10 151

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["10 151"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X12,6,13,5 X9,17,10,16 X17,1,18,20 X13,19,14,18 X19,15,20,14 X15,11,16,10 X6,12,7,11 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 -12 2 16 -6 18 10 20 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[(21,2)(21,2-)]

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,−1,3,−2,1,3,−2})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_151.

A1 Invariants.

Weight	Invariant
1	−q5 + 2q3−2q + 2q−1 + q−5 + 2q−7−2q−9 + 2q−11−2q−13
2	q16−2q14−2q12 + 7q10−2q8−10q6 + 10q4 + 5q2−14 + 5q−2 + 10q−4−9q−6−q−8 + 9q−10−6q−14 + 2q−16 + 10q−18−10q−20−6q−22 + 15q−24−6q−26−10q−28 + 9q−30−5q−34 + 2q−36 + q−38
3	−q33 + 2q31 + 2q29−3q27−7q25 + 2q23 + 17q21 + 2q19−25q17−17q15 + 28q13 + 38q11−22q9−55q7 + 5q5 + 64q3 + 19q−67q−1−37q−3 + 56q−5 + 52q−7−40q−9−55q−11 + 26q−13 + 57q−15−10q−17−47q−19−5q−21 + 40q−23 + 21q−25−31q−27−39q−29 + 17q−31 + 56q−33−3q−35−65q−37−20q−39 + 71q−41 + 35q−43−61q−45−50q−47 + 41q−49 + 52q−51−20q−53−45q−55 + 5q−57 + 28q−59 + 6q−61−14q−63−6q−65 + 7q−67 + 2q−69−2q−73

A2 Invariants.

Weight	Invariant
1,0	−q6 + q4−q2 + 2q−2−q−4 + 3q−6 + 2q−10 + q−12−q−14 + q−16−2q−18−q−20
1,1	q20−4q18 + 12q16−28q14 + 52q12−86q10 + 128q8−170q6 + 196q4−206q2 + 188−134q−2 + 52q−4 + 48q−6−152q−8 + 254q−10−324q−12 + 378q−14−380q−16 + 360q−18−295q−20 + 208q−22−108q−24 + 92q−28−166q−30 + 200q−32−208q−34 + 188q−36−154q−38 + 108q−40−70q−42 + 40q−44−18q−46 + 6q−48−2q−50 + 2q−54
2,0	q18−q16−2q14 + 2q12 + 2q10−2q8−4q6 + 2q4 + 5q2−5−3q−2 + 6q−4 + q−6−3q−8 + 5q−12 + q−16 + 6q−18 + 3q−20−2q−22 + 4q−24 + 4q−26−7q−28−3q−30 + 3q−32−q−34−6q−36−4q−38 + 3q−40−q−42−3q−44 + q−46 + 2q−48 + 2q−50

A3 Invariants.

Weight	Invariant
0,1,0	q14−2q12 + q10 + 3q8−7q6 + 3q4 + 3q2−11 + 6q−2 + 5q−4−8q−6 + 6q−8 + 7q−10−q−12 + q−14 + 4q−16 + 4q−18−q−20−3q−22 + 9q−24−6q−26−9q−28 + 8q−30−7q−32−7q−34 + 7q−36−q−38−2q−40 + 3q−42
1,0,0	−q7 + q5−2q3 + q−q−1 + 2q−3 + 2q−7 + 2q−9 + q−11 + 2q−13 + 2q−17−2q−19 + q−21−2q−23−q−25−q−27

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q14 + 3q10−2q8−4q6 + 4q4−q2−9−q−2 + 6q−4−3q−6−7q−8 + 9q−10 + 10q−12−4q−14 + 2q−16 + 16q−18−q−20−2q−22 + 12q−24 + 5q−26−6q−28 + 3q−30 + 6q−32−8q−34−11q−36−2q−40−12q−42−3q−44 + 5q−46−q−50 + 3q−52 + 2q−54 + q−56
1,0,0,0	−q8 + q6−2q4−q−2 + 2q−4 + 3q−8 + q−10 + 3q−12 + q−14 + 2q−16 + q−20 + q−22−2q−24 + q−26−2q−28−q−30−q−32−q−34

B2 Invariants.

