10 51

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 51"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 16 18 20 6 12 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[32,21,2]

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,{1,1,2,−1,2,2,−3,2,2,−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[{9, 4}, {3, 7}, {6, 8}, {7, 9}, {8, 11}, {5, 10}, {4, 6}, {2, 5}, {12, 3}, {11, 13}, {1, 12}, {13, 2}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_51.

A1 Invariants.

Weight	Invariant
1	−q5 + 2q3−3q + 3q−1−q−3 + 2q−5 + 2q−7−2q−9 + 3q−11−3q−13 + q−15−q−17
2	q16−2q14−q12 + 7q10−6q8−8q6 + 17q4−4q2−19 + 20q−2 + 5q−4−22q−6 + 12q−8 + 12q−10−12q−12−2q−14 + 11q−16 + 5q−18−17q−20 + 5q−22 + 19q−24−21q−26−5q−28 + 21q−30−13q−32−9q−34 + 13q−36−3q−38−5q−40 + 4q−42−q−46 + q−48
3	−q33 + 2q31 + q29−3q27−4q25 + 6q23 + 11q21−11q19−21q17 + 11q15 + 37q13−4q11−59q9−10q7 + 77q5 + 35q3−85q−72q−1 + 85q−3 + 101q−5−66q−7−124q−9 + 42q−11 + 130q−13−6q−15−122q−17−17q−19 + 99q−21 + 45q−23−69q−25−64q−27 + 34q−29 + 82q−31 + 5q−33−95q−35−36q−37 + 95q−39 + 75q−41−95q−43−102q−45 + 72q−47 + 118q−49−48q−51−121q−53 + 13q−55 + 111q−57 + 10q−59−84q−61−29q−63 + 57q−65 + 34q−67−31q−69−26q−71 + 12q−73 + 18q−75−4q−77−8q−79 + q−81 + 4q−83−q−85−q−87 + q−91−q−93
4	q56−2q54−q52 + 3q50 + 4q46−9q44−6q42 + 12q40 + 6q38 + 16q36−32q34−34q32 + 24q30 + 38q28 + 69q26−62q24−122q22−23q20 + 85q18 + 233q16−5q14−255q12−233q10 + 14q8 + 468q6 + 275q4−234q2−548−326q−2 + 524q−4 + 672q−6 + 92q−8−666q−10−758q−12 + 238q−14 + 831q−16 + 522q−18−431q−20−920q−22−168q−24 + 621q−26 + 707q−28−54q−30−718q−32−418q−34 + 245q−36 + 613q−38 + 240q−40−361q−42−505q−44−105q−46 + 421q−48 + 458q−50 + 7q−52−554q−54−431q−56 + 210q−58 + 644q−60 + 385q−62−521q−64−729q−66−100q−68 + 692q−70 + 752q−72−277q−74−832q−76−471q−78 + 445q−80 + 895q−82 + 126q−84−580q−86−660q−88 + 12q−90 + 666q−92 + 376q−94−143q−96−496q−98−258q−100 + 255q−102 + 301q−104 + 122q−106−185q−108−219q−110 + 7q−112 + 98q−114 + 117q−116−12q−118−79q−120−26q−122 + 40q−126 + 10q−128−14q−130−3q−132−8q−134 + 7q−136 + 2q−138−3q−140 + 2q−142−2q−144 + q−146−q−150 + q−152
5	−q85 + 2q83 + q81−3q79−q73 + 4q71 + 5q69−8q67−9q65 + 2q63 + 9q61 + 17q59 + 9q57−21q55−51q53−24q51 + 46q49 + 93q47 + 76q45−41q43−179q41−201q39−6q37 + 282q35 + 401q33 + 181q31−317q29−706q27−557q25 + 209q23 + 1028q21 + 1132q19 + 211q17−1202q15−1891q13−1017q11 + 1049q9 + 2644q7 + 2157q5−384q3−3097q−3513q−1−821q−3 + 3066q−5 + 4732q−7 + 2373q−9−2350q−11−5499q−13−4052q−15 + 1114q−17 + 5615q−19 + 5393q−21 + 456q−23−5006q−25−6196q−27−1970q−29 + 3889q−31 + 6262q−33 + 3182q−35−2507q−37−5732q−39−3854q−41 + 1140q−43 + 4764q−45 + 4068q−47 + 10q−49−3660q−51−3884q−53−874q−55 + 2552q−57 + 3565q−59 + 1528q−61−1627q−63−3235q−65−2069q−67 + 824q−69 + 3006q−71 + 2651q−73−67q−75−2927q−77−3322q−79−714q−81 + 2802q−83 + 4097q−85 + 1704q−87−2597q−89−4878q−91−2805q−93 + 2041q−95 + 5433q−97 + 4067q−99−1140q−101−5593q−103−5197q−105−139q−107 + 5156q−109 + 6002q−111 + 1594q−113−4116q−115−6198q−117−2986q−119 + 2618q−121 + 5729q−123 + 3923q−125−915q−127−4596q−129−4283q−131−600q−133 + 3131q−135 + 3936q−137 + 1626q−139−1575q−141−3105q−143−2061q−145 + 340q−147 + 2051q−149 + 1922q−151 + 436q−153−1051q−155−1462q−157−745q−159 + 343q−161 + 910q−163 + 686q−165 + 55q−167−443q−169−495q−171−183q−173 + 157q−175 + 276q−177 + 169q−179−19q−181−128q−183−103q−185−24q−187 + 43q−189 + 57q−191 + 20q−193−14q−195−17q−197−12q−199−4q−201 + 11q−203 + 6q−205−3q−207−3q−213 + q−215 + 2q−217−q−219 + q−223−q−225

