10 67

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 67"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 12 18 2 6 20 8 16

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,3,21]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_67.

A1 Invariants.

Weight	Invariant
1	q19−2q17 + 2q15−3q13 + 2q11 + 2q5−3q3 + 3q−q−1 + q−3
2	q54−2q52−2q50 + 7q48−2q46−10q44 + 12q42 + 5q40−18q38 + 8q36 + 12q34−18q32−q30 + 15q28−7q26−7q24 + 9q22 + 6q20−11q18−5q16 + 17q14−8q12−13q10 + 19q8−q6−12q4 + 10q2 + 1−5q−2 + 3q−4−q−8 + q−10
3	q105−2q103−2q101 + 3q99 + 7q97−2q95−16q93−2q91 + 24q89 + 15q87−30q85−35q83 + 26q81 + 57q79−10q77−72q75−18q73 + 78q71 + 46q69−67q67−71q65 + 48q63 + 90q61−28q59−94q57 + 2q55 + 92q53 + 16q51−82q49−33q47 + 69q45 + 47q43−45q41−61q39 + 17q37 + 70q35 + 13q33−74q31−48q29 + 67q27 + 75q25−47q23−90q21 + 27q19 + 90q17−6q15−73q13−12q11 + 55q9 + 16q7−33q5−12q3 + 16q + 9q−1−8q−3−3q−5 + 4q−7 + q−9−2q−11 + q−13−q−19 + q−21
4	q172−2q170−2q168 + 3q166 + 3q164 + 7q162−9q160−16q158−q156 + 10q154 + 42q152 + 2q150−47q148−48q146−19q144 + 102q142 + 87q140−17q138−124q136−167q134 + 65q132 + 206q130 + 175q128−58q126−344q124−180q122 + 128q120 + 394q118 + 257q116−277q114−444q112−224q110 + 342q108 + 572q106 + 74q104−434q102−562q100 + 32q98 + 608q96 + 406q94−193q92−638q90−253q88 + 423q86 + 518q84 + 27q82−520q80−355q78 + 221q76 + 477q74 + 153q72−350q70−384q68 + 21q66 + 402q64 + 285q62−129q60−412q58−261q56 + 237q54 + 443q52 + 228q50−328q48−568q46−89q44 + 447q42 + 590q40−31q38−646q36−430q34 + 181q32 + 666q30 + 292q28−389q26−488q24−132q22 + 407q20 + 357q18−73q16−272q14−211q12 + 117q10 + 193q8 + 44q6−59q4−114q2 + 4 + 52q−2 + 23q−4 + 6q−6−34q−8−q−10 + 8q−12 + 8q−16−8q−18 + q−20 + q−22−2q−24 + 4q−26−2q−28−q−34 + q−36
5	q255−2q253−2q251 + 3q249 + 3q247 + 3q245−9q241−16q239−q237 + 20q235 + 28q233 + 20q231−16q229−61q227−68q225 + q223 + 92q221 + 133q219 + 78q217−81q215−235q213−226q211−2q209 + 288q207 + 424q205 + 242q203−218q201−625q199−605q197−61q195 + 661q193 + 1013q191 + 603q189−404q187−1291q185−1297q183−232q181 + 1217q179 + 1929q177 + 1190q175−641q173−2252q171−2248q169−388q167 + 2023q165 + 3083q163 + 1729q161−1218q159−3460q157−3022q155−16q153 + 3215q151 + 3976q149 + 1433q147−2453q145−4423q143−2674q141 + 1383q139 + 4290q137 + 3552q135−249q133−3777q131−3940q129−670q127 + 3027q125 + 3893q123 + 1304q121−2285q119−3584q117−1591q115 + 1651q113 + 3157q111 + 1679q109−1195q107−2755q105−1692q103 + 841q101 + 2466q99 + 1776q97−490q95−2248q93−2015q91−24q89 + 2028q87 + 2419q85 + 764q83−1652q81−2864q79−1751q77 + 974q75 + 3181q73 + 2890q71 + 33q69−3165q67−3921q65−1324q63 + 2651q61 + 4629q59 + 2678q57−1701q55−4748q53−3787q51 + 415q49 + 4236q47 + 4407q45 + 873q43−3228q41−4365q39−1845q37 + 1936q35 + 3761q33 + 2345q31−761q29−2797q27−2290q25−107q23 + 1750q21 + 1888q19 + 557q17−894q15−1320q13−638q11 + 322q9 + 777q7 + 535q5−33q3−396q−349q−1−60q−3 + 164q−5 + 191q−7 + 66q−9−57q−11−86q−13−45q−15 + 17q−17 + 38q−19 + 16q−21−4q−23−8q−25−10q−27−2q−29 + 10q−31 + q−33−4q−35 + 2q−37−2q−39−2q−41 + 4q−43 + q−45−2q−47−q−53 + q−55

A2 Invariants.

