10 60

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 60"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 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]=

[211,211,2]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_60.

A1 Invariants.

Weight	Invariant
1	q13−2q11 + 3q9−4q7 + 3q5−q3 + 3q−1−3q−3 + 4q−5−3q−7 + q−9
2	q38−2q36−2q34 + 8q32−4q30−12q28 + 20q26 + q24−30q22 + 24q20 + 16q18−38q16 + 12q14 + 26q12−25q10−7q8 + 20q6 + 2q4−20q2 + 3 + 29q−2−23q−4−17q−6 + 39q−8−13q−10−24q−12 + 28q−14−2q−16−15q−18 + 9q−20 + q−22−3q−24 + q−26
3	q75−2q73−2q71 + 3q69 + 8q67−4q65−19q63 + 2q61 + 35q59 + 10q57−56q55−36q53 + 74q51 + 77q49−79q47−130q45 + 61q43 + 187q41−22q39−224q37−39q35 + 239q33 + 105q31−228q29−159q27 + 186q25 + 200q23−132q21−216q19 + 68q17 + 212q15−3q13−185q11−66q9 + 151q7 + 127q5−96q3−183q + 32q−1 + 223q−3 + 37q−5−239q−7−105q−9 + 230q−11 + 154q−13−187q−15−181q−17 + 136q−19 + 177q−21−80q−23−150q−25 + 38q−27 + 105q−29−8q−31−68q−33−3q−35 + 38q−37 + 2q−39−14q−41−3q−43 + 6q−45 + q−47−3q−49 + q−51
4	q124−2q122−2q120 + 3q118 + 3q116 + 8q114−11q112−19q110 + 2q108 + 17q106 + 53q104−13q102−80q100−55q98 + 18q96 + 191q94 + 87q92−146q90−262q88−158q86 + 363q84 + 436q82 + 32q80−526q78−711q76 + 212q74 + 881q72 + 712q70−382q68−1405q66−552q64 + 848q62 + 1571q60 + 454q58−1559q56−1531q54 + 65q52 + 1887q50 + 1500q48−920q46−1987q44−950q42 + 1434q40 + 2035q38 + 23q36−1731q34−1557q32 + 643q30 + 1904q28 + 747q26−1098q24−1650q22−125q20 + 1387q18 + 1223q16−330q14−1451q12−878q10 + 624q8 + 1548q6 + 608q4−963q2−1604−430q−2 + 1540q−4 + 1568q−6−71q−8−1901q−10−1520q−12 + 929q−14 + 2007q−16 + 961q−18−1423q−20−2014q−22 + 1563q−26 + 1455q−28−508q−30−1607q−32−557q−34 + 679q−36 + 1159q−38 + 104q−40−801q−42−489q−44 + 82q−46 + 562q−48 + 199q−50−247q−52−206q−54−64q−56 + 174q−58 + 86q−60−51q−62−41q−64−35q−66 + 35q−68 + 18q−70−10q−72−2q−74−6q−76 + 6q−78 + q−80−3q−82 + q−84
5	q185−2q183−2q181 + 3q179 + 3q177 + 3q175 + q173−11q171−19q169 + 2q167 + 26q165 + 35q163 + 21q161−37q159−96q157−74q155 + 49q153 + 178q151 + 192q149 + 14q147−285q145−436q143−199q141 + 354q139 + 784q137 + 623q135−217q133−1192q131−1379q129−298q127 + 1450q125 + 2397q123 + 1412q121−1197q119−3475q117−3188q115 + 99q113 + 4143q111 + 5399q109 + 2083q107−3824q105−7540q103−5277q101 + 2111q99 + 8914q97 + 8920q95 + 1106q93−8786q91−12301q89−5470q87 + 6927q85 + 14531q83 + 10158q81−3404q79−15031q77−14349q75−1132q73 + 13740q71 + 17219q69 + 5794q67−10917q65−18399q63−9867q61 + 7313q59 + 17977q57 + 12704q55−3602q53−16274q51−14229q49 + 318q47 + 13932q45 + 14556q43 + 2264q41−11369q39−14124q37−4186q35 + 8885q33 + 13315q31 + 5752q29−6516q27−12480q25−7229q23 + 4096q21 + 11575q19 + 8978q17−1323q15−10563q13−10941q11−1993q9 + 9010q7 + 12920q5 + 5974q3−6650q−14461q−1−10268q−3 + 3254q−5 + 14958q−7 + 14366q−9 + 1074q−11−14016q−13−17544q−15−5758q−17 + 11455q−19 + 19058q−21 + 10134q−23−7620q−25−18599q−27−13299q−29 + 3218q−31 + 16217q−33 + 14720q−35 + 880q−37−12534q−39−14220q−41−3941q−43 + 8410q−45 + 12217q−47 + 5483q−49−4658q−51−9318q−53−5678q−55 + 1833q−57 + 6393q−59 + 4845q−61−169q−63−3844q−65−3594q−67−599q−69 + 2057q−71 + 2369q−73 + 716q−75−970q−77−1356q−79−574q−81 + 372q−83 + 713q−85 + 381q−87−149q−89−329q−91−184q−93 + 31q−95 + 137q−97 + 92q−99−6q−101−61q−103−32q−105 + 9q−107 + 15q−109 + 6q−111 + 2q−113−5q−115−6q−117 + 6q−119 + q−121−3q−123 + q−125

