10 84

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 84"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 16 14 2 8 18 20 12 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.22.2]

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(4,{1,1,1,2,−1,−3,2,2,−3,2,−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]=

{ 4, 11, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_84.

A1 Invariants.

Weight	Invariant
1	−q5 + 2q3−3q + 5q−1−2q−3 + 2q−5 + q−7−3q−9 + 3q−11−4q−13 + 3q−15−q−17
2	q16−2q14−q12 + 6q10−8q8−4q6 + 22q4−15q2−19 + 40q−2−7q−4−35q−6 + 33q−8 + 10q−10−30q−12 + 6q−14 + 19q−16−7q−18−23q−20 + 19q−22 + 19q−24−39q−26 + 8q−28 + 35q−30−33q−32−8q−34 + 31q−36−12q−38−12q−40 + 11q−42−3q−46 + q−48
3	−q33 + 2q31 + q29−2q27−3q25 + 5q23 + 4q21−15q19−7q17 + 30q15 + 22q13−55q11−56q9 + 77q7 + 116q5−81q3−185q + 47q−1 + 263q−3 + 12q−5−297q−7−98q−9 + 294q−11 + 178q−13−248q−15−228q−17 + 171q−19 + 247q−21−81q−23−235q−25−9q−27 + 207q−29 + 85q−31−161q−33−167q−35 + 116q−37 + 228q−39−55q−41−284q−43−10q−45 + 310q−47 + 90q−49−299q−51−166q−53 + 253q−55 + 216q−57−172q−59−232q−61 + 80q−63 + 207q−65−5q−67−155q−69−34q−71 + 89q−73 + 47q−75−40q−77−36q−79 + 15q−81 + 16q−83−2q−85−7q−87 + 3q−91−q−93
4	q56−2q54−q52 + 2q50−q48 + 6q46−5q44 + 9q40−14q38−q36−23q34 + 22q32 + 77q30−9q28−67q26−171q24−2q22 + 301q20 + 235q18−56q16−607q14−444q12 + 443q10 + 925q8 + 579q6−930q4−1509q2−245 + 1445q−2 + 1983q−4−232q−6−2323q−8−1791q−10 + 793q−12 + 3002q−14 + 1352q−16−1799q−18−2874q−20−729q−22 + 2579q−24 + 2432q−26−351q−28−2573q−30−1800q−32 + 1199q−34 + 2311q−36 + 854q−38−1459q−40−1961q−42−114q−44 + 1595q−46 + 1507q−48−353q−50−1761q−52−1176q−54 + 880q−56 + 2026q−58 + 701q−60−1513q−62−2235q−64−9q−66 + 2367q−68 + 1922q−70−797q−72−2995q−74−1313q−76 + 1876q−78 + 2831q−80 + 610q−82−2624q−84−2367q−86 + 390q−88 + 2514q−90 + 1840q−92−1100q−94−2163q−96−965q−98 + 1077q−100 + 1793q−102 + 276q−104−940q−106−1105q−108−115q−110 + 828q−112 + 531q−114−8q−116−475q−118−325q−120 + 130q−122 + 200q−124 + 148q−126−69q−128−119q−130−12q−132 + 15q−134 + 46q−136 + 2q−138−19q−140−2q−142−2q−144 + 7q−146−3q−150 + q−152
5	−q85 + 2q83 + q81−2q79 + q77−2q75−6q73 + q71 + 6q69 + q67 + 13q65 + 13q63−14q61−35q59−39q57−13q55 + 64q53 + 142q51 + 100q49−98q47−311q45−330q43−q41 + 549q39 + 850q37 + 387q35−741q33−1659q31−1355q29 + 479q27 + 2674q25 + 3154q23 + 695q21−3376q19−5677q17−3348q15 + 2943q13 + 8388q11 + 7598q9−475q7−10214q5−12893q3−4415q + 9795q−1 + 17931q−3 + 11490q−5−6365q−7−21186q−9−19189q−11−93q−13 + 21141q−15 + 25898q−17 + 8431q−19−17601q−21−29735q−23−16708q−25 + 11076q−27 + 29878q−29 + 23196q−31−3242q−33−26528q−35−26612q−37−4145q−39 + 20728q−41 + 26751q−43 + 9861q−45−14068q−47−24316q−49−13301q−51 + 7817q−53 + 20429q−55 + 14786q−57−2721q−59−16324q−61−15064q−63−1184q−65 + 12810q−67 + 15086q−69 + 4250q−71−10052q−73−15511q−75−7310q−77 + 7856q−79 + 16703q−81 + 10795q−83−5503q−85−18203q−87−15247q−89 + 2294q−91 + 19488q−93 + 20227q−95 + 2311q−97−19372q−99−25126q−101−8404q−103 + 17105q−105 + 28708q−107 + 15262q−109−12185q−111−29724q−113−21717q−115 + 5021q−117 + 27387q−119 + 26228q−121 + 3107q−123−21689q−125−27471q−127−10525q−129 + 13702q−131 + 25036q−133 + 15499q−135−5220q−137−19530q−139−17038q−141−1869q−143 + 12466q−145 + 15290q−147 + 6291q−149−5787q−151−11362q−153−7627q−155 + 794q−157 + 6798q−159 + 6695q−161 + 1905q−163−3044q−165−4579q−167−2577q−169 + 609q−171 + 2454q−173 + 2134q−175 + 454q−177−998q−179−1275q−181−623q−183 + 198q−185 + 588q−187 + 465q−189 + 54q−191−220q−193−215q−195−82q−197 + 45q−199 + 86q−201 + 55q−203−12q−205−33q−207−12q−209 + 2q−211 + 5q−213 + 6q−215 + 2q−217−7q−219 + 3q−223−q−225

