5 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["5 1"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3849 X5,10,6,1 X7283 X9,4,10,5

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -2, 5, -3, 1, -4, 2, -5, 3

In[6]:=

DTCode[K]

Out[6]=

6 8 10 2 4

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[5]

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(2,{−1,−1,−1,−1,−1})

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]=

{ 2, 5, 2 }

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[{7, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 5_1.

A1 Invariants.

Weight	Invariant
1	−q15 + q7 + q5 + q3
2	q40−q32−q30−q28 + q14 + q12 + q10 + q8 + q6
3	−q75 + q67 + q65 + q63−q49−q47−q45−q43−q41 + q21 + q19 + q17 + q15 + q13 + q11 + q9
4	q120−q112−q110−q108 + q94 + q92 + q90 + q88 + q86−q66−q64−q62−q60−q58−q56−q54 + q28 + q26 + q24 + q22 + q20 + q18 + q16 + q14 + q12
5	−q175 + q167 + q165 + q163−q149−q147−q145−q143−q141 + q121 + q119 + q117 + q115 + q113 + q111 + q109−q83−q81−q79−q77−q75−q73−q71−q69−q67 + q35 + q33 + q31 + q29 + q27 + q25 + q23 + q21 + q19 + q17 + q15
6	q240−q232−q230−q228 + q214 + q212 + q210 + q208 + q206−q186−q184−q182−q180−q178−q176−q174 + q148 + q146 + q144 + q142 + q140 + q138 + q136 + q134 + q132−q100−q98−q96−q94−q92−q90−q88−q86−q84−q82−q80 + q42 + q40 + q38 + q36 + q34 + q32 + q30 + q28 + q26 + q24 + q22 + q20 + q18
8	q400−q392−q390−q388 + q374 + q372 + q370 + q368 + q366−q346−q344−q342−q340−q338−q336−q334 + q308 + q306 + q304 + q302 + q300 + q298 + q296 + q294 + q292−q260−q258−q256−q254−q252−q250−q248−q246−q244−q242−q240 + q202 + q200 + q198 + q196 + q194 + q192 + q190 + q188 + q186 + q184 + q182 + q180 + q178−q134−q132−q130−q128−q126−q124−q122−q120−q118−q116−q114−q112−q110−q108−q106 + q56 + q54 + q52 + q50 + q48 + q46 + q44 + q42 + q40 + q38 + q36 + q34 + q32 + q30 + q28 + q26 + q24

A2 Invariants.

Weight	Invariant
1,0	−q22−q20−q18 + q14 + q12 + 2q10 + q8 + q6
1,1	q60−2q36−2q34−4q32−4q30−3q28 + 2q24 + 4q22 + 5q20 + 4q18 + 4q16 + 2q14 + q12
2,0	q54 + q52 + 2q50 + q48−2q44−3q42−3q40−3q38−2q36−q34 + q28 + q26 + 2q24 + 2q22 + 3q20 + 2q18 + 2q16 + q14 + q12
3,0	−q96−q94−2q92−2q90−q88 + q86 + 3q84 + 4q82 + 5q80 + 4q78 + 4q76 + 2q74 + q72−q70−2q68−3q66−4q64−5q62−5q60−5q58−4q56−3q54−2q52−q50 + q42 + q40 + 2q38 + 2q36 + 3q34 + 3q32 + 4q30 + 3q28 + 3q26 + 2q24 + 2q22 + q20 + q18

A3 Invariants.

Weight	Invariant
0,1,0	q50−q36−2q34−3q32−3q30−2q28−q26 + 2q24 + 3q22 + 4q20 + 3q18 + 3q16 + q14 + q12
1,0,0	−q29−q27−2q25−q23 + q19 + 2q17 + 2q15 + 2q13 + q11 + q9
1,0,1	q80 + q58 + q56 + 3q54 + 3q52 + 2q50−q48−5q46−9q44−12q42−12q40−9q38−3q36 + 2q34 + 7q32 + 10q30 + 11q28 + 10q26 + 7q24 + 5q22 + 2q20 + q18

A4 Invariants.

Weight	Invariant
0,1,0,0	q64 + q62 + q60 + q58 + q56−q50−2q48−4q46−5q44−6q42−6q40−4q38−q36 + 2q34 + 4q32 + 7q30 + 6q28 + 6q26 + 4q24 + 3q22 + q20 + q18
1,0,0,0	−q36−q34−2q32−2q30−q28 + q24 + 2q22 + 3q20 + 2q18 + 2q16 + q14 + q12

B2 Invariants.

Weight	Invariant
0,1	−q50−q36−q32−q30 + q26 + q22 + 2q20 + q18 + q16 + q14 + q12
1,0	q80−q58−q56−q54−q52−2q50−q48−q46−q44 + q38 + q36 + 2q34 + q32 + 2q30 + q28 + 2q26 + q24 + q22 + q18

B3 Invariants.

Weight	Invariant
1,0,0	q120−q86−q82−q80−2q78−q76−2q74−q72−2q70−q68−q66−q64 + q58 + q56 + 2q54 + q52 + 3q50 + q48 + 3q46 + q44 + 2q42 + q40 + 2q38 + q34 + q30

B4 Invariants.

