8 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.org/wiki/KnotTheory.

In[3]:=

K = Knot["8 1"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 16 14 12 2 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[62]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 8_1.

A1 Invariants.

Weight	Invariant
1	q13−q7 + q−1 + q−5
2	q38−q34−q28 + q24 + q12 + q10−1 + q−8 + q−14
3	q75−q71−q69 + q65−q61 + q57 + q55−q51 + q37 + q35−q31−q25−q23−q17 + q13 + 2q11 + q9 + q7−q5 + q + q−1−q−5 + q−9−q−13−q−15 + q−19 + q−27
4	q124−q120−q118−q116 + q114 + q112 + q110−2q106 + q102 + q100 + q98−q96−q94−q92 + q88 + q74 + q72−q68−2q66 + q62−q58−2q56 + 2q52 + q50−q46 + 2q42 + q40 + q32−q28−q26 + q22−q18−q16−q14−q12 + q10 + 2q8 + q6−q4−2q2 + 2q−2 + 4q−4 + q−6−2q−8−q−10 + q−12 + 3q−14−2q−18−q−20 + 2q−24−q−28−q−30−q−32 + q−34 + q−44
5	q185−q181−q179−q177 + q173 + 2q171 + q169−q165−2q163−q161 + q159 + 2q157 + q155−q151−2q149−q147 + q143 + q141 + q139−q135 + q123 + q121−q117−2q115−2q113 + 2q109 + 2q107−2q103−2q101−q99 + 2q97 + 4q95 + 2q93−q91−2q89−2q87 + 2q83 + 2q81−2q77−2q75 + q71 + 2q69 + q67−q65−2q63−q61 + q57 + q55−q53−2q51−q49 + q45−q41 + q37 + 3q35 + 2q33−q25 + q23 + q21 + q19 + q17−3q13−3q11−q9 + q7 + 3q5 + 2q3−3q−1−3q−3 + 3q−7 + 3q−9−2q−13−2q−15 + 3q−19 + 2q−21−2q−25 + 2q−29 + 2q−31 + q−33−q−35−2q−37−q−39 + q−41 + q−43−q−49−q−51 + q−65
6	q258−q254−q252−q250 + 2q244 + 2q242 + q240−q236−2q234−3q232 + q228 + 2q226 + 2q224 + q222−3q218−2q216−q214 + q210 + 2q208 + 2q206−q200−q198−q196 + q192 + q184 + q182−q178−2q176−2q174−2q172 + q170 + 3q168 + 3q166 + 2q164−q162−3q160−4q158−q156 + 2q154 + 4q152 + 5q150 + 2q148−2q146−5q144−4q142−2q140 + q138 + 4q136 + 3q134 + q132−2q130−3q128−3q126−q124 + 3q122 + 3q120 + 3q118 + q116−q114−3q112−4q110−q108 + q106 + 2q104 + 2q102 + q100−2q98−4q96−2q94 + q92 + 3q90 + 3q88 + 2q86−q84−4q82−q80 + 2q78 + 3q76 + 3q74 + 2q72−q70−3q68−q66 + q64 + q62−q58−2q56−2q54 + q50−q46−2q44−2q42−q40 + q38 + 2q36 + 3q34 + 2q32 + q30−q26−3q24−3q22 + 3q18 + 4q16 + 4q14 + 3q12−2q10−4q8−4q6−q4 + 2q2 + 5 + 6q−2 + q−4−2q−6−5q−8−4q−10−2q−12 + 2q−14 + 4q−16 + q−18−2q−22−q−24−q−26 + q−30−q−32 + q−34 + q−36 + 2q−38−q−42−q−44−2q−46 + q−48 + 2q−50 + 3q−52 + q−54−q−58−2q−60 + q−66−q−74−q−78 + q−90

A2 Invariants.

Weight	Invariant
1,0	q20 + q18−q12−q10 + q−2 + q−6 + q−8
1,1	q52 + 2q48−2q46 + 2q44−4q42 + 2q40−2q38−2q32 + 2q30−3q28 + 4q26−2q24 + 4q22−2q20 + 4q18 + 2q14 + 2q12−2q6−2q4−2q2−2 + q−4 + 2q−8 + 2q−12 + 2q−16 + q−20
2,0	q52 + q50 + q48−q46−q44−q42−q40−q38−q36 + q34 + q32 + q30 + q18 + 2q16 + q14 + q12 + q10−q4−2q2−2−q−2 + q−4 + q−10 + q−12 + q−16 + q−18 + q−20

A3 Invariants.

Weight	Invariant
0,1,0	q40 + q36−q32−q30−q28 + q24 + 2q22 + q20 + 2q18−q14−q12−q10−q8−q6 + q−4 + q−8 + 2q−10 + q−12 + q−16
1,0,0	q27 + q25 + q23−q17−q15−q13 + q−3 + q−7 + q−9 + q−11

B2 Invariants.

Weight	Invariant
0,1	q40 + q36 + q32−q30 + q28−q24−q20−2q16 + q14−q12 + q10−q8 + q6 + q−4 + q−8 + q−12 + q−16
1,0	q66 + q58−q54−q52−q46−q44 + q40 + q38 + q36 + q32 + q30 + q28−q18−q16−q10−q8 + q−6 + q−14 + q−16 + q−18 + q−26

G2 Invariants.

Weight	Invariant
1,0	q94 + q90−q88 + 2q80−2q78 + q76 + q74 + q70−q68 + q64 + q54−q52−q46−q42 + q40−q38 + q36−q34−2q32 + q30−q28−q22 + q18−q12 + q8 + q−2−q−6 + q−10 + q−14 + q−20 + q−24 + q−28 + q−34 + q−38

.

