10 133

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["10 133"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X14,9,15,10 X5,13,6,12 X13,7,14,6 X18,11,19,12 X20,15,1,16 X16,19,17,20 X10,17,11,18 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[23,21,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_133.

A1 Invariants.

Weight	Invariant
1	q19−q17−q13 + 2q5 + q
2	q54−q52−2q50 + 2q48 + q46−2q44 + 2q40−q36 + q34 + q32−2q30 + q26−2q24−q22 + q20−2q16 + q14 + 2q12 + q6 + q4 + q2
3	q105−q103−2q101 + 3q97 + 3q95−3q93−4q91 + 4q87 + 3q85−2q83−4q81−q79 + 3q77 + 4q75−2q73−6q71−2q69 + 6q67 + 4q65−5q63−4q61 + 5q59 + 5q57−3q55−3q53 + 3q51 + 3q49−4q47−3q45 + q43 + 3q41−2q37−5q35 + 2q33 + 6q31 + q29−8q27−6q25 + 7q23 + 7q21−3q19−6q17 + q15 + 5q13 + 2q11−2q9 + q5 + 2q3

A2 Invariants.

Weight	Invariant
1,0	q28−2q20−q18−q16 + q12 + q10 + 2q8 + q6 + q2
1,1	q76−2q74 + 4q72−8q70 + 11q68−14q66 + 14q64−12q62 + 8q60−8q56 + 14q54−19q52 + 22q50−20q48 + 20q46−16q44 + 12q42−6q40 + 6q36−10q34 + 12q32−14q30 + 6q28−8q26−2q24−2q20 + 2q18 + 4q16 + 2q14 + 6q12 + 4q8 + q4
2,0	q72−q68−q66 + q62−q60−q58 + q52 + q50 + 2q48 + 3q46 + q44−2q40−q38−2q36−2q34−2q32−q30−q28−q26−2q22 + 2q20 + 2q18 + 2q16 + q14 + 2q12 + 3q10 + q8 + q4

A3 Invariants.

Weight	Invariant
0,1,0	q60−q58−2q52−q48 + q46 + 2q44 + 3q42 + 3q40 + 3q38−3q34−4q32−6q30−4q28−3q26 + 2q22 + 2q20 + 3q18 + 3q16 + q14 + 2q12 + 2q10 + q8 + q4
1,0,0	q37 + q33−2q27−2q25−2q23−q21 + q17 + 2q15 + q13 + 2q11 + q9 + q7 + q3

A4 Invariants.

Weight	Invariant
0,1,0,0	q78 + q72−q70−3q68−2q66−2q64−4q62−q60 + 4q58 + 7q56 + 6q54 + 9q52 + 9q50 + 2q48−3q46−5q44−11q42−13q40−9q38−7q36−4q34 + 5q30 + 5q28 + 4q26 + 5q24 + 5q22 + 2q20 + 2q18 + 2q16 + 2q14 + 2q12 + q10 + q6
1,0,0,0	q46 + q42 + q40−2q34−2q32−3q30−2q28−q26 + q22 + 2q20 + 2q18 + q16 + 2q14 + q12 + q10 + q8 + q4

B2 Invariants.

Weight	Invariant
0,1	q60−q58 + 2q56−2q54 + 2q52−2q50 + q48−q46 + q42−3q40 + 3q38−4q36 + 3q34−4q32 + 2q30−2q28 + q26 + 2q20−q18 + 3q16−q14 + 2q12 + q8 + q4
1,0	q98−q94−q92 + q90 + q88−2q86−2q84 + q82 + 2q80−q78−2q76 + 3q72 + 2q70 + 2q64 + 2q62 + q60−q58−q56−q52−4q50−3q48−q46−2q42−3q40 + 2q36 + q34−q32 + q30 + 2q28 + 3q26 + q20 + 2q18 + q16 + q14 + q6

D4 Invariants.

