10 131

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["10 131"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X6,14,7,13 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[311,21,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_131.

A1 Invariants.

Weight	Invariant
1	q19−q17 + q15−2q13 + 2q5−q3 + 2q
2	q54−q52−2q50 + 3q48 + q46−5q44 + 2q42 + 4q40−5q38 + 5q34−2q32−3q30 + 4q28 + q26−4q24 + 3q20−2q18−5q16 + 5q14 + q12−5q10 + 4q8 + 2q6−2q4 + 2q2 + 1
3	q105−q103−2q101 + 4q97 + 3q95−5q93−7q91 + 3q89 + 11q87 + 2q85−13q83−9q81 + 11q79 + 14q77−4q75−17q73−q71 + 15q69 + 8q67−14q65−12q63 + 11q61 + 14q59−8q57−16q55 + 6q53 + 16q51−3q49−16q47 + q45 + 13q43 + 7q41−11q39−11q37 + 3q35 + 16q33 + 3q31−17q29−10q27 + 15q25 + 14q23−11q21−14q19 + 5q17 + 10q15−q13−6q11 + q9 + 4q7−q5 + q3 + 2q−1

A2 Invariants.

Weight	Invariant
1,0	q28 + q22−2q20−q18−q16−q14 + q12 + 2q8 + q6 + 2q2
1,1	q76−2q74 + 6q72−14q70 + 21q68−32q66 + 44q64−48q62 + 47q60−40q58 + 28q56−6q54−22q52 + 38q50−60q48 + 76q46−84q44 + 88q42−74q40 + 66q38−40q36 + 18q34 + 4q32−26q30 + 36q28−46q26 + 36q24−38q22 + 28q20−24q18 + 14q16−8q14 + 13q12 + 4q8 + 2q4 + 2q2
2,0	q72−q68−q66 + q64 + 2q62−2q60−2q58 + q56 + q54−2q52−3q50 + 2q48 + 4q46 + 2q40 + 2q38−q30−2q28−q26−4q24−6q22 + q20 + 2q18−q16 + 5q12 + 5q10−q6 + 3q4 + q2

A3 Invariants.

Weight	Invariant
0,1,0	q60−q58 + q56−4q52 + q50−q48−2q46 + 5q44 + 3q42 + 6q38 + q36−4q34−2q32−3q30−3q28−4q26−q24 + 2q22−3q20 + 5q16−2q14 + 2q12 + 6q10 + 3q4
1,0,0	q37 + q33 + q29−2q27−q25−2q23−q21−q19 + q15 + 2q11 + q9 + 2q7 + 2q3

A4 Invariants.

Weight	Invariant
0,1,0,0	q78 + 2q72−4q68−2q66−q64−5q62−5q60 + 3q58 + 8q56 + 3q54 + 6q52 + 12q50 + 6q48−3q46−q44−4q42−10q40−8q38−4q36−4q34−6q32 + q30 + 3q28−3q26−q24 + 5q22 + 3q20 + 3q16 + 6q14 + 4q12 + q10 + q8 + 3q6
1,0,0,0	q46 + q42 + q40 + q36−2q34−q32−2q30−2q28−q26−q24 + q18 + 2q14 + q12 + 2q10 + 2q8 + 2q4

B2 Invariants.

Weight	Invariant
0,1	q60−q58 + 3q56−4q54 + 4q52−5q50 + 5q48−4q46 + 3q44−q42−2q40 + 4q38−7q36 + 8q34−10q32 + 9q30−9q28 + 6q26−5q24 + 2q22 + q20−2q18 + 5q16−4q14 + 6q12−4q10 + 4q8−2q6 + 3q4
1,0	q98−q94−q92 + 2q90 + 2q88−3q86−4q84 + 4q80 + q78−5q76−3q74 + 4q72 + 6q70 + q68−4q66 + 5q62 + 4q60−2q58−3q56 + q54 + 2q52−3q50−5q48−q46 + 3q44−q42−5q40−3q38 + 3q36 + 3q34−3q32−4q30 + q28 + 6q26 + q24−3q22−2q20 + 4q18 + 5q16 + q14−2q12−q10 + q8 + 3q6

D4 Invariants.

Weight	Invariant
1,0,0,0	q82−q80 + 2q78−3q76 + 3q74−5q72 + 2q70−5q68 + 4q66−3q64 + 3q62 + q60 + 4q58 + 5q56 + 6q52−5q50 + 5q48−10q46 + 4q44−11q42 + 3q40−9q38 + 3q36−4q34 + 2q32−q30−q28 + 2q26−2q24 + 5q22−3q20 + 5q18−q16 + 7q14 + 3q10−q8 + 3q6

G2 Invariants.