A2 Invariants.

Weight	Invariant
1,0	−q6 + q4−q2−1 + 2q−2−2q−4 + 3q−6 + q−8 + 2q−10 + 3q−12−q−14 + 2q−16−2q−18−2q−20−q−24
1,1	q20−4q18 + 12q16−28q14 + 56q12−98q10 + 160q8−240q6 + 325q4−414q2 + 488−532q−2 + 511q−4−436q−6 + 294q−8−86q−10−161q−12 + 436q−14−670q−16 + 884q−18−1004q−20 + 1056q−22−1000q−24 + 866q−26−664q−28 + 408q−30−160q−32−88q−34 + 284q−36−428q−38 + 496q−40−502q−42 + 467q−44−398q−46 + 310q−48−230q−50 + 161q−52−108q−54 + 66q−56−40q−58 + 25q−60−12q−62 + 6q−64−2q−66 + q−68
2,0	q18−q16−2q14 + 3q12 + 3q10−4q8−5q6 + 5q4 + 6q2−10−6q−2 + 10q−4 + 2q−6−10q−8 + 8q−12−2q−14−2q−16 + 8q−18 + 7q−20−3q−22 + 9q−24 + 9q−26−7q−28−3q−30 + 9q−32−12q−36−4q−38 + 5q−40−4q−42−11q−44−q−46 + 4q−48−q−52 + q−54 + 2q−56 + q−58 + q−62

A3 Invariants.

Weight	Invariant
0,1,0	q14−2q12 + q10 + 4q8−8q6 + 2q4 + 8q2−16 + 4q−2 + 13q−4−19q−6 + 2q−8 + 14q−10−10q−12 + 13q−16 + 6q−18 + 2q−20 + 2q−22 + 14q−24−5q−26−16q−28 + 12q−30−6q−32−20q−34 + 13q−36−12q−40 + 8q−42 + q−44−4q−46 + 3q−48 + q−50−q−52 + q−54
1,0,0	−q7 + q5−2q3 + q−2q−1 + 2q−3−2q−5 + 2q−7 + q−9 + 2q−11 + 3q−13 + 2q−15 + 4q−17−q−19 + 2q−21−3q−23−q−25−3q−27−q−31

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q14 + 3q10−q8−4q6 + 3q4 + 2q2−8−5q−2 + 8q−4 + 2q−6−15q−8 + 12q−12−9q−14−14q−16 + 12q−18 + 9q−20−4q−22 + 12q−24 + 26q−26 + 10q−28 + 3q−30 + 19q−32 + 7q−34−19q−36−7q−38 + 2q−40−19q−42−21q−44 + q−46 + 2q−48−10q−50−4q−52 + 8q−54 + 4q−56−3q−58 + 3q−60 + 4q−62 + q−68
1,0,0,0	−q8 + q6−2q4−2q−2 + 2q−4−2q−6 + 2q−8 + 2q−12 + 2q−14 + 3q−16 + 3q−18 + 3q−20 + 4q−22−q−24 + 2q−26−3q−28−2q−30−2q−32−3q−34−q−38

B2 Invariants.