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−3t + 7−3t−1

In[5]:=

Conway[K][z]

Out[5]=

1−3z2

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]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q2−q + 2−2q−1 + 2q−2−2q−3 + q−4−q−5 + q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a6−z2a4−a4−z2a2−z2 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a5z7 + a3z7 + a6z6 + 2a4z6 + a2z6−5a5z5−4a3z5 + az5−5a6z4−8a4z4−2a2z4 + z4 + 7a5z3 + 5a3z3−az3 + z3a−1 + 6a6z2 + 7a4z2 + z2a−2−3a5z−3a3z−a6−a4−a−2

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 1"];

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]=

{ −3t + 7−3t−1, q2−q + 2−2q−1 + 2q−2−2q−3 + q−4−q−5 + q−6 }

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]=

{K11n70,}

n	Jn
2	q6−q5 + 2q3−2q2 + 2−3q−1 + q−2 + 2q−3−3q−4 + q−5 + 3q−6−3q−7 + 3q−9−3q−10 + 3q−12−2q−13−q−14 + 2q−15−q−16−q−17 + q−18
3	q12−q11 + 2q8−2q7−q6 + 3q4−q3−2q2−q + 4−2q−2−2q−3 + 3q−4 + 2q−5−2q−6−q−7 + 2q−8 + q−9−3q−10 + 2q−12−3q−14 + q−15 + 2q−16−q−17−2q−18 + 2q−19 + 2q−20−2q−21−2q−22 + 2q−23 + 2q−24−2q−25−2q−26 + q−27 + 3q−28−q−29−2q−30 + 2q−32−q−34−q−35 + q−36
4	q20−q19 + 2q15−3q14 + q11 + 4q10−5q9−q8−q7 + 3q6 + 7q5−7q4−3q3−2q2 + 6q + 10−9q−1−5q−2−4q−3 + 7q−4 + 12q−5−8q−6−6q−7−6q−8 + 7q−9 + 12q−10−8q−11−5q−12−5q−13 + 6q−14 + 11q−15−8q−16−4q−17−4q−18 + 5q−19 + 11q−20−8q−21−3q−22−3q−23 + 3q−24 + 10q−25−7q−26−2q−27−2q−28 + q−29 + 8q−30−6q−31−q−32−q−33 + 6q−35−5q−36 + 5q−40−5q−41 + 5q−45−4q−46−q−47−q−48 + 5q−50−2q−51−q−52−q−53−q−54 + 3q−55−q−58−q−59 + q−60
5	q30−q29 + q24−2q23 + q21 + q19 + q18−4q17−q16 + 2q15 + 2q14 + 2q13 + q12−6q11−3q10 + 4q9 + 4q8 + 3q7−2q6−8q5−3q4 + 6q3 + 7q2 + 3q−5−11q−1−3q−2 + 9q−3 + 9q−4 + 4q−5−7q−6−13q−7−5q−8 + 9q−9 + 12q−10 + 5q−11−7q−12−13q−13−5q−14 + 7q−15 + 13q−16 + 5q−17−7q−18−11q−19−4q−20 + 5q−21 + 12q−22 + 4q−23−6q−24−10q−25−5q−26 + 4q−27 + 11q−28 + 5q−29−4q−30−10q−31−6q−32 + 3q−33 + 10q−34 + 6q−35−2q−36−9q−37−7q−38 + 2q−39 + 8q−40 + 6q−41−7q−43−7q−44 + 6q−46 + 6q−47 + q−48−4q−49−5q−50−2q−51 + 3q−52 + 5q−53 + q−54−2q−55−3q−56−2q−57 + q−58 + 3q−59 + q−60−q−61−2q−62−q−63 + q−64 + 2q−65 + q−66−q−67−2q−68−q−69 + 3q−71 + 2q−72−q−73−2q−74−2q−75−q−76 + 2q−77 + 3q−78−q−80−q−81−2q−82 + 2q−84 + q−85−q−88−q−89 + q−90
6	q42−q41−q36 + 2q35−2q34 + q33 + q30−2q29 + 2q28−4q27 + 2q26 + q25 + q24 + 3q23−3q22 + q21−7q20 + 3q19 + q18 + 2q17 + 5q16−4q15 + q14−9q13 + 5q12 + q10 + 5q9−5q8 + 3q7−8q6 + 8q5−2q4−3q3 + 3q2−6q + 6−5q−1 + 13q−2−3q−3−6q−4−9q−6 + 6q−7−3q−8 + 18q−9−2q−10−6q−11−q−12−12q−13 + 3q−14−3q−15 + 20q−16−q−17−5q−18−11q−20 + 2q−21−4q−22 + 18q−23−2q−24−5q−25 + q−26−10q−27 + 3q−28−5q−29 + 16q−30−2q−31−4q−32 + 2q−33−9q−34 + 3q−35−7q−36 + 14q−37−q−39 + 3q−40−9q−41 + 2q−42−10q−43 + 11q−44 + 3q−45 + 2q−46 + 4q−47−9q−48−13q−50 + 9q−51 + 5q−52 + 5q−53 + 5q−54−9q−55−q−56−15q−57 + 6q−58 + 6q−59 + 7q−60 + 7q−61−7q−62−q−63−15q−64 + 2q−65 + 4q−66 + 7q−67 + 8q−68−4q−69 + q−70−14q−71−q−72 + q−73 + 5q−74 + 7q−75−q−76 + 5q−77−11q−78−2q−79−q−80 + 2q−81 + 4q−82 + 7q−84−8q−85−q−86−q−87 + q−89 + 7q−91−7q−92 + 7q−98−6q−99−q−100−q−101 + q−104 + 7q−105−4q−106−q−107−2q−108−q−109−q−110 + 6q−112−q−113−q−115−q−116−2q−117−q−118 + 3q−119 + q−121−q−124−q−125 + q−126