10 103

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["10 103"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X18,6,19,5 X20,13,1,14 X16,7,17,8 X10,3,11,4 X4,11,5,12 X14,9,15,10 X8,15,9,16 X12,19,13,20 X2,18,3,17

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 18 16 14 4 20 8 2 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[30:2:2]

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,−2,1,3,−2,−2,3,−2,−2,3})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_103.

A1 Invariants.

Weight	Invariant
1	−q17 + 2q15−3q13 + 3q11−3q9 + q7 + 2q5−q3 + 4q−3q−1 + 2q−3−q−5
2	q48−2q46 + 6q42−8q40−4q38 + 19q36−10q34−19q32 + 26q30−27q26 + 20q24 + 11q22−20q20−q18 + 13q16−q14−19q12 + 13q10 + 19q8−26q6 + 2q4 + 27q2−19−9q−2 + 19q−4−5q−6−7q−8 + 6q−10−q−12−2q−14 + q−16
3	−q93 + 2q91−3q87 + 7q83 + q81−18q79−5q77 + 34q75 + 22q73−47q71−58q69 + 48q67 + 103q65−24q63−143q61−25q59 + 165q57 + 82q55−158q53−132q51 + 125q49 + 164q47−83q45−170q43 + 36q41 + 153q39 + 16q37−131q35−50q33 + 94q31 + 91q29−64q27−128q25 + 20q23 + 159q21 + 25q19−175q17−73q15 + 168q13 + 124q11−134q9−151q7 + 82q5 + 160q3−29q−131q−1−19q−3 + 95q−5 + 37q−7−53q−9−34q−11 + 22q−13 + 22q−15−8q−17−12q−19 + 5q−21 + 4q−23−3q−25−2q−27 + q−29 + 2q−31−q−33
4	q152−2q150 + 3q146−3q144 + q142−5q140 + 7q138 + 16q136−17q134−17q132−28q130 + 31q128 + 93q126 + 6q124−76q122−177q120−34q118 + 246q116 + 248q114 + 47q112−424q110−459q108 + 95q106 + 597q104 + 677q102−224q100−974q98−687q96 + 363q94 + 1355q92 + 666q90−793q88−1447q86−557q84 + 1248q82 + 1443q80 + 59q78−1390q76−1282q74 + 494q72 + 1410q70 + 729q68−738q66−1277q64−178q62 + 874q60 + 884q58−132q56−910q54−543q52 + 378q50 + 870q48 + 316q46−600q44−909q42−79q40 + 938q38 + 866q36−236q34−1315q32−732q30 + 778q28 + 1434q26 + 479q24−1294q22−1398q20 + 57q18 + 1430q16 + 1246q14−530q12−1410q10−784q8 + 623q6 + 1304q4 + 351q2−631−903q−2−218q−4 + 619q−6 + 523q−8 + 85q−10−394q−12−359q−14 + 53q−16 + 187q−18 + 189q−20−26q−22−128q−24−41q−26−7q−28 + 58q−30 + 17q−32−16q−34 + 2q−36−13q−38 + 6q−40−q−42−4q−44 + 6q−46−q−48 + 2q−50−q−52−2q−54 + q−56

A2 Invariants.

Weight	Invariant
1,0	−q24 + q22−q20−q18 + 2q16−3q14 + q12 + q8 + 4q6−q4 + 3q2−1−q−2 + q−4−q−6
1,1	q68−4q66 + 10q64−22q62 + 46q60−78q58 + 124q56−202q54 + 301q52−406q50 + 528q48−644q46 + 711q44−702q42 + 606q40−428q38 + 145q36 + 198q34−556q32 + 896q30−1167q28 + 1354q26−1420q24 + 1360q22−1198q20 + 912q18−578q16 + 228q14 + 115q12−388q10 + 596q8−672q6 + 682q4−632q2 + 534−418q−2 + 311q−4−224q−6 + 146q−8−92q−10 + 54q−12−28q−14 + 12q−16−4q−18 + q−20
2,0	q62−q60 + 2q56−q54−2q52−q50 + 7q48 + q46−11q44 + 9q40−q38−13q36 + 2q34 + 12q32−2q30−10q28 + 5q26 + 3q24−11q22 + 3q20 + 5q18−3q16 + 12q12 + 3q10−10q8 + 4q6 + 14q4−6q2−11 + 7q−2 + 7q−4−5q−6−6q−8 + 2q−10 + 3q−12−2q−14−q−16 + q−18

A3 Invariants.

