10 120

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["10 120"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X5,18,6,19 X13,20,14,1 X11,16,12,17 X3,10,4,11 X7,12,8,13 X9,4,10,5 X15,8,16,9 X19,14,20,15 X17,2,18,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 18 12 4 16 20 8 2 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*20::20]

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,−2,1,3,2,−1,−4,−3,−2,−2,−3,−3,−4})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_120.

A1 Invariants.

Weight	Invariant
1	q25−3q23 + 4q21−5q19 + 3q17−2q15−q13 + 4q11−3q9 + 6q7−3q5 + q3
2	q70−3q68−q66 + 13q64−10q62−20q60 + 35q58 + 3q56−52q54 + 36q52 + 33q50−58q48 + 10q46 + 45q44−33q42−20q40 + 31q38 + 8q36−39q34−q32 + 49q30−34q28−33q26 + 62q24−9q22−41q20 + 36q18 + 5q16−20q14 + 9q12 + 3q10−3q8 + q6
3	q135−3q133−q131 + 8q129 + 8q127−18q125−34q123 + 28q121 + 81q119−5q117−149q115−69q113 + 203q111 + 194q109−198q107−344q105 + 112q103 + 474q101 + 35q99−525q97−217q95 + 496q93 + 374q91−400q89−477q87 + 262q85 + 518q83−120q81−507q79−9q77 + 462q75 + 135q73−379q71−260q69 + 287q67 + 369q65−143q63−477q61−25q59 + 518q57 + 214q55−507q53−381q51 + 404q49 + 479q47−253q45−490q43 + 100q41 + 415q39 + 18q37−294q35−57q33 + 165q31 + 68q29−87q27−39q25 + 40q23 + 18q21−13q19−8q17 + 6q15 + 3q13−3q11 + q9

A2 Invariants.

Weight	Invariant
1,0	q38 + q36−3q34 + q32−5q28 + 2q26−2q24 + q22 + 2q20−q18 + 5q16−2q14 + 2q12 + 3q10−3q8 + q6
2,0	q96 + q94−2q92−4q90 + 10q86−q84−15q82−5q80 + 20q78 + 14q76−27q74−12q72 + 28q70 + 22q68−21q66−16q64 + 25q62 + 8q60−22q58−13q56 + 8q54−9q52−7q50 + 4q48−10q46−9q44 + 19q42 + 20q40−26q38−9q36 + 38q34 + 13q32−33q30−8q28 + 29q26 + 10q24−19q22−5q20 + 11q18−3q14 + q12

A3 Invariants.

Weight	Invariant
0,1,0	q80−3q78 + 2q76 + 5q74−15q72 + 11q70 + 11q68−32q66 + 26q64 + 17q62−40q60 + 29q58 + 15q56−39q54 + 9q52 + 14q50−18q48−9q46 + 3q44 + 14q42−21q40−14q38 + 43q36−22q34−17q32 + 49q30−15q28−18q26 + 31q24−8q22−12q20 + 11q18−3q14 + q12
1,0,0	q51 + q49 + q47−3q45 + q43−3q41−5q37 + 2q35−3q33 + q31 + q29 + 2q27 + 2q25 + 5q21−2q19 + 3q17−q15 + 3q13−3q11 + q9

B2 Invariants.

Weight	Invariant
0,1	q80−3q78 + 8q76−15q74 + 25q72−37q70 + 45q68−52q66 + 54q64−47q62 + 34q60−15q58−11q56 + 37q54−65q52 + 84q50−100q48 + 103q46−97q44 + 80q42−57q40 + 32q38−3q36−18q34 + 39q32−49q30 + 55q28−50q26 + 43q24−32q22 + 24q20−13q18 + 6q16−3q14 + q12
1,0	q130−3q126−3q124 + 5q122 + 10q120−3q118−20q116−9q114 + 27q112 + 27q110−20q108−45q106−q104 + 55q102 + 29q100−43q98−47q96 + 21q94 + 56q92 + 3q90−49q88−21q86 + 34q84 + 25q82−25q80−31q78 + 16q76 + 32q74−12q72−41q70 + q68 + 41q66 + 8q64−44q62−26q60 + 40q58 + 42q56−23q54−53q52 + 5q50 + 56q48 + 26q46−35q44−37q42 + 13q40 + 37q38 + 9q36−20q34−18q32 + 5q30 + 12q28 + 3q26−3q24−3q22 + q18

G2 Invariants.