Weight	Invariant
0,1	−q14 + 2q12−5q10 + 8q8−12q6 + 16q4−20q2 + 20−20q−2 + 17q−4−9q−6 + 2q−8 + 10q−10−18q−12 + 30q−14−35q−16 + 40q−18−38q−20 + 36q−22−28q−24 + 19q−26−8q−28−2q−30 + 10q−32−16q−34 + 19q−36−20q−38 + 18q−40−16q−42 + 11q−44−8q−46 + 5q−48−3q−50 + q−52−q−54
1,0	q24−2q20−2q18 + 3q16 + 6q14−q12−10q10−6q8 + 10q6 + 14q4−6q2−20−6q−2 + 19q−4 + 16q−6−13q−8−21q−10 + q−12 + 20q−14 + 6q−16−14q−18−9q−20 + 13q−22 + 13q−24−5q−26−10q−28 + 8q−30 + 16q−32 + q−34−13q−36 + 16q−40 + 6q−42−17q−44−14q−46 + 11q−48 + 17q−50−7q−52−24q−54−8q−56 + 17q−58 + 14q−60−9q−62−17q−64−2q−66 + 12q−68 + 6q−70−4q−72−6q−74 + 4q−78 + 2q−80−q−82−q−84 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−2q16 + 3q14−4q12 + 7q10−10q8 + 10q6−14q4 + 15q2−18 + 14q−2−15q−4 + 14q−6−10q−8 + 2q−10 + q−12−4q−14 + 13q−16−19q−18 + 24q−20−21q−22 + 37q−24−24q−26 + 35q−28−22q−30 + 32q−32−17q−34 + 13q−36−15q−38−q−40−q−42−12q−44 + 4q−46−17q−48 + 15q−50−15q−52 + 14q−54−15q−56 + 13q−58−9q−60 + 7q−62−6q−64 + 5q−66−2q−68 + 2q−70−q−72 + q−74

G2 Invariants.

Weight

Invariant

1,0

q32−2q30 + 5q28−8q26 + 8q24−6q22−3q20 + 16q18−31q16 + 42q14−44q12 + 24q10 + 7q8−48q6 + 87q4−101q2 + 88−42q−2−28q−4 + 91q−6−127q−8 + 120q−10−70q−12−2q−14 + 67q−16−98q−18 + 82q−20−25q−22−44q−24 + 92q−26−95q−28 + 44q−30 + 39q−32−116q−34 + 163q−36−147q−38 + 84q−40 + 20q−42−116q−44 + 180q−46−180q−48 + 130q−50−33q−52−57q−54 + 120q−56−126q−58 + 89q−60−17q−62−50q−64 + 83q−66−73q−68 + 18q−70 + 51q−72−103q−74 + 113q−76−78q−78 + 6q−80 + 62q−82−114q−84 + 123q−86−94q−88 + 37q−90 + 16q−92−60q−94 + 74q−96−66q−98 + 43q−100−15q−102−7q−104 + 19q−106−24q−108 + 20q−110−14q−112 + 8q−114−q−116−3q−118 + 4q−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 51"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−7t2 + 15t−19 + 15t−1−7t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 5z4 + 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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 2q7−5q6 + 8q5−10q4 + 12q3−10q2 + 9q−6 + 3q−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 + 3z4a−2 + 4z4a−4−z4a−6−z4 + 3z2a−2 + 7z2a−4−3z2a−6−2z2 + a−2 + 4a−4−3a−6−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z9a−3 + z9a−5 + 3z8a−2 + 6z8a−4 + 3z8a−6 + 4z7a−1 + 6z7a−3 + 5z7a−5 + 3z7a−7−z6a−2−12z6a−4−6z6a−6 + 2z6a−8 + 3z6 + az5−6z5a−1−16z5a−3−16z5a−5−6z5a−7 + z5a−9−6z4a−2 + 13z4a−4 + 9z4a−6−4z4a−8−6z4−2az3 + 15z3a−3 + 21z3a−5 + 5z3a−7−3z3a−9 + 4z2a−2−8z2a−4−8z2a−6 + z2a−8 + 3z2 + az−5za−3−9za−5−3za−7 + 2za−9−a−2 + 4a−4 + 3a−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 51"];

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]=

{ 2t3−7t2 + 15t−19 + 15t−1−7t−2 + 2t−3, −q8 + 2q7−5q6 + 8q5−10q4 + 12q3−10q2 + 9q−6 + 3q−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]=