Weight	Invariant
0,1,0	q54−2q52 + 4q48−6q46 + 2q44 + 11q42−15q40 + 2q38 + 17q36−22q34−q32 + 17q30−16q28−4q26 + 10q24−3q22−6q20−q18 + 12q16−10q12 + 22q10 + 5q8−20q6 + 18q4 + 2q2−17 + 9q−2 + q−4−8q−6 + 4q−8 + q−10−2q−12 + q−14
1,0,0	−q31 + q29−2q27 + q25−2q23 + 2q21−3q19 + q17−q15 + q13 + 2q11 + 2q9 + 4q7−q5 + 3q3−2q + q−1−2q−3 + q−5−q−7

A4 Invariants.

Weight	Invariant
0,1,0,0	q68−q66−2q64 + 4q62 + q60−8q58 + 4q56 + 11q54−6q52−8q50 + 12q48 + 9q46−18q44−10q42 + 14q40−6q38−22q36 + 8q34 + 10q32−18q30−2q28 + 15q26−3q24−9q22 + 17q20 + 19q18−9q16 + 2q14 + 22q12 + q10−16q8 + 5q6 + 7q4−8q2−8 + 2q−2 + 2q−4−4q−6−q−8 + 3q−10−q−14 + q−16
1,0,0,0	−q38 + q36−2q34−2q28 + 2q26−3q24 + q22−q20 + q16 + 2q14 + 3q12 + 2q10 + 4q8−q6 + 3q4−2q2−2q−4 + q−6−q−8

B2 Invariants.

Weight	Invariant
0,1	−q54 + 2q52−4q50 + 8q48−12q46 + 16q44−23q42 + 25q40−26q38 + 23q36−16q34 + 9q32 + 3q30−16q28 + 30q26−42q24 + 47q22−50q20 + 47q18−42q16 + 32q14−16q12 + 6q10 + 9q8−14q6 + 24q4−26q2 + 25−23q−2 + 17q−4−12q−6 + 8q−8−5q−10 + 2q−12−q−14
1,0	q88−2q84−2q82 + 2q80 + 6q78−9q74−6q72 + 9q70 + 17q68−3q66−23q64−10q62 + 21q60 + 22q58−12q56−28q54−3q52 + 25q50 + 12q48−19q46−17q44 + 11q42 + 17q40−7q38−18q36 + q34 + 17q32 + 2q30−17q28−6q26 + 18q24 + 14q22−14q20−16q18 + 13q16 + 28q14−q12−26q10−12q8 + 23q6 + 21q4−10q2−23−4q−2 + 16q−4 + 9q−6−7q−8−10q−10−q−12 + 6q−14 + 3q−16−2q−18−2q−20 + q−24

D4 Invariants.

Weight	Invariant
1,0,0,0	q74−2q72 + 2q70−4q68 + 7q66−9q64 + 11q62−13q60 + 19q58−20q56 + 20q54−19q52 + 19q50−17q48 + 6q46−7q44−q42 + 8q40−21q38 + 22q36−29q34 + 37q32−38q30 + 36q28−38q26 + 38q24−27q22 + 26q20−17q18 + 15q16 + 4q14 + 10q10−14q8 + 20q6−20q4 + 18q2−22 + 17q−2−15q−4 + 10q−6−10q−8 + 7q−10−4q−12 + 3q−14−2q−16 + q−18

G2 Invariants.