A2 Invariants.

Weight	Invariant
1,0	−q6 + q4−q2−1 + 4q−2−q−4 + 4q−6 + q−8−q−10 + q−12−4q−14 + 2q−16−q−18−q−20 + 2q−22−q−24
1,1	q20−4q18 + 12q16−28q14 + 56q12−102q10 + 168q8−272q6 + 409q4−590q2 + 804−1024q−2 + 1229q−4−1340q−6 + 1342q−8−1158q−10 + 792q−12−252q−14−410q−16 + 1118q−18−1798q−20 + 2348q−22−2706q−24 + 2824q−26−2693q−28 + 2326q−30−1764q−32 + 1078q−34−351q−36−330q−38 + 882q−40−1262q−42 + 1445q−44−1444q−46 + 1300q−48−1064q−50 + 804q−52−560q−54 + 354q−56−204q−58 + 107q−60−50q−62 + 20q−64−6q−66 + q−68
2,0	q18−q16−2q14 + 3q12 + 2q10−7q8−5q6 + 9q4 + 6q2−14−5q−2 + 22q−4 + 5q−6−15q−8 + 5q−10 + 16q−12−3q−14−12q−16 + 6q−18 + 4q−20−14q−22 + 4q−24 + 8q−26−10q−28−4q−30 + 16q−32−18q−36 + q−38 + 15q−40−q−42−16q−44 + 6q−46 + 11q−48−4q−50−6q−52 + 5q−56−q−58−2q−60 + q−62

A3 Invariants.

Weight	Invariant
0,1,0	q14−2q12 + q10 + 4q8−9q6 + q4 + 12q2−20 + 27q−4−25q−6 + 3q−8 + 34q−10−17q−12−3q−14 + 18q−16−7q−18−12q−20−4q−22 + 11q−24−5q−26−19q−28 + 24q−30 + 5q−32−30q−34 + 23q−36 + 7q−38−26q−40 + 16q−42 + 4q−44−12q−46 + 7q−48 + q−50−3q−52 + q−54
1,0,0	−q7 + q5−2q3 + q−2q−1 + 4q−3−q−5 + 5q−7 + 2q−9 + 2q−11−2q−15−4q−19 + 2q−21−2q−23 + 2q−25−2q−27 + 2q−29−q−31

