10 68

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 68"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X12,4,13,3 X20,13,1,14 X16,5,17,6 X8,19,9,20 X18,9,19,10 X10,17,11,18 X14,7,15,8 X6,15,7,16 X2,12,3,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 14 18 2 20 6 10 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,3,3]

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,{1,1,−2,1,−2,−2,−3,2,2,−4,3,−2,−4,−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]=

{ 5, 14, 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[{3, 11}, {2, 7}, {6, 8}, {1, 3}, {10, 12}, {11, 9}, {7, 10}, {9, 5}, {4, 6}, {5, 2}, {12, 4}, {8, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_68.

A1 Invariants.

Weight	Invariant
1	−q15 + q13−2q11 + 3q9−q7 + q5−q + 3q−1−2q−3 + 2q−5−q−7
2	q44−q42−2q40 + 4q38−8q34 + 6q32 + 6q30−12q28 + 3q26 + 12q24−10q22−5q20 + 11q18−4q16−9q14 + 6q12 + 6q10−6q8−2q6 + 11q4−q2−11 + 10q−2 + 4q−4−12q−6 + 6q−8 + 4q−10−6q−12 + 3q−14−2q−18 + q−20
3	−q87 + q85 + 2q83−5q79−2q77 + 8q75 + 8q73−10q71−17q69 + 6q67 + 27q65 + 4q63−33q61−20q59 + 30q57 + 37q55−20q53−45q51 + q49 + 52q47 + 16q45−47q43−32q41 + 37q39 + 42q37−25q35−51q33 + 13q31 + 53q29−2q27−52q25−11q23 + 51q21 + 23q19−41q17−34q15 + 25q13 + 42q11−5q9−44q7−18q5 + 42q3 + 37q−28q−1−45q−3 + 17q−5 + 47q−7−7q−9−37q−11−q−13 + 25q−15 + q−17−14q−19−q−21 + 8q−23−2q−25−q−27−q−33 + 2q−37−q−39
4	q144−q142−2q140 + q136 + 7q134−8q130−8q128−6q126 + 22q124 + 20q122−3q120−28q118−47q116 + 14q114 + 54q112 + 56q110−q108−104q106−71q104 + 13q102 + 122q100 + 125q98−51q96−145q94−147q92 + 41q90 + 221q88 + 133q86−44q84−252q82−171q80 + 119q78 + 248q76 + 176q74−154q72−299q70−101q68 + 173q66 + 312q64 + 44q62−255q60−245q58 + 27q56 + 301q54 + 170q52−147q50−273q48−65q46 + 242q44 + 215q42−81q40−273q38−109q36 + 192q34 + 247q32−8q30−255q28−182q26 + 77q24 + 267q22 + 148q20−133q18−246q16−150q14 + 158q12 + 296q10 + 115q8−172q6−349q4−81q2 + 268 + 307q−2 + 33q−4−335q−6−243q−8 + 98q−10 + 279q−12 + 159q−14−170q−16−201q−18−22q−20 + 127q−22 + 128q−24−52q−26−87q−28−27q−30 + 30q−32 + 55q−34−17q−36−19q−38−5q−40 + 19q−44−8q−46−3q−48−3q−52 + 6q−54−2q−56 + q−58−2q−62 + q−64
5	−q215 + q213 + 2q211−q207−3q205−5q203 + 10q199 + 10q197 + 3q195−8q193−24q191−23q189 + 6q187 + 40q185 + 50q183 + 23q181−38q179−94q177−84q175 + 3q173 + 116q171 + 168q169 + 98q167−76q165−239q163−253q161−63q159 + 225q157 + 400q155 + 300q153−55q151−446q149−569q147−265q145 + 296q143 + 723q141 + 665q139 + 90q137−651q135−1000q133−610q131 + 291q129 + 1074q127 + 1134q125 + 316q123−842q121−1452q119−978q117 + 282q115 + 1434q113 + 1554q111 + 441q109−1088q107−1837q105−1162q103 + 473q101 + 1802q99 + 1708q97 + 222q95−1474q93−1985q91−855q89 + 983q87 + 1988q85 + 1313q83−464q81−1795q79−1541q77 + 40q75 + 1504q73 + 1573q71 + 238q69−1233q67−1481q65−353q63 + 1039q61 + 1358q59 + 358q57−944q55−1285q53−357q51 + 931q49 + 1305q47 + 412q45−894q43−1389q41−622q39 + 743q37 + 1488q35 + 965q33−394q31−1462q29−1378q27−197q25 + 1208q23 + 1742q21 + 923q19−683q17−1868q15−1658q13−83q11 + 1695q9 + 2208q7 + 912q5−1198q3−2404q−1630q−1 + 521q−3 + 2236q−5 + 2054q−7 + 153q−9−1772q−11−2102q−13−671q−15 + 1186q−17 + 1855q−19 + 911q−21−647q−23−1424q−25−904q−27 + 256q−29 + 969q−31 + 737q−33−34q−35−596q−37−516q−39−35q−41 + 318q−43 + 308q−45 + 55q−47−163q−49−174q−51−25q−53 + 80q−55 + 71q−57 + 16q−59−32q−61−35q−63−q−65 + 19q−67 + 11q−69−6q−71−8q−73−q−75 + 5q−79 + 4q−81−3q−83−4q−85 + 2q−87−q−89 + 2q−93−q−95

