10 74

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 74"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,18,12,19 X9,20,10,1 X19,10,20,11 X17,6,18,7 X7,16,8,17 X15,8,16,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 14 16 20 18 2 8 6 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,21+]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_74.

A1 Invariants.

Weight	Invariant
1	q19−q17 + 2q15−4q13 + q11−q9 + q7 + 3q5−2q3 + 3q−2q−1 + q−3
2	q54−q52−q50 + 4q48−3q46−7q44 + 10q42−15q38 + 15q36 + 9q34−19q32 + 7q30 + 13q28−15q26−4q24 + 9q22−2q20−10q18 + 2q16 + 16q14−12q12−7q10 + 22q8−8q6−12q4 + 15q2−2−7q−2 + 6q−4−2q−8 + q−10
4	q172−q170−q168 + q166 + 3q162−5q160−5q158 + 5q156 + 4q154 + 14q152−11q150−25q148−3q146 + 14q144 + 58q142 + 4q140−63q138−64q136−17q134 + 134q132 + 102q130−43q128−170q126−176q124 + 130q122 + 259q120 + 146q118−161q116−404q114−75q112 + 277q110 + 411q108 + 81q106−467q104−374q102 + 51q100 + 517q98 + 392q96−291q94−518q92−239q90 + 397q88 + 538q86−38q84−462q82−397q80 + 212q78 + 508q76 + 138q74−318q72−409q70 + 40q68 + 390q66 + 261q64−144q62−374q60−166q58 + 203q56 + 385q54 + 100q52−279q50−404q48−79q46 + 446q44 + 387q42−52q40−530q38−401q36 + 308q34 + 537q32 + 241q30−411q28−548q26 + 48q24 + 407q22 + 377q20−144q18−418q16−111q14 + 149q12 + 283q10 + 23q8−188q6−84q4 + q2 + 121 + 35q−2−57q−4−20q−6−19q−8 + 38q−10 + 10q−12−19q−14 + 2q−16−8q−18 + 12q−20 + 2q−22−7q−24 + 2q−26−2q−28 + 3q−30−2q−34 + q−36
5	q255−q253−q251 + q249 + q243−3q241−4q239 + 5q237 + 8q235 + 2q233−2q231−13q229−19q227 + 2q225 + 30q223 + 37q221 + 15q219−35q217−82q215−63q213 + 29q211 + 134q209 + 151q207 + 32q205−172q203−290q201−181q199 + 139q197 + 441q195 + 434q193 + 32q191−520q189−758q187−391q185 + 414q183 + 1059q181 + 923q179−46q177−1159q175−1491q173−631q171 + 930q169 + 1967q167 + 1453q165−321q163−2062q161−2313q159−635q157 + 1782q155 + 2910q153 + 1681q151−1044q149−3120q147−2686q145 + 80q143 + 2905q141 + 3357q139 + 952q137−2321q135−3650q133−1827q131 + 1588q129 + 3577q127 + 2409q125−841q123−3222q121−2686q119 + 227q117 + 2763q115 + 2664q113 + 215q111−2276q109−2552q107−470q105 + 1853q103 + 2322q101 + 698q99−1469q97−2210q95−902q93 + 1125q91 + 2072q89 + 1258q87−648q85−2030q83−1699q81 + 69q79 + 1857q77 + 2240q75 + 741q73−1547q71−2742q69−1673q67 + 964q65 + 3033q63 + 2636q61−112q59−3014q57−3457q55−861q53 + 2573q51 + 3889q49 + 1862q47−1802q45−3874q43−2586q41 + 840q39 + 3373q37 + 2925q35 + 82q33−2553q31−2804q29−770q27 + 1624q25 + 2348q23 + 1081q21−797q19−1682q17−1092q15 + 214q13 + 1063q11 + 870q9 + 77q7−545q5−586q3−178q + 235q−1 + 342q−3 + 149q−5−81q−7−161q−9−95q−11 + 14q−13 + 72q−15 + 53q−17−6q−19−31q−21−16q−23 + 2q−25 + 10q−27 + 7q−29−10q−33−2q−35 + 7q−37 + q−39−2q−41 + q−43−q−45−2q−47 + 3q−49−2q−53 + q−55

A2 Invariants.

Weight	Invariant
1,0	q28 + 2q22−3q20−2q18−2q14 + 2q12 + 2q8 + 2q6−q4 + 3q2−1−q−2 + q−4
2,0	q72 + 2q64−q62−6q60−2q58 + 5q56 + q54−10q52−2q50 + 13q48 + 11q46−9q44−3q42 + 11q40 + 4q38−9q36−6q34 + 5q32−4q30−7q28−q26−3q24−5q22 + 9q20 + 7q18−7q16 + 14q12 + 4q10−14q8−3q6 + 14q4 + 2q2−9−q−2 + 5q−4 + 2q−6−2q−8−q−10 + q−12

A3 Invariants.

