8 3

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["8 3"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X14,10,15,9 X10,5,11,6 X12,3,13,4 X4,11,5,12 X2,13,3,14 X16,8,1,7 X8,16,9,15

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 12 10 16 14 4 2 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[44]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 8_3.

A1 Invariants.

Weight	Invariant
1	q9 + q5−q3−q−3 + q−5 + q−9
2	q26 + q20−q18−2q16 + q14−2q10 + 2q8 + q6−q4 + q2 + 1 + q−2−q−4 + q−6 + 2q−8−2q−10 + q−14−2q−16−q−18 + q−20 + q−26
3	q51−q41−q39 + q35−q33−2q31 + 3q27 + 3q25−2q23−2q21 + 2q19 + 4q17−2q15−3q13 + q11 + 3q9−2q5−2q−5 + 3q−9 + q−11−3q−13−2q−15 + 4q−17 + 2q−19−2q−21−2q−23 + 3q−25 + 3q−27−2q−31−q−33 + q−35−q−39−q−41 + q−51
4	q84−q76−q72 + q68−q66−2q62−q60 + 3q58 + 2q56 + 3q54−2q52−3q50−q48 + q46 + 7q44 + 2q42−3q40−6q38−5q36 + 6q34 + 6q32 + q30−6q28−9q26 + 4q24 + 6q22 + 3q20−3q18−7q16 + q14 + 3q12 + 3q10−2q6 + q4 + q2 + 1 + q−2 + q−4−2q−6 + 3q−10 + 3q−12 + q−14−7q−16−3q−18 + 3q−20 + 6q−22 + 4q−24−9q−26−6q−28 + q−30 + 6q−32 + 6q−34−5q−36−6q−38−3q−40 + 2q−42 + 7q−44 + q−46−q−48−3q−50−2q−52 + 3q−54 + 2q−56 + 3q−58−q−60−2q−62−q−66 + q−68−q−72−q−76 + q−84
5	q125−q117−q115 + q107−2q103−q101 + q97 + 3q95 + 3q93−3q89−3q87−2q85 + 3q83 + 5q81 + 5q79 + q77−5q75−8q73−6q71 + 8q67 + 10q65 + 5q63−6q61−14q59−11q57 + 12q53 + 16q51 + 7q49−11q47−18q45−10q43 + 6q41 + 18q39 + 16q37−3q35−15q33−12q31 + 11q27 + 12q25 + 2q23−7q21−8q19−2q17 + 4q15 + 4q13 + 2q11−q9−3q7−3q−7−q−9 + 2q−11 + 4q−13 + 4q−15−2q−17−8q−19−7q−21 + 2q−23 + 12q−25 + 11q−27−12q−31−15q−33−3q−35 + 16q−37 + 18q−39 + 6q−41−10q−43−18q−45−11q−47 + 7q−49 + 16q−51 + 12q−53−11q−57−14q−59−6q−61 + 5q−63 + 10q−65 + 8q−67−6q−71−8q−73−5q−75 + q−77 + 5q−79 + 5q−81 + 3q−83−2q−85−3q−87−3q−89 + 3q−93 + 3q−95 + q−97−q−101−2q−103 + q−107−q−115−q−117 + q−125

A2 Invariants.

Weight	Invariant
1,0	q14 + q12 + q8−q4−1−q−4 + q−8 + q−12 + q−14
1,1	q36 + 2q32−2q30 + 4q28−2q26 + 4q24−6q22 + 5q20−6q18 + 4q16−6q14−q12−6q8 + 8q6−8q4 + 16q2−6 + 16q−2−8q−4 + 8q−6−6q−8−q−12−6q−14 + 4q−16−6q−18 + 5q−20−6q−22 + 4q−24−2q−26 + 4q−28−2q−30 + 2q−32 + q−36
2,0	q36 + q34 + q32 + q28 + q26−q24−3q22−2q20−2q14 + 2q10 + q4 + 2q2 + 2 + 2q−2 + q−4 + 2q−10−2q−14−2q−20−3q−22−q−24 + q−26 + q−28 + q−32 + q−34 + q−36

A3 Invariants.