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q14 + 3q10−q8−5q6 + q4 + 4q2−8−12q−2 + 8q−4 + 12q−6−17q−8−2q−10 + 32q−12 + 10q−14−15q−16 + 17q−18 + 25q−20−14q−22−18q−24 + 13q−26−4q−28−33q−30 + 5q−32 + 18q−34−22q−36−7q−38 + 30q−40−23q−44 + 8q−46 + 19q−48−12q−50−15q−52 + 13q−54 + 8q−56−11q−58−q−60 + 7q−62−2q−64−2q−66 + q−68
1,0,0,0	−q8 + q6−2q4−2q−2 + 4q−4−q−6 + 5q−8 + 3q−10 + 3q−12 + 2q−14−q−18−3q−20−4q−24 + 2q−26−2q−28 + q−30 + q−32−2q−34 + 2q−36−q−38

B2 Invariants.

Weight	Invariant
0,1	−q14 + 2q12−5q10 + 8q8−13q6 + 19q4−26q2 + 32−34q−2 + 35q−4−27q−6 + 19q−8−2q−10−13q−12 + 33q−14−48q−16 + 61q−18−68q−20 + 68q−22−63q−24 + 49q−26−33q−28 + 14q−30 + 3q−32−18q−34 + 29q−36−35q−38 + 36q−40−34q−42 + 28q−44−20q−46 + 13q−48−7q−50 + 3q−52−q−54
1,0	q24−2q20−2q18 + 3q16 + 6q14−q12−11q10−8q8 + 10q6 + 19q4−2q2−28−15q−2 + 24q−4 + 33q−6−9q−8−37q−10−7q−12 + 37q−14 + 24q−16−20q−18−26q−20 + 13q−22 + 28q−24−3q−26−27q−28−4q−30 + 22q−32 + 6q−34−23q−36−15q−38 + 19q−40 + 19q−42−17q−44−27q−46 + 11q−48 + 35q−50 + 4q−52−36q−54−21q−56 + 28q−58 + 34q−60−11q−62−35q−64−8q−66 + 25q−68 + 17q−70−11q−72−16q−74 + 10q−78 + 4q−80−3q−82−3q−84 + q−88

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−2q16 + 3q14−4q12 + 7q10−11q8 + 11q6−16q4 + 20q2−27 + 23q−2−26q−4 + 30q−6−23q−8 + 20q−10−7q−12 + 13q−14 + 13q−16−15q−18 + 26q−20−33q−22 + 45q−24−53q−26 + 45q−28−57q−30 + 51q−32−46q−34 + 37q−36−34q−38 + 22q−40−6q−42 + q−44 + 5q−46−17q−48 + 25q−50−26q−52 + 27q−54−30q−56 + 28q−58−22q−60 + 19q−62−16q−64 + 11q−66−6q−68 + 4q−70−3q−72 + q−74

G2 Invariants.

Weight

Invariant

1,0

q32−2q30 + 5q28−8q26 + 8q24−7q22−2q20 + 15q18−30q16 + 43q14−49q12 + 39q10−14q8−31q6 + 88q4−135q2 + 155−130q−2 + 45q−4 + 73q−6−192q−8 + 270q−10−257q−12 + 158q−14 + 9q−16−171q−18 + 266q−20−245q−22 + 128q−24 + 45q−26−178q−28 + 213q−30−130q−32−29q−34 + 208q−36−308q−38 + 285q−40−137q−42−84q−44 + 291q−46−414q−48 + 398q−50−255q−52 + 33q−54 + 188q−56−340q−58 + 366q−60−262q−62 + 73q−64 + 112q−66−230q−68 + 224q−70−106q−72−61q−74 + 203q−76−245q−78 + 172q−80−12q−82−172q−84 + 292q−86−302q−88 + 206q−90−47q−92−113q−94 + 215q−96−231q−98 + 180q−100−86q−102−8q−104 + 69q−106−96q−108 + 82q−110−51q−112 + 23q−114 + 2q−116−13q−118 + 15q−120−13q−122 + 7q−124−3q−126 + q−128

.