Weight

Invariant

1,0

q142−q140 + 3q138−5q136 + 3q134−2q132−4q130 + 10q128−14q126 + 14q124−7q122−4q120 + 14q118−19q116 + 18q114−8q112−2q110 + 13q108−15q106 + 12q104 + q102−8q100 + 14q98−11q96 + 2q94 + 7q92−15q90 + 19q88−18q86 + 9q84 + 2q82−17q80 + 21q78−25q76 + 14q74−4q72−11q70 + 16q68−18q66 + 11q64 + q62−12q60 + 14q58−9q56−q54 + 11q52−15q50 + 14q48−4q46−3q44 + 10q42−14q40 + 14q38−7q36 + 2q34 + 4q32−7q30 + 8q28−5q26 + 6q24−q22 + 2q18−2q16 + 3q14−q12 + 2q10 + q8

.

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["10 131"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t2 + 8t−11 + 8t−1−2t−2

In[5]:=

Conway[K][z]

Out[5]=

1−2z4

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]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

2q−1−3q−2 + 5q−3−5q−4 + 5q−5−5q−6 + 3q−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−z4a6−2z2a6−2a6−z4a4−z2a4 + 2z2a2 + 2a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z6a10−4z4a10 + 4z2a10 + 2z7a9−8z5a9 + 9z3a9−3za9 + z8a8−z6a8−4z4a8 + 2z2a8 + a8 + 4z7a7−12z5a7 + 10z3a7−5za7 + z8a6−2z4a6−3z2a6 + 2a6 + 2z7a5−3z5a5 + 2z3a5−za5 + 2z6a4−2z4a4 + 2z2a4 + z5a3 + z3a3 + za3 + 3z2a2−2a2

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["10 131"];

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]=

{ −2t2 + 8t−11 + 8t−1−2t−2, 2q−1−3q−2 + 5q−3−5q−4 + 5q−5−5q−6 + 3q−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]=

{8_14, 9_8,}

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−1 + q−2−4q−3 + 5q−4 + 3q−5−13q−6 + 11q−7 + 7q−8−23q−9 + 14q−10 + 12q−11−26q−12 + 10q−13 + 17q−14−23q−15 + 3q−16 + 18q−17−16q−18−2q−19 + 13q−20−7q−21−4q−22 + 6q−23−q−24−2q−25 + q−26
3	2q−1−2q−2 + q−3−2q−4 + 7q−5−5q−6−6q−7 + 3q−8 + 18q−9−10q−10−25q−11 + 6q−12 + 43q−13−9q−14−50q−15−q−16 + 63q−17 + 4q−18−63q−19−15q−20 + 63q−21 + 22q−22−57q−23−27q−24 + 46q−25 + 35q−26−38q−27−37q−28 + 24q−29 + 43q−30−16q−31−40q−32 + q−33 + 41q−34 + 6q−35−33q−36−15q−37 + 25q−38 + 19q−39−15q−40−18q−41 + 5q−42 + 15q−43−9q−45−3q−46 + 5q−47 + 2q−48−q−49−2q−50 + q−51
4	1 + q−1−2q−2−q−3 + 4q−4−2q−5 + 2q−6−7q−7−4q−8 + 22q−9−5q−11−37q−12−17q−13 + 73q−14 + 34q−15−11q−16−108q−17−72q−18 + 132q−19 + 115q−20 + 21q−21−184q−22−170q−23 + 150q−24 + 192q−25 + 92q−26−207q−27−256q−28 + 119q−29 + 211q−30 + 154q−31−174q−32−282q−33 + 76q−34 + 172q−35 + 181q−36−117q−37−262q−38 + 39q−39 + 112q−40 + 186q−41−57q−42−222q−43 + q−44 + 44q−45 + 180q−46 + 12q−47−168q−48−38q−49−29q−50 + 151q−51 + 74q−52−89q−53−50q−54−94q−55 + 85q−56 + 95q−57−5q−58−17q−59−112q−60 + 8q−61 + 58q−62 + 36q−63 + 32q−64−73q−65−27q−66 + 5q−67 + 21q−68 + 46q−69−21q−70−16q−71−15q−72−3q−73 + 25q−74−7q−77−7q−78 + 6q−79 + q−80 + 2q−81−q−82−2q−83 + q−84