Weight	Invariant
0,1,0	q28 + q24 + q22 + q18 + 2q16−2q14−q12−3q8−q6 + q4 + 2q2 + 2 + 2q−2 + q−4−q−6−3q−8−q−12−2q−14 + 2q−16 + q−18 + q−22 + q−24 + q−28
1,0,0	q19 + q17 + q15 + q11−q5−q−q−1−q−5 + q−11 + q−15 + q−17 + q−19

B2 Invariants.

Weight	Invariant
0,1	q28 + q24−q22 + 2q20−q18 + 2q16 + q12−q8 + q6−3q4 + 2q2−4 + 2q−2−3q−4 + q−6−q−8 + q−12 + 2q−16−q−18 + 2q−20−q−22 + q−24 + q−28
1,0	q46 + q38 + q36−q32 + q28 + 2q26−q24−2q22−q20 + q18−2q14−q12 + q8 + q2 + 3 + q−2 + q−8−q−12−2q−14 + q−18−q−20−2q−22−q−24 + 2q−26 + q−28−q−32 + q−36 + q−38 + q−46

G2 Invariants.

Weight	Invariant
1,0	q66 + q62−q60 + q58 + q56−q54 + 2q52−q50 + 2q48−q46 + q42−2q40 + 4q38−3q36 + q34 + q32−2q30 + 3q28−2q26 + q24 + 2q22−2q20 + q18−2q14 + 4q12−4q10 + q8−3q4 + 3q2−5 + 3q−2−3q−4 + q−8−4q−10 + 4q−12−2q−14 + q−18−2q−20 + 2q−22 + q−24−2q−26 + 3q−28−2q−30 + q−32 + q−34−3q−36 + 4q−38−2q−40 + q−42−q−46 + 2q−48−q−50 + 2q−52−q−54 + q−56 + q−58−q−60 + q−62 + q−66

.

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["8 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−4t + 9−4t−1

In[5]:=

Conway[K][z]

Out[5]=

1−4z2

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q4−q3 + 2q2−3q + 3−3q−1 + 2q−2−q−3 + q−4

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a4−z2a2−2z2−1−z2a−2 + a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

az7 + z7a−1 + a2z6 + z6a−2 + 2z6 + a3z5−4az5−4z5a−1 + z5a−3 + a4z4−2a2z4−2z4a−2 + z4a−4−6z4−2a3z3 + 8az3 + 8z3a−1−2z3a−3−3a4z2 + a2z2 + z2a−2−3z2a−4 + 8z2−4az−4za−1 + a4 + a−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["8 3"];