A2 Invariants.

Weight	Invariant
1,0	q20 + q18−2q16−3q10 + 3q8 + q4 + 2q2−2 + 3q−2−3q−4 + q−6 + 2q−8−2q−10 + q−12
2,0	q52 + q50−q48−5q46−2q44 + 6q42 + 5q40−6q38−6q36 + 11q34 + 12q32−12q30−15q28 + 10q26 + 11q24−13q22−12q20 + 15q18 + 10q16−11q14−q12 + 9q10−6q8−2q6 + 9q4−6q2−10 + 11q−2 + 15q−4−17q−6−12q−8 + 20q−10 + 10q−12−18q−14−5q−16 + 14q−18 + 3q−20−8q−22−3q−24 + 5q−26−2q−30 + q−32

A3 Invariants.

Weight	Invariant
0,1,0	q40−2q38 + 2q36 + 3q34−9q32 + 7q30 + 6q28−20q26 + 14q24 + 11q22−28q20 + 17q18 + 14q16−27q14 + 8q12 + 13q10−12q8−5q6 + 7q4 + 11q2−11−6q−2 + 27q−4−15q−6−17q−8 + 30q−10−11q−12−17q−14 + 22q−16−4q−18−10q−20 + 9q−22−3q−26 + q−28
1,0,0	q27 + q25 + q23−2q21−3q17−3q13 + 4q11 + 3q7 + q5 + q3−2q−1 + 2q−3−3q−5 + 2q−7−q−9 + 3q−11−2q−13 + q−15

B2 Invariants.

Weight	Invariant
0,1	q40−2q38 + 6q36−9q34 + 15q32−21q30 + 26q28−32q26 + 32q24−31q22 + 22q20−13q18−2q16 + 19q14−34q12 + 51q10−58q8 + 65q6−61q4 + 55q2−43 + 26q−2−9q−4−7q−6 + 19q−8−28q−10 + 33q−12−33q−14 + 30q−16−24q−18 + 18q−20−11q−22 + 6q−24−3q−26 + q−28
1,0	q66−2q62−2q60 + 4q58 + 6q56−3q54−12q52−4q50 + 16q48 + 15q46−12q44−27q42−q40 + 32q38 + 19q36−25q34−31q32 + 9q30 + 36q28 + 7q26−30q24−17q22 + 19q20 + 20q18−13q16−20q14 + 7q12 + 21q10−4q8−22q6−q4 + 25q2 + 10−23q−2−17q−4 + 22q−6 + 27q−8−12q−10−34q−12−3q−14 + 33q−16 + 19q−18−22q−20−29q−22 + 5q−24 + 26q−26 + 9q−28−14q−30−14q−32 + 3q−34 + 10q−36 + 3q−38−3q−40−3q−42 + q−46

G2 Invariants.