Weight

Invariant

1,0

q128−2q126 + 4q124−7q122 + 7q120−6q118 + 12q114−22q112 + 35q110−42q108 + 35q106−15q104−25q102 + 72q100−108q98 + 118q96−88q94 + 18q92 + 74q90−153q88 + 184q86−149q84 + 53q82 + 55q80−142q78 + 160q76−106q74 + 11q72 + 92q70−143q68 + 115q66−28q64−92q62 + 180q60−204q58 + 150q56−38q54−94q52 + 208q50−252q48 + 218q46−117q44−25q42 + 148q40−206q38 + 189q36−98q34−12q32 + 112q30−142q28 + 98q26−2q24−100q22 + 159q20−140q18 + 58q16 + 52q14−136q12 + 176q10−149q8 + 80q6 + 3q4−78q2 + 111−107q−2 + 78q−4−34q−6−3q−8 + 28q−10−42q−12 + 39q−14−29q−16 + 15q−18−3q−20−6q−22 + 8q−24−8q−26 + 5q−28−2q−30 + q−32

.

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["10 103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−8t2 + 17t−21 + 17t−1−8t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 4z4 + 3z2 + 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]=

{5,t + 1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 75, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q2 + 3q−6 + 10q−1−11q−2 + 13q−3−12q−4 + 9q−5−6q−6 + 3q−7−q−8

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4a6−2z2a6−a6 + z6a4 + 3z4a4 + 3z2a4 + z6a2 + 3z4a2 + 4z2a2 + 3a2−z4−2z2−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z5a9−2z3a9 + 3z6a8−6z4a8 + z2a8 + 5z7a7−12z5a7 + 10z3a7−4za7 + 5z8a6−12z6a6 + 13z4a6−6z2a6 + a6 + 2z9a5 + 3z7a5−16z5a5 + 21z3a5−6za5 + 9z8a4−23z6a4 + 25z4a4−8z2a4 + 2z9a3 + 2z7a3−9z5a3 + 9z3a3−2za3 + 4z8a2−5z6a2 + 2z2a2−3a2 + 4z7a−5z5a−2z3a + za + 3z6−6z4 + 3z2−1 + z5a−1−2z3a−1 + za−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["10 103"];

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]=

{ 2t3−8t2 + 17t−21 + 17t−1−8t−2 + 2t−3, −q2 + 3q−6 + 10q−1−11q−2 + 13q−3−12q−4 + 9q−5−6q−6 + 3q−7−q−8 }