{}

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 + q21 + 5q20−11q19 + 3q18 + 21q17−33q16−q15 + 55q14−59q13−17q12 + 95q11−73q10−39q9 + 117q8−67q7−52q6 + 107q5−43q4−52q3 + 73q2−16q−37 + 34q−1−q−2−16q−3 + 9q−4 + q−5−3q−6 + q−7
3	−q45 + 2q44−q43−q42−q41 + 7q40−4q39−10q38 + 3q37 + 29q36−10q35−48q34−2q33 + 94q32 + 13q31−134q30−57q29 + 188q28 + 114q27−232q26−191q25 + 261q24 + 280q23−278q22−365q21 + 268q20 + 450q19−258q18−496q17 + 209q16 + 550q15−181q14−544q13 + 111q12 + 545q11−67q10−490q9−5q8 + 440q7 + 49q6−354q5−93q4 + 274q3 + 107q2−187q−109 + 117q−1 + 94q−2−67q−3−67q−4 + 30q−5 + 45q−6−12q−7−26q−8 + 4q−9 + 13q−10−2q−11−4q−12−q−13 + 3q−14−q−15
4	q74−2q73 + q72 + q71−3q70 + 5q69−7q68 + 6q67 + 6q66−18q65 + 10q64−18q63 + 30q62 + 36q61−58q60−16q59−71q58 + 97q57 + 165q56−77q55−107q54−297q53 + 131q52 + 472q51 + 102q50−153q49−810q48−107q47 + 825q46 + 621q45 + 137q44−1464q43−779q42 + 905q41 + 1327q40 + 906q39−1914q38−1695q37 + 544q36 + 1882q35 + 1935q34−1974q33−2487q32−85q31 + 2090q30 + 2841q29−1715q28−2921q27−726q26 + 1967q25 + 3402q24−1264q23−2958q22−1252q21 + 1567q20 + 3546q19−663q18−2585q17−1620q16 + 904q15 + 3246q14 + q13−1824q12−1706q11 + 115q10 + 2494q9 + 490q8−871q7−1397q6−478q5 + 1498q4 + 582q3−113q2−817q−626 + 648q−1 + 360q−2 + 201q−3−308q−4−433q−5 + 194q−6 + 113q−7 + 179q−8−58q−9−195q−10 + 46q−11 + 5q−12 + 80q−13 + 2q−14−64q−15 + 15q−16−9q−17 + 22q−18 + 4q−19−16q−20 + 5q−21−3q−22 + 4q−23 + q−24−3q−25 + q−26
5	−q110 + 2q109−q108−q107 + 3q106−q105−5q104 + 5q103−q102−4q101 + 12q100 + 4q99−20q98−3q97−6q96−q95 + 46q94 + 41q93−34q92−70q91−85q90−26q89 + 155q88 + 229q87 + 73q86−189q85−425q84−338q83 + 207q82 + 727q81 + 704q80 + 35q79−992q78−1426q77−510q76 + 1138q75 + 2191q74 + 1521q73−863q72−3137q71−2911q70 + 94q69 + 3721q68 + 4722q67 + 1447q66−3942q65−6642q64−3589q63 + 3408q62 + 8403q61 + 6285q60−2136q59−9753q58−9193q57 + 196q56 + 10485q55 + 11995q54 + 2272q53−10599q52−14488q51−4862q50 + 10089q49 + 16448q48 + 7479q47−9233q46−17880q45−9708q44 + 8016q43 + 18729q42 + 11780q41−6840q40−19175q39−13224q38 + 5408q37 + 19124q36 + 14640q35−4122q34−18820q33−15406q32 + 2515q31 + 17958q30 + 16248q29−967q28−16783q27−16419q26−911q25 + 14948q24 + 16472q23 + 2703q22−12725q21−15723q20−4535q19 + 9954q18 + 14594q17 + 5928q16−7036q15−12643q14−6908q13 + 4095q12 + 10368q11 + 7118q10−1574q9−7706q8−6686q7−406q6 + 5202q5 + 5671q4 + 1575q3−2983q2−4327q−2072 + 1315q−1 + 2979q−2 + 1991q−3−270q−4−1786q−5−1585q−6−280q−7 + 913q−8 + 1117q−9 + 419q−10−373q−11−664q−12−384q−13 + 94q−14 + 351q−15 + 270q−16 + 16q−17−166q−18−164q−19−25q−20 + 63q−21 + 75q−22 + 38q−23−28q−24−47q−25−8q−26 + 16q−27 + 5q−28 + 11q−29 + 2q−30−17q−31 + 8q−33−2q−34 + 3q−36−4q−37−q−38 + 3q−39−q−40