Weight	Invariant
0,1	−q14 + 2q12−5q10 + 7q8−9q6 + 11q4−11q2 + 9−6q−2 + 3q−4 + 4q−6−8q−8 + 15q−10−17q−12 + 21q−14−20q−16 + 18q−18−13q−20 + 9q−22−3q−24−2q−26 + 7q−28−10q−30 + 11q−32−11q−34 + 9q−36−7q−38 + 4q−40−3q−42
1,0	q24−2q20−2q18 + 3q16 + 5q14−2q12−8q10−3q8 + 9q6 + 7q4−8q2−11 + 2q−2 + 12q−4 + 4q−6−10q−8−5q−10 + 7q−12 + 8q−14−3q−16−5q−18 + 3q−20 + 9q−22 + q−24−7q−26−q−28 + 9q−30 + 5q−32−6q−34−7q−36 + 6q−38 + 9q−40−4q−42−13q−44−2q−46 + 10q−48 + 4q−50−9q−52−10q−54 + 2q−56 + 8q−58 + 2q−60−4q−62−3q−64 + q−66 + 3q−68

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−2q16 + 3q14−4q12 + 6q10−8q8 + 7q6−10q4 + 8q2−9 + 4q−2−4q−4 + 3q−6 + 3q−8−4q−10 + 12q−12−6q−14 + 16q−16−13q−18 + 17q−20−13q−22 + 17q−24−12q−26 + 10q−28−7q−30 + 7q−32−q−34−4q−36 + q−38−9q−40 + 7q−42−11q−44 + 5q−46−10q−48 + 8q−50−4q−52 + 4q−54−3q−56 + 3q−58

G2 Invariants.

Weight

Invariant

1,0

q32−2q30 + 5q28−8q26 + 7q24−4q22−6q20 + 19q18−30q16 + 34q14−27q12 + q10 + 27q8−52q6 + 62q4−46q2 + 15 + 23q−2−51q−4 + 55q−6−34q−8 + 33q−12−46q−14 + 35q−16−35q−20 + 62q−22−62q−24 + 40q−26−q−28−43q−30 + 75q−32−80q−34 + 64q−36−24q−38−18q−40 + 56q−42−68q−44 + 57q−46−25q−48−11q−50 + 42q−52−45q−54 + 24q−56 + 13q−58−42q−60 + 55q−62−42q−64 + 5q−66 + 29q−68−56q−70 + 60q−72−45q−74 + 15q−76 + 13q−78−34q−80 + 34q−82−28q−84 + 15q−86−2q−88−7q−90 + 7q−92−8q−94 + 6q−96−2q−98 + q−100 + q−102

.

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["10 151"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−4t2 + 10t−13 + 10t−1−4t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + 2z4 + 3z2 + 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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−2q6 + 4q5−6q4 + 8q3−7q2 + 7q−5 + 3q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−2 + z8a−4 + 3z7a−1 + 5z7a−3 + 2z7a−5 + 5z6a−2 + 3z6a−4 + z6a−6 + 3z6 + az5−4z5a−1−7z5a−3−2z5a−5−15z4a−2−6z4a−4 + 2z4a−6−7z4−2az3−3z3a−1 + z3a−3 + 5z3a−5 + 3z3a−7 + 10z2a−2 + 4z2a−4−2z2a−6 + 4z2 + az + 2za−1 + za−3−3za−5−3za−7−3a−2 + a−6−1

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["10 151"];

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]=

{ t3−4t2 + 10t−13 + 10t−1−4t−2 + t−3, −2q6 + 4q5−6q4 + 8q3−7q2 + 7q−5 + 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]=

