10 137

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["10 137"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 -14 2 -16 -18 -6 -20 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,211,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_137.

A1 Invariants.

Weight	Invariant
1	q13−q11 + q9−q7 + 2q−1−q−3 + q−5
2	q38−q36−2q34 + 3q32 + q30−4q28 + q26 + 3q24−2q22−2q20 + 3q18 + q16−3q14 + 2q12 + 2q10−3q8−q6 + 3q4−3 + 3q−2 + 2q−4−3q−6 + 2q−10
3	q75−q73−2q71 + 4q67 + 3q65−5q63−6q61 + 2q59 + 8q57 + 3q55−7q53−7q51 + 2q49 + 10q47 + 5q45−10q43−10q41 + 5q39 + 14q37−2q35−15q33−q31 + 14q29 + 4q27−12q25−4q23 + 11q21 + 5q19−10q17−5q15 + 6q13 + 7q11−3q9−8q7−4q5 + 10q3 + 11q−5q−1−15q−3 + 18q−7 + 3q−9−14q−11−7q−13 + 9q−15 + 8q−17−4q−19−5q−21 + q−23 + 2q−25 + q−27−q−29

A2 Invariants.

Weight	Invariant
1,0	q20 + q18−q16−q12−q10 + q8 + q4 + q−2−q−4 + q−6 + q−8
2,0	q52 + q50−3q46−2q44 + q42 + 2q40−q36 + 2q34 + 2q32−3q28−q26 + q22 + q18 + 3q16−2q10−q8 + q4−q2−1 + 2q−2 + 3q−4−3q−8 + q−10 + 2q−12 + q−20

A3 Invariants.

Weight	Invariant
0,1,0	q40−q38 + q36 + q34−2q32 + q30−q28−q26 + 2q24 + 2q18−q14−q10−q6 + 2q2−1 + q−2 + 2q−4−2q−6 + q−8 + 2q−10 + q−16
1,0,0	q27 + q25 + q23−q21−2q17−q15−q13 + q11 + q9 + q7 + q5−q−1 + q−3−q−5 + q−7 + q−9 + q−11

B2 Invariants.

Weight	Invariant
0,1	q40−q38 + 3q36−3q34 + 4q32−3q30 + 3q28−3q26−4q20 + 4q18−6q16 + 7q14−6q12 + 7q10−4q8 + 3q6 + 1−3q−2 + 4q−4−4q−6 + 3q−8−2q−10 + 2q−12 + q−16
1,0	q66−q62−q60 + 2q58 + 2q56−2q54−3q52 + q50 + 3q48−4q44−2q42 + 4q40 + 3q38−q36−3q34 + q32 + 3q30 + q28−2q26−q24 + 2q22 + q20−2q18−3q16 + q14 + 3q12−q10−4q8 + 3q4 + 2q2−2−2q−2 + 2q−4 + 4q−6−q−8−3q−10 + 2q−14 + 2q−16−q−20 + q−26

G2 Invariants.

Weight	Invariant
1,0	q94−q92 + 3q90−4q88 + 3q86−q84−4q82 + 10q80−10q78 + 10q76−4q74−4q72 + 11q70−12q68 + 8q66−q64−6q62 + 8q60−6q58−2q56 + 9q54−13q52 + 10q50−5q48−6q46 + 12q44−15q42 + 13q40−9q38 + 3q36 + 5q34−9q32 + 11q30−9q28 + 5q26 + 3q24−6q22 + 6q20−2q18−3q16 + 10q14−11q12 + 7q10 + q8−10q6 + 14q4−13q2 + 7 + q−2−7q−4 + 7q−6−5q−8 + 3q−10 + q−12−2q−14 + q−16−2q−20 + 3q−22 + 2q−28−q−30 + q−32 + q−38

.

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["10 137"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−6t + 11−6t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

z4−2z2 + 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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a6−2z2a4−2a4 + z4a2 + 2z2a2 + 2a2−2z2−1 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a4z8 + a2z8 + 2a5z7 + 4a3z7 + 2az7 + a6z6−a4z6−a2z6 + z6−8a5z5−15a3z5−7az5−4a6z4−7a4z4−5a2z4−2z4 + 8a5z3 + 15a3z3 + 9az3 + 2z3a−1 + 4a6z2 + 8a4z2 + 7a2z2 + z2a−2 + 4z2−3a5z−5a3z−3az−za−1−a6−2a4−2a2−a−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["10 137"];

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]=

{ t2−6t + 11−6t−1 + t−2, q2−2q + 4−4q−1 + 4q−2−4q−3 + 3q−4−2q−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]=