Weight

Invariant

1,0

q94−2q92 + 6q90−10q88 + 12q86−11q84 + 22q80−45q78 + 69q76−72q74 + 47q72 + 7q70−83q68 + 155q66−189q64 + 162q62−75q60−59q58 + 184q56−259q54 + 248q52−149q50−2q48 + 141q46−220q44 + 196q42−90q40−48q38 + 160q36−186q34 + 112q32 + 35q30−189q28 + 289q26−278q24 + 162q22 + 33q20−231q18 + 361q16−373q14 + 267q12−75q10−130q8 + 270q6−303q4 + 226q2−75−81q−2 + 172q−4−168q−6 + 73q−8 + 61q−10−170q−12 + 207q−14−146q−16 + 18q−18 + 122q−20−225q−22 + 252q−24−194q−26 + 83q−28 + 41q−30−136q−32 + 177q−34−161q−36 + 108q−38−38q−40−20q−42 + 53q−44−68q−46 + 57q−48−35q−50 + 17q−52 + q−54−8q−56 + 10q−58−10q−60 + 6q−62−3q−64 + q−66

.

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 60"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 7t2−20t + 29−20t−1 + 7t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6 + z4−z2 + 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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q4−4q3 + 8q2−11q + 14−14q−1 + 13q−2−10q−3 + 6q−4−3q−5 + q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a6−3z2a4−3a4 + 3z4a2 + 6z2a2 + 4a2−z6−3z4−5z2−2 + z4a−2 + z2a−2 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a3z9 + az9 + 3a4z8 + 8a2z8 + 5z8 + 3a5z7 + 10a3z7 + 16az7 + 9z7a−1 + a6z6−3a4z6−7a2z6 + 8z6a−2 + 5z6−9a5z5−32a3z5−38az5−11z5a−1 + 4z5a−3−3a6z4−8a4z4−17a2z4−9z4a−2 + z4a−4−22z4 + 9a5z3 + 27a3z3 + 25az3 + 5z3a−1−2z3a−3 + 3a6z2 + 11a4z2 + 18a2z2 + 4z2a−2 + 14z2−3a5z−7a3z−6az−2za−1−a6−3a4−4a2−a−2−2

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 60"];

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]=

{ −t3 + 7t2−20t + 29−20t−1 + 7t−2−t−3, q4−4q3 + 8q2−11q + 14−14q−1 + 13q−2−10q−3 + 6q−4−3q−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]=