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 84"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t3−9t2 + 20t−25 + 20t−1−9t−2 + 2t−3

In[5]:=

Conway[K][z]

Out[5]=

2z6 + 3z4 + 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]=

{ 87, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q8 + 4q7−8q6 + 11q5−14q4 + 15q3−13q2 + 11q−6 + 3q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + z6a−4 + 3z4a−2 + 2z4a−4−z4a−6−z4 + 5z2a−2−z2a−6−2z2 + 4a−2−2a−4−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−3 + 2z9a−5 + 4z8a−2 + 10z8a−4 + 6z8a−6 + 4z7a−1 + 5z7a−3 + 8z7a−5 + 7z7a−7−2z6a−2−17z6a−4−8z6a−6 + 4z6a−8 + 3z6 + az5−4z5a−1−11z5a−3−20z5a−5−13z5a−7 + z5a−9−5z4a−2 + 9z4a−4 + 2z4a−6−6z4a−8−6z4−2az3−2z3a−1 + 4z3a−3 + 11z3a−5 + 6z3a−7−z3a−9 + 7z2a−2 + z2a−4−z2a−6 + z2a−8 + 4z2 + az + 2za−1 + 2za−3−za−7−4a−2−2a−4−1

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 84"];

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]=

{ 2t3−9t2 + 20t−25 + 20t−1−9t−2 + 2t−3, −q8 + 4q7−8q6 + 11q5−14q4 + 15q3−13q2 + 11q−6 + 3q−1−q−2 }

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]=

{K11a46, K11n184,}

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	q23−4q22 + 3q21 + 12q20−27q19 + 3q18 + 55q17−66q16−22q15 + 123q14−93q13−69q12 + 181q11−93q10−111q9 + 197q8−67q7−124q6 + 161q5−27q4−101q3 + 93q2 + q−54 + 34q−1 + 5q−2−17q−3 + 8q−4 + q−5−3q−6 + q−7
3	−q45 + 4q44−3q43−7q42 + 4q41 + 22q40−4q39−58q38 + 109q36 + 38q35−181q34−121q33 + 259q32 + 250q31−308q30−433q29 + 319q28 + 638q27−271q26−852q25 + 186q24 + 1027q23−51q22−1172q21−88q20 + 1256q19 + 232q18−1284q17−371q16 + 1262q15 + 478q14−1162q13−587q12 + 1036q11 + 632q10−834q9−663q8 + 637q7 + 612q6−408q5−547q4 + 245q3 + 413q2−99q−296 + 29q−1 + 181q−2 + 5q−3−99q−4−10q−5 + 48q−6 + 6q−7−22q−8−2q−9 + 11q−10−2q−11−3q−12−q−13 + 3q−14−q−15
4	q74−4q73 + 3q72 + 7q71−9q70 + q69−21q68 + 24q67 + 51q66−40q65−26q64−128q63 + 74q62 + 268q61 + 12q60−96q59−583q58−76q57 + 735q56 + 551q55 + 201q54−1526q53−1066q52 + 900q51 + 1767q50 + 1718q49−2242q48−3108q47−298q46 + 2830q45 + 4658q44−1568q43−5232q42−3055q41 + 2573q40 + 7892q39 + 653q38−6187q37−6244q36 + 891q35 + 10090q34 + 3372q33−5742q32−8620q31−1335q30 + 10812q29 + 5586q28−4417q27−9766q26−3391q25 + 10227q24 + 6994q23−2557q22−9678q21−5100q20 + 8380q19 + 7496q18−244q17−8221q16−6212q15 + 5381q14 + 6723q13 + 1978q12−5438q11−6065q10 + 2073q9 + 4578q8 + 3053q7−2287q6−4415q5−136q4 + 1994q3 + 2521q2−196q−2200−674q−1 + 304q−2 + 1257q−3 + 383q−4−691q−5−328q−6−178q−7 + 370q−8 + 220q−9−140q−10−37q−11−112q−12 + 67q−13 + 51q−14−36q−15 + 21q−16−26q−17 + 12q−18 + 6q−19−14q−20 + 8q−21−3q−22 + 3q−23 + q−24−3q−25 + q−26