Weight	Invariant
1,0,0,0	q82−q80 + q78−2q76 + 2q74−3q72−2q68 + q66 + q62 + 3q60 + 2q58 + 6q56 + q54 + 4q52−3q50 + q48−7q46−3q44−8q42−4q40−5q38−q36 + q32 + 4q30 + 2q28 + 4q26 + q24 + 4q22 + 2q18 + q16 + 2q14 + q12 + q10 + q6

G2 Invariants.

Weight	Invariant
1,0	q142−q140 + 2q138−3q136 + q134−q132−3q130 + 6q128−6q126 + 4q124−q122−2q120 + 5q118−4q116 + 2q114 + 3q112−3q110 + 4q108 + q106−3q104 + 8q102−6q100 + 3q98 + q96−4q94 + 5q92−6q90 + 3q88−4q86−q82−5q80 + q78−4q76−3q70 + q66−4q64 + 6q62−5q60 + 2q58 + 4q56−5q54 + 7q52−2q50 + q48 + 3q46−2q44 + q42 + 2q40 + 2q36 + q34−q32 + 2q30−q28 + q26 + q24−q22 + 2q20 + q14 + q10

.

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["10 133"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t2 + 5t−7 + 5t−1−t−2

In[5]:=

Conway[K][z]

Out[5]=

−z4 + z2 + 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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−1−q−2 + 3q−3−3q−4 + 3q−5−3q−6 + 2q−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−z4a6−3z2a6−3a6 + 2z2a4 + 2a4 + z2a2 + a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a10−4z4a10 + 3z2a10 + 2z7a9−9z5a9 + 10z3a9−3za9 + z8a8−3z6a8 + z2a8 + a8 + 3z7a7−13z5a7 + 16z3a7−7za7 + z8a6−4z6a6 + 6z4a6−6z2a6 + 3a6 + z7a5−4z5a5 + 7z3a5−4za5 + 2z4a4−3z2a4 + 2a4 + z3a3 + z2a2−a2

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["10 133"];

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]=

{ −t2 + 5t−7 + 5t−1−t−2, q−1−q−2 + 3q−3−3q−4 + 3q−5−3q−6 + 2q−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]=

{7_6,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n27,}

n	Jn
2	q−2 + 2q−7−q−8−3q−9 + 4q−10−5q−12 + 3q−13 + 3q−14−6q−15 + q−16 + 6q−17−6q−18−q−19 + 7q−20−4q−21−3q−22 + 5q−23−q−24−2q−25 + q−26
3	2q−3−q−4−q−5−2q−6 + 6q−7 + 2q−8−5q−9−9q−10 + 9q−11 + 12q−12−5q−13−22q−14 + 7q−15 + 21q−16−26q−18 + 24q−20 + 2q−21−23q−22−2q−23 + 20q−24 + q−25−16q−26−2q−27 + 14q−28 + q−29−8q−30−2q−31 + 5q−32 + q−34−3q−36−4q−37 + 5q−38 + 6q−39−4q−40−8q−41 + 2q−42 + 8q−43 + q−44−7q−45−2q−46 + 4q−47 + 2q−48−q−49−2q−50 + q−51
4	q−3 + q−4−2q−5−q−7 + 4q−8 + 3q−9−7q−10−3q−11−2q−12 + 16q−13 + 10q−14−18q−15−19q−16−12q−17 + 40q−18 + 35q−19−22q−20−48q−21−40q−22 + 58q−23 + 69q−24−11q−25−65q−26−70q−27 + 55q−28 + 87q−29 + 5q−30−63q−31−83q−32 + 45q−33 + 88q−34 + 10q−35−54q−36−79q−37 + 34q−38 + 82q−39 + 12q−40−43q−41−73q−42 + 19q−43 + 73q−44 + 17q−45−26q−46−64q−47−2q−48 + 57q−49 + 21q−50−5q−51−47q−52−18q−53 + 33q−54 + 14q−55 + 12q−56−21q−57−19q−58 + 14q−59−3q−60 + 12q−61−q−62−7q−63 + 12q−64−15q−65 + 19q−69−9q−70−5q−71−7q−72−4q−73 + 16q−74−5q−77−6q−78 + 5q−79 + q−80 + 2q−81−q−82−2q−83 + q−84