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]=

{ −4t + 9−4t−1, q4−q3 + 2q2−3q + 3−3q−1 + 2q−2−q−3 + q−4 }

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]=

{10_1,}

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	q12−q11 + 2q9−3q8−q7 + 5q6−4q5−3q4 + 9q3−5q2−5q + 11−5q−1−5q−2 + 9q−3−3q−4−4q−5 + 5q−6−q−7−3q−8 + 2q−9−q−11 + q−12
3	q24−q23 + q20−2q19 + q17 + 2q16−4q15−q14 + 3q13 + 5q12−4q11−6q10 + 3q9 + 9q8−2q7−12q6 + 2q5 + 13q4−15q2 + 15−15q−2 + 13q−4 + 2q−5−12q−6−2q−7 + 9q−8 + 3q−9−6q−10−4q−11 + 5q−12 + 3q−13−q−14−4q−15 + 2q−16 + q−17−2q−19 + q−20−q−23 + q−24
4	q40−q39−q36 + 2q35−2q34 + q33 + q32−3q31 + 3q30−4q29 + 2q28 + 5q27−4q26 + 4q25−9q24 + q23 + 7q22−2q21 + 10q20−14q19−4q18 + 4q17−q16 + 21q15−14q14−9q13−3q12−4q11 + 34q10−12q9−12q8−9q7−8q6 + 42q5−10q4−12q3−12q2−10q + 45−10q−1−12q−2−12q−3−10q−4 + 42q−5−8q−6−9q−7−12q−8−12q−9 + 34q−10−4q−11−3q−12−9q−13−14q−14 + 21q−15−q−16 + 4q−17−4q−18−14q−19 + 10q−20−2q−21 + 7q−22 + q−23−9q−24 + 4q−25−4q−26 + 5q−27 + 2q−28−4q−29 + 3q−30−3q−31 + q−32 + q−33−2q−34 + 2q−35−q−36−q−39 + q−40
5	q60−q59−q56 + 2q54−q53 + q51−2q50−2q49 + 3q48 + q46 + 3q45−2q44−5q43 + 2q40 + 8q39−5q37−4q36−6q35−q34 + 10q33 + 6q32 + 3q31−2q30−11q29−12q28 + 2q27 + 9q26 + 14q25 + 10q24−7q23−21q22−16q21 + 2q20 + 22q19 + 26q18 + 5q17−23q16−35q15−10q14 + 25q13 + 38q12 + 16q11−22q10−45q9−19q8 + 24q7 + 44q6 + 22q5−22q4−47q3−22q2 + 22q + 47 + 22q−1−22q−2−47q−3−22q−4 + 22q−5 + 44q−6 + 24q−7−19q−8−45q−9−22q−10 + 16q−11 + 38q−12 + 25q−13−10q−14−35q−15−23q−16 + 5q−17 + 26q−18 + 22q−19 + 2q−20−16q−21−21q−22−7q−23 + 10q−24 + 14q−25 + 9q−26 + 2q−27−12q−28−11q−29−2q−30 + 3q−31 + 6q−32 + 10q−33−q−34−6q−35−4q−36−5q−37 + 8q−39 + 2q−40−5q−43−2q−44 + 3q−45 + q−46 + 3q−48−2q−49−2q−50 + q−51−q−53 + 2q−54−q−56−q−59 + q−60
6	q84−q83−q80 + 3q77−2q76 + q74−2q73−q72−q71 + 6q70−2q69 + 3q67−4q66−3q65−4q64 + 9q63−q62 + q61 + 7q60−4q59−7q58−11q57 + 10q56−2q55 + 2q54 + 15q53 + 2q52−6q51−17q50 + 7q49−13q48−5q47 + 20q46 + 12q45 + 7q44−11q43 + 13q42−29q41−25q40 + 8q39 + 12q38 + 21q37 + 10q36 + 39q35−32q34−44q33−21q32−8q31 + 19q30 + 29q29 + 82q28−15q27−50q26−50q25−40q24 + q23 + 36q22 + 124q21 + 9q20−45q19−68q18−65q17−19q16 + 34q15 + 151q14 + 25q13−38q12−76q11−76q10−31q9 + 31q8 + 163q7 + 29q6−34q5−78q4−78q3−34q2 + 29q + 167 + 29q−1−34q−2−78q−3−78q−4−34q−5 + 29q−6 + 163q−7 + 31q−8−31q−9−76q−10−76q−11−38q−12 + 25q−13 + 151q−14 + 34q−15−19q−16−65q−17−68q−18−45q−19 + 9q−20 + 124q−21 + 36q−22 + q−23−40q−24−50q−25−50q−26−15q−27 + 82q−28 + 29q−29 + 19q−30−8q−31−21q−32−44q−33−32q−34 + 39q−35 + 10q−36 + 21q−37 + 12q−38 + 8q−39−25q−40−29q−41 + 13q−42−11q−43 + 7q−44 + 12q−45 + 20q−46−5q−47−13q−48 + 7q−49−17q−50−6q−51 + 2q−52 + 15q−53 + 2q−54−2q−55 + 10q−56−11q−57−7q−58−4q−59 + 7q−60 + q−61−q−62 + 9q−63−4q−64−3q−65−4q−66 + 3q−67−2q−69 + 6q−70−q−71−q−72−2q−73 + q−74−2q−76 + 3q−77−q−80−q−83 + q−84