.

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 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−6t2 + 10t−11 + 10t−1−6t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 6z4 + 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]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 2q7−4q6 + 6q5−7q4 + 8q3−7q2 + 6q−3 + 2q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + z6a−4 + 4z4a−2 + 4z4a−4−z4a−6−z4 + 5z2a−2 + 5z2a−4−3z2a−6−3z2 + 2a−2 + 2a−4−2a−6−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z9a−3 + z9a−5 + 2z8a−2 + 4z8a−4 + 2z8a−6 + 2z7a−1−z7a−3−z7a−5 + 2z7a−7−5z6a−2−14z6a−4−5z6a−6 + 2z6a−8 + 2z6 + az5−4z5a−1−z5a−3−3z5a−7 + z5a−9 + 4z4a−2 + 23z4a−4 + 8z4a−6−5z4a−8−6z4−3az3 + 5z3a−3 + 4z3a−5−z3a−7−3z3a−9 + 2z2a−2−12z2a−4−8z2a−6 + 2z2a−8 + 4z2 + az + za−1−za−3−3za−5 + 2za−9−2a−2 + 2a−4 + 2a−6−1

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 12"];

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]=

{ 2t3−6t2 + 10t−11 + 10t−1−6t−2 + 2t−3, −q8 + 2q7−4q6 + 6q5−7q4 + 8q3−7q2 + 6q−3 + 2q−1−q−2 }

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]=

{10_54,}

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	q23−2q22 + 5q20−7q19−q18 + 15q17−16q16−7q15 + 31q14−23q13−18q12 + 46q11−24q10−28q9 + 52q8−20q7−30q6 + 44q5−11q4−25q3 + 27q2−2q−15 + 12q−1 + q−2−7q−3 + 4q−4−2q−6 + q−7
3	−q45 + 2q44−q42−3q41 + 4q40 + 2q39−4q38−5q37 + 10q36 + 5q35−13q34−14q33 + 24q32 + 21q31−28q30−38q29 + 32q28 + 58q27−31q26−79q25 + 23q24 + 101q23−14q22−116q21−3q20 + 133q19 + 11q18−135q17−28q16 + 141q15 + 31q14−128q13−46q12 + 122q11 + 47q10−98q9−57q8 + 82q7 + 52q6−50q5−57q4 + 37q3 + 42q2−13q−37 + 7q−1 + 24q−2−16q−4−q−5 + 10q−6−q−7−5q−8 + 4q−10−2q−11−q−12 + 2q−14−q−15
4	q74−2q73 + q71−q70 + 6q69−6q68 + q66−8q65 + 16q64−11q63 + 6q62 + 7q61−24q60 + 22q59−29q58 + 18q57 + 34q56−30q55 + 29q54−84q53 + 7q52 + 74q51 + q50 + 80q49−159q48−61q47 + 68q46 + 51q45 + 214q44−191q43−165q42−21q41 + 61q40 + 394q39−142q38−237q37−163q36 + 5q35 + 549q34−48q33−253q32−286q31−78q30 + 633q29 + 35q28−226q27−354q26−151q25 + 645q24 + 91q23−173q22−368q21−210q20 + 581q19 + 131q18−84q17−330q16−263q15 + 439q14 + 142q13 + 33q12−227q11−284q10 + 244q9 + 102q8 + 125q7−90q6−238q5 + 82q4 + 22q3 + 135q2 + 15q−141 + 9q−1−39q−2 + 86q−3 + 44q−4−60q−5 + 6q−6−47q−7 + 34q−8 + 27q−9−22q−10 + 14q−11−26q−12 + 9q−13 + 9q−14−11q−15 + 12q−16−8q−17 + 2q−18 + 2q−19−6q−20 + 5q−21−q−22 + q−23−2q−25 + q−26