{K11n54, K11n129,}

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	q18 + q17−7q16 + 6q15 + 10q14−26q13 + 10q12 + 31q11−47q10 + 6q9 + 51q8−55q7−2q6 + 57q5−46q4−12q3 + 49q2−27q−17 + 30q−1−8q−2−12q−3 + 10q−4−3q−6 + q−7
3	−2q35 + 2q34 + 2q33 + 5q32−15q31−6q30 + 22q29 + 27q28−38q27−56q26 + 47q25 + 99q24−49q23−147q22 + 36q21 + 195q20−13q19−238q18−9q17 + 257q16 + 46q15−277q14−65q13 + 265q12 + 98q11−258q10−110q9 + 223q8 + 135q7−191q6−141q5 + 142q4 + 150q3−99q2−137q + 49 + 120q−1−13q−2−92q−3−10q−4 + 60q−5 + 20q−6−32q−7−20q−8 + 15q−9 + 12q−10−5q−11−5q−12 + 3q−14−q−15
4	q58 + q57−3q56−6q55 + 4q54 + 7q53 + 17q52−7q51−51q50−9q49 + 27q48 + 107q47 + 37q46−168q45−131q44−18q43 + 315q42 + 271q41−260q40−414q39−290q38 + 515q37 + 725q36−142q35−700q34−794q33 + 519q32 + 1196q31 + 179q30−795q29−1295q28 + 328q27 + 1457q26 + 514q25−685q24−1592q23 + 79q22 + 1475q21 + 735q20−473q19−1657q18−148q17 + 1305q16 + 848q15−195q14−1540q13−368q12 + 978q11 + 873q10 + 140q9−1245q8−553q7 + 519q6 + 756q5 + 449q4−784q3−583q2 + 62q + 461 + 558q−1−294q−2−394q−3−191q−4 + 121q−5 + 409q−6−130q−8−175q−9−60q−10 + 171q−11 + 52q−12 + 11q−13−64q−14−61q−15 + 38q−16 + 15q−17 + 20q−18−8q−19−19q−20 + 5q−21 + 5q−23−3q−25 + q−26
5	−2q86 + 2q85 + 4q83 + 5q82−7q81−22q80−q79 + 10q78 + 34q77 + 60q76−18q75−115q74−113q73−24q72 + 158q71 + 325q70 + 168q69−262q68−572q67−473q66 + 175q65 + 958q64 + 1030q63 + 70q62−1290q61−1796q60−678q59 + 1477q58 + 2730q57 + 1616q56−1360q55−3665q54−2850q53 + 886q52 + 4421q51 + 4240q50−67q49−4887q48−5597q47−990q46 + 5020q45 + 6757q44 + 2119q43−4843q42−7587q41−3238q40 + 4461q39 + 8157q38 + 4098q37−3939q36−8344q35−4864q34 + 3383q33 + 8418q32 + 5307q31−2827q30−8185q29−5723q28 + 2252q27 + 7951q26 + 5910q25−1656q24−7456q23−6134q22 + 971q21 + 6930q20 + 6187q19−209q18−6116q17−6228q16−640q15 + 5202q14 + 6016q13 + 1502q12−4009q11−5657q10−2276q9 + 2758q8 + 4966q7 + 2820q6−1410q5−4063q4−3078q3 + 267q2 + 2944q + 2942 + 649q−1−1809q−2−2498q−3−1166q−4 + 806q−5 + 1834q−6 + 1306q−7−79q−8−1114q−9−1148q−10−340q−11 + 528q−12 + 819q−13 + 458q−14−126q−15−464q−16−406q−17−76q−18 + 212q−19 + 268q−20 + 116q−21−54q−22−132q−23−106q−24−6q−25 + 62q−26 + 56q−27 + 12q−28−12q−29−24q−30−20q−31 + 8q−32 + 12q−33 + 2q−34−5q−37 + 3q−39−q−40
6	q120 + q119−3q118−2q117−2q116 + q115 + 2q114 + 13q113 + 21q112−8q111−41q110−48q109−17q108 + 6q107 + 113q106 + 183q105 + 92q104−146q103−357q102−341q101−243q100 + 327q99 + 915q98 + 970q97 + 231q96−928q95−1710q94−2000q93−429q92 + 2055q91 + 3803q90 + 3118q89 + 94q88−3635q87−6799q86−5086q85 + 735q84 + 7577q83 + 10229q82 + 6481q81−2214q80−12818q79−15176q78−7357q77 + 7382q76 + 18698q75 + 19187q74 + 6932q73−14315q72−26637q71−22143q70−864q69 + 22214q68 + 32959q67 + 22463q66−7769q65−32841q64−37349q63−14781q62 + 17946q61 + 41183q60 + 37568q59 + 3743q58−31588q57−46734q56−27737q55 + 9250q54 + 42256q53 + 46837q52 + 14301q51−26107q50−49357q49−35575q48 + 1077q47 + 39177q46 + 50048q45 + 20995q44−20269q43−47977q42−38806q41−4769q40 + 34917q39 + 49793q38 + 24848q37−14993q36−44855q35−39861q34−9656q33 + 29765q32 + 47813q31 + 28065q30−8734q29−39836q28−39980q27−15505q26 + 22101q25 + 43550q24 + 31297q23 + 47q22−31223q21−37992q20−22209q19 + 10784q18 + 35021q17 + 32471q16 + 10341q15−18165q14−31284q13−26575q12−2302q11 + 21452q10 + 28161q9 + 17824q8−3343q7−19019q6−24437q5−11863q4 + 6209q3 + 17586q2 + 17892q + 7230−5189q−1−15429q−2−13277q−3−4234q−4 + 5443q−5 + 10909q−6 + 9321q−7 + 3615q−8−5026q−9−7871q−10−6366q−11−1807q−12 + 2900q−13 + 5254q−14 + 4911q−15 + 740q−16−1901q−17−3355q−18−2721q−19−952q−20 + 1066q−21 + 2402q−22 + 1415q−23 + 571q−24−589q−25−1072q−26−1070q−27−375q−28 + 490q−29 + 457q−30 + 526q−31 + 183q−32−73q−33−351q−34−274q−35 + q−36 + 3q−37 + 131q−38 + 102q−39 + 72q−40−56q−41−65q−42−7q−43−29q−44 + 12q−45 + 15q−46 + 29q−47−8q−48−12q−49 + 5q−50−7q−51 + 5q−54−3q−56 + q−57