Weight

Invariant

1,0

q190−3q188 + 8q186−16q184 + 21q182−23q180 + 9q178 + 25q176−74q174 + 131q172−159q170 + 124q168−16q166−151q164 + 326q162−410q160 + 358q158−152q156−149q154 + 426q152−569q150 + 496q148−227q146−123q144 + 405q142−489q140 + 345q138−47q136−260q134 + 434q132−410q130 + 162q128 + 180q126−491q124 + 639q122−549q120 + 251q118 + 157q116−531q114 + 731q112−704q110 + 430q108−19q106−371q104 + 597q102−569q100 + 321q98 + 41q96−335q94 + 427q92−308q90 + 21q88 + 288q86−464q84 + 445q82−224q80−79q78 + 349q76−482q74 + 440q72−267q70 + 46q68 + 153q66−266q64 + 286q62−215q60 + 119q58−14q56−56q54 + 84q52−91q50 + 68q48−38q46 + 13q44 + 7q42−12q40 + 12q38−10q36 + 6q34−3q32 + q30

.

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["10 120"];

In[4]:=

Alexander[K][t]

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

Out[4]=

8t2−26t + 37−26t−1 + 8t−2

In[5]:=

Conway[K][z]

Out[5]=

8z4 + 6z2 + 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]=

{ 105, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−2−4q−3 + 10q−4−13q−5 + 17q−6−18q−7 + 16q−8−13q−9 + 8q−10−4q−11 + q−12

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a12−4z2a10−3a10 + 3z4a8 + 3z2a8 + 4z4a6 + 7z2a6 + 3a6 + z4a4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a14−2z4a14 + z2a14 + 4z7a13−10z5a13 + 8z3a13−2za13 + 6z8a12−13z6a12 + 6z4a12 + z2a12 + a12 + 3z9a11 + 7z7a11−33z5a11 + 29z3a11−8za11 + 16z8a10−33z6a10 + 17z4a10−7z2a10 + 3a10 + 3z9a9 + 16z7a9−44z5a9 + 26z3a9−4za9 + 10z8a8−9z6a8−3z4a8 + 13z7a7−17z5a7 + 5z3a7 + 2za7 + 10z6a6−11z4a6 + 7z2a6−3a6 + 4z5a5 + z4a4

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["10 120"];

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]=

{ 8t2−26t + 37−26t−1 + 8t−2, q−2−4q−3 + 10q−4−13q−5 + 17q−6−18q−7 + 16q−8−13q−9 + 8q−10−4q−11 + q−12 }

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]=

{}

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	q−4−4q−5 + 6q−6 + 7q−7−33q−8 + 31q−9 + 38q−10−110q−11 + 63q−12 + 109q−13−205q−14 + 62q−15 + 192q−16−255q−17 + 24q−18 + 239q−19−232q−20−27q−21 + 226q−22−154q−23−62q−24 + 158q−25−63q−26−59q−27 + 70q−28−8q−29−27q−30 + 15q−31 + 2q−32−4q−33 + q−34
3	q−6−4q−7 + 6q−8 + 3q−9−13q−10−9q−11 + 37q−12 + 25q−13−92q−14−57q−15 + 192q−16 + 122q−17−314q−18−294q−19 + 504q−20 + 519q−21−629q−22−884q−23 + 741q−24 + 1251q−25−704q−26−1669q−27 + 615q−28 + 1972q−29−400q−30−2212q−31 + 163q−32 + 2306q−33 + 112q−34−2294q−35−384q−36 + 2187q−37 + 626q−38−1967q−39−855q−40 + 1689q−41 + 1013q−42−1329q−43−1111q−44 + 950q−45 + 1090q−46−555q−47−989q−48 + 237q−49 + 782q−50 + 5q−51−550q−52−125q−53 + 326q−54 + 151q−55−158q−56−116q−57 + 54q−58 + 71q−59−14q−60−30q−61 + q−62 + 9q−63 + 2q−64−4q−65 + q−66