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_40,}

In[6]:=

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

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

Out[6]=

{10_40,}

n	Jn
2	q7−3q6 + q5 + 8q4−16q3 + 3q2 + 32q−44−7q−1 + 78q−2−69q−3−35q−4 + 123q−5−75q−6−67q−7 + 141q−8−61q−9−81q−10 + 122q−11−30q−12−72q−13 + 75q−14−3q−15−46q−16 + 30q−17 + 6q−18−17q−19 + 7q−20 + 2q−21−3q−22 + q−23
3	−q15 + 3q14−q13−3q12−2q11 + 10q10−20q8 + 2q7 + 40q6−76q4−17q3 + 130q2 + 58q−190−129q−1 + 232q−2 + 247q−3−268q−4−362q−5 + 249q−6 + 505q−7−224q−8−603q−9 + 147q−10 + 705q−11−90q−12−742q−13−q−14 + 769q−15 + 65q−16−739q−17−145q−18 + 688q−19 + 212q−20−602q−21−262q−22 + 482q−23 + 299q−24−355q−25−301q−26 + 225q−27 + 273q−28−115q−29−218q−30 + 35q−31 + 155q−32 + 4q−33−91q−34−20q−35 + 49q−36 + 15q−37−22q−38−8q−39 + 10q−40 + 2q−41−3q−42−2q−43 + 3q−44−q−45
4	q26−3q25 + q24 + 3q23−3q22 + 8q21−13q20 + 4q19 + 10q18−22q17 + 23q16−31q15 + 37q14 + 51q13−87q12−11q11−118q10 + 139q9 + 266q8−89q7−145q6−530q5 + 104q4 + 745q3 + 349q2−49q−1367−581q−1 + 1017q−2 + 1331q−3 + 904q−4−2048q−5−1988q−6 + 391q−7 + 2211q−8 + 2680q−9−1864q−10−3361q−11−1064q−12 + 2315q−13 + 4453q−14−909q−15−4017q−16−2574q−17 + 1732q−18 + 5532q−19 + 193q−20−3945q−21−3591q−22 + 902q−23 + 5841q−24 + 1109q−25−3391q−26−4083q−27−19q−28 + 5474q−29 + 1887q−30−2375q−31−4093q−32−1071q−33 + 4375q−34 + 2426q−35−908q−36−3412q−37−1987q−38 + 2599q−39 + 2318q−40 + 541q−41−2028q−42−2182q−43 + 794q−44 + 1428q−45 + 1195q−46−569q−47−1493q−48−198q−49 + 378q−50 + 908q−51 + 181q−52−592q−53−278q−54−124q−55 + 354q−56 + 216q−57−121q−58−77q−59−126q−60 + 74q−61 + 73q−62−20q−63 + 5q−64−39q−65 + 12q−66 + 14q−67−9q−68 + 5q−69−6q−70 + 3q−71 + 2q−72−3q−73 + q−74
5	−q40 + 3q39−q38−3q37 + 3q36−3q35−5q34 + 9q33 + 6q32−6q31 + 11q30−5q29−31q28−12q27 + 8q26 + 27q25 + 72q24 + 56q23−65q22−174q21−161q20−q19 + 298q18 + 459q17 + 219q16−385q15−899q14−761q13 + 192q12 + 1400q11 + 1756q10 + 564q9−1671q8−3127q7−2139q6 + 1184q5 + 4518q4 + 4662q3 + 529q2−5289q−7834−3775q−1 + 4744q−2 + 10896q−3 + 8472q−4−2221q−5−13154q−6−14015q−7−2144q−8 + 13593q−9 + 19434q−10 + 8337q−11−12153q−12−23999q−13−15026q−14 + 8685q−15 + 26814q−16 + 21869q−17−4042q−18−28061q−19−27448q−20−1273q−21 + 27591q−22 + 31996q−23 + 6247q−24−26227q−25−34797q−26−10731q−27 + 24132q−28 + 36701q−29 + 14253q−30−21994q−31−37381q−32−17209q−33 + 19591q−34 + 37727q−35 + 19581q−36−17184q−37−37280q−38−21810q−39 + 14232q−40 + 36397q−41 + 23911q−42−10754q−43−34576q−44−25775q−45 + 6436q−46 + 31610q−47 + 27167q−48−1485q−49−27347q−50−27508q−51−3652q−52 + 21653q−53 + 26442q−54 + 8385q−55−15075q−56−23661q−57−11847q−58 + 8232q−59 + 19270q−60 + 13533q−61−2099q−62−13857q−63−13171q−64−2526q−65 + 8308q−66 + 11063q−67 + 5126q−68−3476q−69−7946q−70−5816q−71 + 105q−72 + 4713q−73 + 4988q−74 + 1705q−75−2036q−76−3525q−77−2172q−78 + 373q−79 + 1989q−80 + 1817q−81 + 431q−82−877q−83−1195q−84−581q−85 + 241q−86 + 640q−87 + 446q−88 + 5q−89−261q−90−266q−91−77q−92 + 109q−93 + 125q−94 + 36q−95−21q−96−41q−97−37q−98 + 12q−99 + 25q−100−2q−101−3q−102 + 3q−103−6q−104−q−105 + 6q−106−3q−107−2q−108 + 3q−109−q−110