{}

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_155, K11n37,}

n	Jn
2	2q4−2q3−3q2 + 7q−1−9q−1 + 10q−2 + 2q−3−13q−4 + 8q−5 + 7q−6−13q−7 + 3q−8 + 11q−9−11q−10−2q−11 + 11q−12−6q−13−4q−14 + 6q−15−q−16−2q−17 + q−18
3	−q13 + 2q12 + q11−q10−7q9 + 3q8 + 13q7−23q5−4q4 + 30q3 + 15q2−41q−19 + 40q−1 + 31q−2−42q−3−33q−4 + 36q−5 + 36q−6−32q−7−34q−8 + 25q−9 + 31q−10−17q−11−28q−12 + 10q−13 + 23q−14−q−15−18q−16−5q−17 + 9q−18 + 12q−19−2q−20−14q−21−6q−22 + 12q−23 + 13q−24−9q−25−14q−26 + 3q−27 + 13q−28 + q−29−9q−30−3q−31 + 5q−32 + 2q−33−q−34−2q−35 + q−36
4	−q22 + 2q21 + q20−3q19−q18−4q17 + 11q16 + 9q15−10q14−17q13−22q12 + 36q11 + 48q10−7q9−58q8−81q7 + 53q6 + 124q5 + 37q4−94q3−174q2 + 29q + 192 + 105q−1−89q−2−241q−3−22q−4 + 210q−5 + 150q−6−58q−7−253q−8−59q−9 + 193q−10 + 154q−11−27q−12−228q−13−79q−14 + 162q−15 + 140q−16 + 5q−17−190q−18−96q−19 + 119q−20 + 118q−21 + 45q−22−136q−23−111q−24 + 62q−25 + 82q−26 + 77q−27−65q−28−99q−29 + 11q−30 + 24q−31 + 74q−32−53q−34−4q−35−28q−36 + 32q−37 + 21q−38−5q−39 + 18q−40−38q−41−7q−42 + 2q−43 + 6q−44 + 35q−45−14q−46−11q−47−13q−48−5q−49 + 23q−50 + q−51−7q−53−7q−54 + 6q−55 + q−56 + 2q−57−q−58−2q−59 + q−60
5	−q30−q29 + 4q28 + 4q27−q26−6q25−13q24−10q23 + 20q22 + 36q21 + 23q20−23q19−73q18−75q17 + 18q16 + 133q15 + 150q14 + 20q13−184q12−269q11−100q10 + 223q9 + 402q8 + 220q7−218q6−534q5−365q4 + 176q3 + 619q2 + 524q−95−679q−1−640q−2 + 3q−3 + 669q−4 + 733q−5 + 94q−6−652q−7−769q−8−158q−9 + 595q−10 + 777q−11 + 212q−12−550q−13−761q−14−234q−15 + 496q−16 + 729q−17 + 259q−18−445q−19−699q−20−276q−21 + 386q−22 + 660q−23 + 308q−24−316q−25−622q−26−344q−27 + 233q−28 + 567q−29 + 385q−30−131q−31−503q−32−415q−33 + 23q−34 + 412q−35 + 428q−36 + 88q−37−302q−38−415q−39−177q−40 + 176q−41 + 361q−42 + 243q−43−54q−44−279q−45−259q−46−47q−47 + 170q−48 + 234q−49 + 113q−50−70q−51−171q−52−129q−53−8q−54 + 89q−55 + 108q−56 + 50q−57−24q−58−59q−59−49q−60−20q−61 + 10q−62 + 27q−63 + 29q−64 + 22q−65 + q−66−16q−67−28q−68−25q−69−q−70 + 23q−71 + 26q−72 + 12q−73−5q−74−21q−75−18q−76−q−77 + 12q−78 + 10q−79 + 5q−80−9q−82−5q−83 + 2q−84 + 2q−85 + q−86 + 2q−87−q−88−2q−89 + q−90
6	q47−2q46−q45 + 2q44 + q43 + q42 + 6q40−11q39−14q38 + 2q37 + 10q36 + 19q35 + 21q34 + 28q33−45q32−84q31−54q30 + 2q29 + 88q28 + 161q27 + 187q26−51q25−274q24−341q23−222q22 + 104q21 + 495q20 + 730q19 + 270q18−401q17−916q16−952q15−318q14 + 746q13 + 1625q12 + 1181q11−19q10−1366q9−2008q8−1358q7 + 433q6 + 2310q5 + 2334q4 + 940q3−1207q2−2730q−2508−396q−1 + 2325q−2 + 3042q−3 + 1903q−4−580q−5−2772q−6−3140q−7−1158q−8 + 1899q−9 + 3101q−10 + 2364q−11−28q−12−2441q−13−3187q−14−1493q−15 + 1503q−16 + 2850q−17 + 2379q−18 + 223q−19−2118q−20−2984q−21−1537q−22 + 1251q−23 + 2574q−24 + 2265q−25 + 360q−26−1835q−27−2759q−28−1574q−29 + 955q−30 + 2268q−31 + 2192q−32 + 615q−33−1423q−34−2495q−35−1720q−36 + 450q−37 + 1799q−38 + 2109q−39 + 