Weight	Invariant
0,1,0	q60−q58 + q56 + q54−6q52 + q50 + 2q48−10q46 + 9q44 + 11q42−10q40 + 14q38 + 9q36−17q34−2q32 + q30−13q28−6q26 + 4q24 + 6q22−3q20−2q18 + 16q16−7q14−8q12 + 18q10−4q8−9q6 + 13q4−q2−6 + 5q−2−2q−6 + q−8
1,0,0	q37 + q33 + 2q29−3q27−q25−3q23−2q19 + q17 + q15 + 2q11 + q9 + 3q7−q5 + 3q3−q−q−3 + q−5

B2 Invariants.

Weight	Invariant
0,1	q60−q58 + 3q56−5q54 + 8q52−11q50 + 14q48−16q46 + 17q44−17q42 + 12q40−8q38−q36 + 9q34−18q32 + 25q30−31q28 + 34q26−34q24 + 30q22−23q20 + 16q18−6q16−q14 + 10q12−14q10 + 18q8−17q6 + 17q4−13q2 + 10−7q−2 + 4q−4−2q−6 + q−8
1,0	q98−q94−q92 + 2q90 + 3q88−2q86−7q84−3q82 + 7q80 + 7q78−7q76−14q74 + 18q70 + 13q68−11q66−14q64 + 7q62 + 22q60 + 5q58−16q56−11q54 + 7q52 + 7q50−9q48−14q46 + 9q42−2q40−12q38−q36 + 14q34 + 7q32−10q30−9q28 + 12q26 + 15q24−5q22−18q20−q18 + 18q16 + 11q14−11q12−15q10 + 2q8 + 15q6 + 6q4−7q2−8 + q−2 + 6q−4 + 2q−6−2q−8−2q−10 + q−14

G2 Invariants.

Weight

Invariant

1,0

q142−q140 + 3q138−5q136 + 4q134−5q132−q130 + 9q128−18q126 + 27q124−29q122 + 20q120 + 3q118−30q116 + 57q114−75q112 + 68q110−34q108−17q106 + 69q104−100q102 + 106q100−58q98 + 7q96 + 44q94−85q92 + 83q90−41q88−16q86 + 55q84−70q82 + 48q80 + 10q78−72q76 + 98q74−107q72 + 67q70−3q68−85q66 + 135q64−145q62 + 115q60−41q58−41q56 + 98q54−119q52 + 99q50−45q48−17q46 + 65q44−65q42 + 38q40 + 19q38−59q36 + 75q34−56q32 + 10q30 + 39q28−80q26 + 98q24−78q22 + 42q20 + 9q18−48q16 + 66q14−66q12 + 51q10−26q8−q6 + 19q4−29q2 + 28−19q−2 + 12q−4−2q−6−3q−8 + 5q−10−6q−12 + 4q−14−2q−16 + q−18

.

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 74"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−4t2 + 16t−23 + 16t−1−4t−2

In[5]:=

Conway[K][z]

Out[5]=

1−4z4

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]=

{3,t + 1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a8 + a8−z4a6−z2a6−2a6−2z4a4−2z2a4−z4a2 + z2a2 + 2a2 + z2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a10−4z4a10 + 4z2a10 + 2z7a9−7z5a9 + 8z3a9−4za9 + 2z8a8−4z6a8 + z2a8 + a8 + z9a7 + 2z7a7−10z5a7 + 11z3a7−8za7 + 5z8a6−9z6a6 + 3z4a6−z2a6 + 2a6 + z9a5 + 5z7a5−12z5a5 + 9z3a5−4za5 + 3z8a4 + z6a4−9z4a4 + 8z2a4 + 5z7a3−6z5a3 + 3z3a3 + 5z6a2−7z4a2 + 5z2a2−2a2 + 3z5a−3z3a + z4−z2

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 74"];

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 + 16t−23 + 16t−1−4t−2, q−3 + 6q−1−8q−2 + 11q−3−10q−4 + 9q−5−8q−6 + 4q−7−2q−8 + q−9 }

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_67, K11n68,}

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	q4−3q3 + 2q2 + 7q−16 + 7q−1 + 24q−2−43q−3 + 11q−4 + 54q−5−72q−6 + 6q−7 + 82q−8−86q−9−6q−10 + 90q−11−75q−12−19q−13 + 79q−14−47q−15−25q−16 + 53q−17−19q−18−19q−19 + 23q−20−4q−21−9q−22 + 6q−23−2q−25 + q−26