A2 Invariants.

Weight	Invariant
1,0	−q22−2q16 + 2q14 + q12 + 2q8−q6 + q4 + 2q−2−2q−4 + q−6 + q−8−q−10
2,0	q58−q54−q52 + 2q50 + 2q48−4q46−4q44 + 3q42 + 5q40−3q38−6q36 + 4q34 + 8q32−3q30−7q28 + q26 + 4q24−3q22−6q20 + 2q18 + 4q16−q14 + 3q12 + 2q10−2q8 + 3q6 + 7q4−2q2−6 + 4q−2 + 8q−4−6q−6−9q−8 + 6q−10 + 6q−12−4q−14−3q−16 + 2q−18 + 2q−20−2q−22−q−24 + q−26

A3 Invariants.

Weight	Invariant
0,1,0	q46−q44 + 2q42 + q40−4q38 + 3q36−2q34−8q32 + 5q30−2q28−7q26 + 10q24−5q20 + 9q18 + 3q16−4q14 + 2q12 + 3q10−6q6 + 2q4 + 7q2−8 + 2q−2 + 10q−4−9q−6−q−8 + 8q−10−5q−12−3q−14 + 5q−16−q−18−2q−20 + q−22
1,0,0	−q29−q25−2q21 + 2q19 + 2q15 + 2q11 + q5 + q−q−1 + 2q−3−2q−5 + q−7 + q−11−q−13

B2 Invariants.

Weight	Invariant
0,1	−q46 + q44−4q42 + 5q40−8q38 + 11q36−12q34 + 14q32−13q30 + 12q28−7q26 + 2q24 + 4q22−11q20 + 17q18−23q16 + 26q14−26q12 + 25q10−20q8 + 16q6−8q4 + 3q2 + 4−8q−2 + 12q−4−13q−6 + 13q−8−12q−10 + 9q−12−7q−14 + 5q−16−3q−18 + 2q−20−q−22
1,0	q76−q72−q70 + 3q68 + 3q66−3q64−6q62 + 8q58 + 3q56−10q54−10q52 + 5q50 + 13q48 + q46−15q44−7q42 + 11q40 + 13q38−4q36−13q34−q32 + 11q30 + 5q28−7q26−4q24 + 7q22 + 6q20−5q18−7q16 + 5q14 + 10q12−2q10−12q8−2q6 + 11q4 + 7q2−9−10q−2 + 5q−4 + 14q−6 + q−8−11q−10−7q−12 + 6q−14 + 10q−16−7q−20−5q−22 + 2q−24 + 5q−26 + q−28−2q−30−2q−32 + q−36

G2 Invariants.