1029q−40−769q−41−2050q−42−1864q−43−244q−44 + 1065q−45 + 1826q−46 + 1422q−47 + 73q−48−1293q−49−1748q−50−885q−51 + 131q−52 + 1167q−53 + 1470q−54 + 810q−55−300q−56−1165q−57−1093q−58−661q−59 + 238q−60 + 966q−61 + 1025q−62 + 519q−63−286q−64−684q−65−867q−66−478q−67 + 169q−68 + 598q−69 + 709q−70 + 334q−71−17q−72−456q−73−556q−74−320q−75−2q−76 + 342q−77 + 337q−78 + 305q−79 + 28q−80−192q−81−254q−82−217q−83−4q−84 + 32q−85 + 170q−86 + 132q−87 + 54q−88−16q−89−80q−90−23q−91−95q−92−11q−93 + 8q−94 + 30q−95 + 33q−96 + 26q−97 + 63q−98−34q−99−19q−100−38q−101−25q−102−18q−103 + 6q−104 + 57q−105 + 8q−106 + 14q−107−8q−108−13q−109−25q−110−13q−111 + 17q−112 + 2q−113 + 11q−114 + 4q−115 + 3q−116−9q−117−7q−118 + 4q−119−2q−120 + 2q−121 + q−122 + 2q−123−q−124−2q−125 + q−126
7	q63−2q62−q61 + 2q60 + 2q59 + 2q58−4q57−2q56 + q55−7q54−4q53 + 9q52 + 18q51 + 24q50−7q49−26q48−32q47−57q46−31q45 + 32q44 + 113q43 + 171q42 + 96q41−46q40−196q39−361q38−327q37−77q36 + 314q35 + 736q34 + 766q33 + 386q32−325q31−1177q30−1526q29−1104q28 + 54q27 + 1636q26 + 2583q25 + 2298q24 + 690q23−1856q22−3765q21−3981q20−2063q19 + 1569q18 + 4835q17 + 5982q16 + 4054q15−635q14−5495q13−7943q12−6433q11−970q10 + 5444q9 + 9541q8 + 8929q7 + 3064q6−4769q5−10520q4−11014q3−5274q2 + 3459q + 10717 + 12593q−1 + 7327q−2−1972q−3−10356q−4−13394q−5−8824q−6 + 461q−7 + 9547q−8 + 13611q−9 + 9820q−10 + 719q−11−8687q−12−13368q−13−10222q−14−1525q−15 + 7866q−16 + 12914q−17 + 10277q−18 + 1973q−19−7261q−20−12424q−21−10106q−22−2157q−23 + 6826q−24 + 11968q−25 + 9874q−26 + 2241q−27−6488q−28−11579q−29−9683q−30−2335q−31 + 6162q−32 + 11209q−33 + 9546q−34 + 2537q−35−5713q−36−10806q−37−9500q−38−2896q−39 + 5114q−40 + 10301q−41 + 9466q−42 + 3436q−43−4273q−44−9642q−45−9446q−46−4111q−47 + 3213q−48 + 8744q−49 + 9326q−50 + 4896q−51−1899q−52−7614q−53−9036q−54−5653q−55 + 418q−56 + 6158q−57 + 8457q−58 + 6312q−59 + 1147q−60−4450q−61−7524q−62−6644q−63−2632q−64 + 2511q−65 + 6174q−66 + 6557q−67 + 3871q−68−556q−69−4471q−70−5923q−71−4609q−72−1234q−73 + 2520q−74 + 4757q−75 + 4746q−76 + 2606q−77−626q−78−3184q−79−4198q−80−3323q−81−979q−82 + 1440q−83 + 3101q−84 + 3336q−85 + 2023q−86 + 110q−87−1703q−88−2692q−89−2345q−90−1223q−91 + 310q−92 + 1662q−93 + 2053q−94 + 1709q−95 + 695q−96−559q−97−1304q−98−1587q−99−1215q−100−304q−101 + 472q−102 + 1075q−103 + 1200q−104 + 731q−105 + 196q−106−441q−107−835q−108−750q−109−535q−110−65q−111 + 390q−112 + 508q−113 + 530q−114 + 296q−115−32q−116−172q−117−350q−118−318q−119−126q−120−29q−121 + 132q−122 + 166q−123 + 109q−124 + 135q−125 + 28q−126−53q−127−40q−128−93q−129−50q−130−37q−131−56q−132 + 34q−133 + 49q−134 + 45q−135 + 65q−136 + 14q−137 + 6q−138−13q−139−69q−140−37q−141−22q−142−2q−143 + 37q−144 + 20q−145 + 24q−146 + 24q−147−10q−148−16q−149−21q−150−17q−151 + 8q−152 + 4q−154 + 13q−155 + 4q−156 + 2q−157−6q−158−7q−159 + 2q−160−2q−162 + 2q−163 + q−164 + 2q−165−q−166−2q−167 + q−168