{K11n165,}

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_86,}

n	Jn
2	q12−4q11 + 4q10 + 9q9−28q8 + 17q7 + 39q6−80q5 + 28q4 + 91q3−136q2 + 22q + 143−162q−1−q−2 + 165q−3−144q−4−28q−5 + 147q−6−93q−7−42q−8 + 97q−9−39q−10−34q−11 + 43q−12−8q−13−15q−14 + 11q−15−3q−17 + q−18
3	q24−4q23 + 4q22 + 5q21−8q20−15q19 + 20q18 + 41q17−49q16−80q15 + 80q14 + 154q13−116q12−268q11 + 150q10 + 411q9−157q8−585q7 + 144q6 + 752q5−81q4−920q3 + 10q2 + 1028q + 105−1111q−1−205q−2 + 1115q−3 + 328q−4−1087q−5−422q−6 + 996q−7 + 510q−8−872q−9−566q−10 + 712q−11 + 594q−12−540q−13−580q−14 + 367q−15 + 525q−16−207q−17−446q−18 + 89q−19 + 340q−20−5q−21−237q−22−37q−23 + 149q−24 + 46q−25−81q−26−40q−27 + 39q−28 + 26q−29−15q−30−15q−31 + 6q−32 + 5q−33−3q−35 + q−36
4	q40−4q39 + 4q38 + 5q37−12q36 + 5q35−12q34 + 32q33 + 22q32−82q31−q30−22q29 + 169q28 + 110q27−320q26−143q25−63q24 + 615q23 + 473q22−800q21−714q20−375q19 + 1520q18 + 1528q17−1280q16−1950q15−1425q14 + 2619q13 + 3491q12−1172q11−3513q10−3439q9 + 3210q8 + 5875q7−126q6−4591q5−5888q4 + 2829q3 + 7705q2 + 1513q−4619−7858q−1 + 1655q−2 + 8346q−3 + 3084q−4−3679q−5−8782q−6 + 153q−7 + 7773q−8 + 4205q−9−2126q−10−8618q−11−1359q−12 + 6248q−13 + 4757q−14−281q−15−7461q−16−2620q−17 + 4048q−18 + 4583q−19 + 1473q−20−5449q−21−3221q−22 + 1664q−23 + 3546q−24 + 2540q−25−3029q−26−2834q−27−158q−28 + 1950q−29 + 2512q−30−1016q−31−1717q−32−881q−33 + 550q−34 + 1659q−35 + 7q−36−623q−37−712q−38−119q−39 + 736q−40 + 192q−41−65q−42−308q−43−192q−44 + 215q−45 + 88q−46 + 51q−47−75q−48−88q−49 + 42q−50 + 15q−51 + 26q−52−8q−53−22q−54 + 6q−55 + 5q−57−3q−59 + q−60
5	q60−4q59 + 4q58 + 5q57−12q56 + q55 + 8q54 + 13q52−q51−53q50−28q49 + 63q48 + 98q47 + 58q46−107q45−268q44−173q43 + 243q42 + 628q41 + 390q40−448q39−1214q38−955q37 + 629q36 + 2314q35 + 2043q34−760q33−3870q32−3950q31 + 379q30 + 5989q29 + 7057q28 + 788q27−8430q26−11461q25−3261q24 + 10649q23 + 17198q22 + 7522q21−12237q20−23812q19−13540q18 + 12335q17 + 30612q16 + 21362q15−10740q14−36811q13−30057q12 + 7035q11 + 41591q10 + 39116q9−1816q8−44414q7−47270q6−4751q5 + 45119q4 + 54206q3 + 11476q2−43822q−58974−18273q−1 + 40926q−2 + 62017q−3 + 24100q−4−36876q−5−62884q−6−29276q−7 + 31978q−8 + 62395q−9 + 33340q−10−26575q−11−60287q−12−36755q−13 + 20653q−14 + 57144q−15 + 39304q−16−14307q−17−52724q−18−41185q−19 + 7582q−20 + 47206q−21 + 42059q−22−674q−23−40432q−24−41809q−25−6032q−26 + 32659q−27 + 40014q−28 + 11998q−29−24126q−30−36536q−31−16696q−32 + 15479q−33 + 31482q−34 + 19480q−35−7415q−36−25111q−37−20175q−38 + 607q−39 + 18265q−40 + 18798q−41 + 4212q−42−11549q−43−15802q−44−6997q−45 + 5868q−46 + 11967q−47 + 7727q−48−1657q−49−7988q−50−7003q−51−935q−52 + 4579q−53 + 5464q−54 + 2059q−55−2081q−56−3687q−57−2191q−58 + 535q−59 + 2177q−60 + 1772q−61 + 197q−62−1078q−63−1206q−64−412q−65 + 429q−66 + 691q−67 + 384q−68−103q−69−366q−70−251q−71−q−72 + 138q−73 + 147q−74 + 48q−75−67q−76−73q−77−15q−78 + 9q−79 + 24q−80 + 26q−81−8q−82−15q−83−q−84 + 5q−87−3q−89 + q−90