Weight

Invariant

1,0

q108−q106 + 4q104−6q102 + 6q100−6q98−q96 + 13q94−24q92 + 32q90−30q88 + 12q86 + 15q84−46q82 + 60q80−60q78 + 34q76 + 4q74−44q72 + 64q70−62q68 + 38q66 + 4q64−38q62 + 45q60−36q58 + 8q56 + 28q54−46q52 + 53q50−28q48−5q46 + 50q44−77q42 + 79q40−53q38 + 8q36 + 41q34−73q32 + 86q30−67q28 + 26q26 + 21q24−55q22 + 56q20−41q18 + 4q16 + 28q14−38q12 + 31q10−8q8−20q6 + 43q4−47q2 + 33−7q−2−20q−4 + 40q−6−43q−8 + 39q−10−21q−12 + 5q−14 + 12q−16−27q−18 + 28q−20−25q−22 + 17q−24−7q−26−q−28 + 8q−30−13q−32 + 13q−34−10q−36 + 7q−38−2q−40−q−42 + 2q−44−4q−46 + 3q−48−2q−50 + q−52

.

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 68"];

In[4]:=

Alexander[K][t]

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

Out[4]=

4t2−14t + 21−14t−1 + 4t−2

In[5]:=

Conway[K][z]

Out[5]=

4z4 + 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]=

{ 57, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q3 + 3q2−5q + 8−9q−1 + 9q−2−8q−3 + 7q−4−4q−5 + 2q−6−q−7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a6−a6 + z4a4 + z2a4 + a4 + 2z4a2 + 3z2a2 + a2 + z4−z2a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a5z9 + a3z9 + 2a6z8 + 6a4z8 + 4a2z8 + a7z7 + a5z7 + 7a3z7 + 7az7−9a6z6−20a4z6−4a2z6 + 7z6−5a7z5−16a5z5−30a3z5−14az5 + 5z5a−1 + 13a6z4 + 17a4z4−9a2z4 + 3z4a−2−10z4 + 8a7z3 + 23a5z3 + 27a3z3 + 8az3−3z3a−1 + z3a−3−7a6z2−5a4z2 + 7a2z2−z2a−2 + 4z2−4a7z−8a5z−6a3z−2az + a6 + a4−a2

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 68"];

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]=

{ 4t2−14t + 21−14t−1 + 4t−2, −q3 + 3q2−5q + 8−9q−1 + 9q−2−8q−3 + 7q−4−4q−5 + 2q−6−q−7 }

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_31,}

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	q9−3q8 + 2q7 + 4q6−12q5 + 12q4 + 6q3−30q2 + 28q + 12−51q−1 + 38q−2 + 24q−3−64q−4 + 34q−5 + 36q−6−64q−7 + 19q−8 + 41q−9−49q−10 + 3q−11 + 36q−12−27q−13−6q−14 + 21q−15−9q−16−6q−17 + 7q−18−q−19−2q−20 + q−21
3	−q18 + 3q17−2q16−q15 + 3q13−3q12−2q11 + 10q10−6q9−16q8 + 13q7 + 34q6−32q5−52q4 + 43q3 + 88q2−62q−114 + 60q−1 + 153q−2−57q−3−174q−4 + 34q−5 + 192q−6−10q−7−191q−8−25q−9 + 185q−10 + 54q−11−163q−12−87q−13 + 144q−14 + 104q−15−108q−16−127q−17 + 80q−18 + 130q−19−41q−20−132q−21 + 11q−22 + 115q−23 + 22q−24−96q−25−40q−26 + 69q−27 + 47q−28−39q−29−47q−30 + 19q−31 + 34q−32−2q−33−24q−34−2q−35 + 11q−36 + 5q−37−6q−38−2q−39 + q−40 + 2q−41−q−42