Weight	Invariant
1,0	q28−q26−q24 + 2q22−2q20 + q16−q14 + 2q12−q10 + q8−2q4 + 3q2 + q−4
2,0	q72−q70−2q68 + 4q64 + 2q62−7q60−2q58 + 8q56 + 6q54−8q52−7q50 + 8q48 + 6q46−10q44−8q42 + 7q40 + 4q38−4q36−2q34 + 6q32 + q30−q28 + 4q26−4q24−6q22 + 7q20 + 5q18−11q16−5q14 + 12q12 + 7q10−11q8−5q6 + 11q4 + 3q2−5−q−2 + 3q−4 + q−6−q−8 + q−12

A3 Invariants.

Weight	Invariant
0,1,0	q60−2q58 + q56 + 2q54−7q52 + 5q50 + 3q48−10q46 + 10q44 + 4q42−13q40 + 10q38 + 5q36−13q34 + 3q32 + 4q30−5q28−4q26 + 2q24 + 7q22−7q20−2q18 + 15q16−9q14−6q12 + 15q10−7q8−7q6 + 10q4−q2−3 + 4q−2 + q−4−q−6 + q−8
1,0,0	q37−q35−q31 + 2q29−2q27 + q25−q23 + q21−q19 + q17 + q15−q13 + q11−q9 + q7−2q5 + 3q3 + q−1 + q−5

B2 Invariants.

Weight	Invariant
0,1	q60−2q58 + 5q56−8q54 + 11q52−15q50 + 17q48−18q46 + 16q44−14q42 + 7q40−9q36 + 19q34−25q32 + 32q30−33q28 + 34q26−30q24 + 23q22−15q20 + 6q18 + q16−9q14 + 14q12−17q10 + 17q8−15q6 + 14q4−9q2 + 7−4q−2 + 3q−4−q−6 + q−8
1,0	q98−2q94−2q92 + 3q90 + 5q88−3q86−9q84−2q82 + 12q80 + 9q78−10q76−15q74 + 4q72 + 19q70 + 7q68−17q66−14q64 + 9q62 + 17q60−q58−16q56−4q54 + 11q52 + 6q50−10q48−7q46 + 8q44 + 9q42−7q40−12q38 + 4q36 + 14q34−15q30−5q28 + 15q26 + 12q24−10q22−16q20 + 3q18 + 18q16 + 6q14−12q12−12q10 + 3q8 + 12q6 + 3q4−5q2−5 + q−2 + 4q−4 + 2q−6−q−8−q−10 + q−14

G2 Invariants.

Weight

Invariant

1,0

q142−2q140 + 5q138−9q136 + 9q134−8q132−2q130 + 19q128−34q126 + 47q124−44q122 + 19q120 + 18q118−60q116 + 90q114−93q112 + 61q110−2q108−57q106 + 98q104−99q102 + 65q100−6q98−47q96 + 72q94−67q92 + 23q90 + 37q88−76q86 + 85q84−53q82−10q80 + 71q78−119q76 + 124q74−94q72 + 23q70 + 56q68−116q66 + 141q64−111q62 + 48q60 + 23q58−76q56 + 93q54−70q52 + 20q50 + 37q48−62q46 + 60q44−20q42−35q40 + 73q38−83q36 + 60q34−20q32−30q30 + 65q28−78q26 + 72q24−41q22 + 6q20 + 20q18−39q16 + 42q14−37q12 + 27q10−9q8−2q6 + 12q4−15q2 + 14−10q−2 + 7q−4−2q−6−q−8 + 3q−10−3q−12 + 3q−14−q−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 67"];

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a8−z4a6−2z4a4−2z2a4−z4a2 + z2 + 1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a10−3z4a10 + 2z2a10 + 3z7a9−10z5a9 + 9z3a9−2za9 + 3z8a8−7z6a8 + 2z4a8 + z9a7 + 5z7a7−21z5a7 + 19z3a7−6za7 + 6z8a6−13z6a6 + 7z4a6−2z2a6 + z9a5 + 6z7a5−19z5a5 + 19z3a5−6za5 + 3z8a4−2z6a4−z4a4 + 2z2a4 + 4z7a3−6z5a3 + 7z3a3−2za3 + 3z6a2−2z4a2 + 2z5a−2z3a + z4−2z2 + 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 67"];

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−2 + 5q−1−8q−2 + 10q−3−10q−4 + 10q−5−8q−6 + 5q−7−3q−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_74, 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	q4−2q3 + q2 + 4q−10 + 7q−1 + 13q−2−32q−3 + 18q−4 + 33q−5−64q−6 + 23q−7 + 58q−8−86q−9 + 17q−10 + 75q−11−83q−12 + q−13 + 75q−14−61q−15−15q−16 + 58q−17−31q−18−19q−19 + 32q−20−8q−21−12q−22 + 10q−23−3q−25 + q−26