Weight	Invariant
1,0,0,0	q160−q114−q110−2q106−q104−2q102−q100−2q98−q96−2q94−q92−2q90−q88−q86 + q78 + 2q74 + q72 + 3q70 + q68 + 3q66 + q64 + 3q62 + q60 + 3q58 + q56 + 2q54 + 2q50 + q46 + q42

C3 Invariants.

Weight	Invariant
1,0,0	−q70−q50−q48−q46−q44−q42−q40 + q34 + q32 + 2q30 + 2q28 + 2q26 + q24 + 2q22 + q20 + q18

C4 Invariants.

Weight	Invariant
1,0,0,0	−q90−q64−q62−2q60−q58−q56−q54−q52−q50−q48 + q44 + 2q42 + 2q40 + 2q38 + 2q36 + 2q34 + 2q32 + 2q30 + 2q28 + q26 + q24

D4 Invariants.

Weight	Invariant
0,1,0,0	q120−q100−q98−3q96−3q94−q92−q90 + 2q88 + 6q86 + 9q84 + 11q82 + 14q80 + 11q78 + 9q76 + 3q74−4q72−12q70−18q68−24q66−27q64−27q62−24q60−17q58−11q56 + 7q52 + 14q50 + 19q48 + 22q46 + 19q44 + 19q42 + 14q40 + 10q38 + 6q36 + 4q34 + q32 + q30
1,0,0,0	q70−q50−q48−3q46−3q44−3q42−3q40−2q38 + q34 + 3q32 + 4q30 + 4q28 + 4q26 + 3q24 + 2q22 + q20 + q18

G2 Invariants.

Weight	Invariant
0,1	q240−q198−q192 + q178 + q176 + q172 + 2q170 + q168 + q166 + q164 + q162 + 2q160 + q158 + q154 + q152−q148−q146−q144−q142−2q140−3q138−2q136−2q134−3q132−4q130−4q128−3q126−3q124−4q122−4q120−3q118−2q116−2q114−3q112−2q110 + q102 + q100 + 2q98 + 3q96 + 2q94 + 2q92 + 4q90 + 3q88 + 3q86 + 4q84 + 3q82 + 3q80 + 4q78 + 2q76 + 2q74 + 3q72 + 2q70 + q68 + 2q66 + q64 + q62 + q60 + q54
1,0	q120−q100−q98−q92−q90−q88−q82−q80−q78−q72 + q58 + q56 + q52 + 2q50 + q48 + q46 + q44 + q42 + 2q40 + q38 + q34 + q32 + 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["5 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−t + 1−t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

z4 + 3z2 + 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]=

{ 5, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−2 + q−4−q−5 + q−6−q−7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a6−2a6 + z4a4 + 4z2a4 + 3a4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

za9 + z2a8 + z3a7−za7 + z4a6−3z2a6 + 2a6 + z3a5−2za5 + z4a4−4z2a4 + 3a4

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["5 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]=

{ t2−t + 1−t−1 + t−2, q−2 + q−4−q−5 + q−6−q−7 }

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_132,}

In[6]:=

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

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

Out[6]=

{10_132,}

n	Jn
2	q−4 + q−7−q−9 + q−10−q−12 + q−13−2q−15 + q−16−q−18 + q−19
3	q−6 + q−10−q−13 + q−14−q−17 + q−18−q−21−q−25 + q−27−q−29 + q−31 + q−35−q−36
4	q−8 + q−13−q−17 + q−18−q−22 + q−23−q−27 + q−28−q−29−q−32 + q−33−q−34 + q−36−q−37 + q−38−q−39 + q−41−q−42 + q−43−q−44 + q−45 + q−46−q−47 + q−48−q−49 + q−51−q−52 + q−53−q−54−q−57 + q−58
5	q−10 + q−16−q−21 + q−22−q−27 + q−28−q−33 + q−34−q−36−q−39 + q−40−q−42 + q−46−q−48 + q−52−q−54 + q−57 + q−58−q−60 + q−63−q−66 + q−69−q−72−q−73 + q−75−q−79 + q−81 + q−84−q−85
6	q−12 + q−19−q−25 + q−26−q−32 + q−33−q−39 + q−40−q−43−q−46 + q−47−q−50−q−53 + 2q−54−q−57−q−60 + 2q−61−q−64−q−67 + 2q−68 + q−69−q−71−q−74 + 2q−75 + q−76−2q−78−q−81 + 2q−82 + q−83−2q−85−q−88 + 2q−89−2q−92−q−95 + 2q−96 + q−97−2q−99−q−102 + 2q−103 + q−104−q−106−q−109 + 2q−110−q−113−q−116 + q−117
7	q−14 + q−22−q−29 + q−30−q−37 + q−38−q−45 + q−46−q−50−q−53 + q−54−q−58−q−61 + q−62 + q−63−q−66−q−69 + q−70 + q−71−q−74−q−77 + q−78 + q−79 + q−81−q−82−q−85 + q−86 + q−87 + q−89−q−90−q−92−q−93 + q−94 + q−95 + q−97−q−98−q−100−q−101 + q−102 + q−103 + q−105−q−106−q−107−q−108−q−109 + q−110 + q−111 + q−113−q−114−q−115−q−117 + q−118 + q−119 + q−121−q−122−q−123−q−125 + q−126 + q−127 + q−128 + q−129−q−130−q−131−q−133 + q−134 + q−136 + q−137−q−138−q−139−q−141 + q−142 + q−145−q−146−q−147 + q−150 + q−153−q−154