7 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["7 1"];

In[4]:=

PD[K]

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

Out[4]=

X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 10 12 14 2 4 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[7]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 7_1.

A1 Invariants.

Weight	Invariant
1	−q21 + q9 + q7 + q5
2	q56−q44−q42−q40 + q18 + q16 + q14 + q12 + q10
3	−q105 + q93 + q91 + q89−q67−q65−q63−q61−q59 + q27 + q25 + q23 + q21 + q19 + q17 + q15
4	q168−q156−q154−q152 + q130 + q128 + q126 + q124 + q122−q90−q88−q86−q84−q82−q80−q78 + q36 + q34 + q32 + q30 + q28 + q26 + q24 + q22 + q20
5	−q245 + q233 + q231 + q229−q207−q205−q203−q201−q199 + q167 + q165 + q163 + q161 + q159 + q157 + q155−q113−q111−q109−q107−q105−q103−q101−q99−q97 + q45 + q43 + q41 + q39 + q37 + q35 + q33 + q31 + q29 + q27 + q25
6	q336−q324−q322−q320 + q298 + q296 + q294 + q292 + q290−q258−q256−q254−q252−q250−q248−q246 + q204 + q202 + q200 + q198 + q196 + q194 + q192 + q190 + q188−q136−q134−q132−q130−q128−q126−q124−q122−q120−q118−q116 + q54 + q52 + q50 + q48 + q46 + q44 + q42 + q40 + q38 + q36 + q34 + q32 + q30
8	q560−q548−q546−q544 + q522 + q520 + q518 + q516 + q514−q482−q480−q478−q476−q474−q472−q470 + q428 + q426 + q424 + q422 + q420 + q418 + q416 + q414 + q412−q360−q358−q356−q354−q352−q350−q348−q346−q344−q342−q340 + q278 + q276 + q274 + q272 + q270 + q268 + q266 + q264 + q262 + q260 + q258 + q256 + q254−q182−q180−q178−q176−q174−q172−q170−q168−q166−q164−q162−q160−q158−q156−q154 + q72 + q70 + q68 + q66 + q64 + q62 + q60 + q58 + q56 + q54 + q52 + q50 + q48 + q46 + q44 + q42 + q40

A2 Invariants.

Weight	Invariant
1,0	−q30−q28−q26 + q18 + q16 + 2q14 + q12 + q10
1,1	q84−2q48−2q46−4q44−4q42−4q40−2q38−q36 + 2q34 + 4q32 + 4q30 + 5q28 + 4q26 + 4q24 + 2q22 + q20
2,0	q74 + q72 + 2q70 + q68 + q66−q62−2q60−3q58−3q56−3q54−2q52−q50 + q36 + q34 + 2q32 + 2q30 + 3q28 + 2q26 + 2q24 + q22 + q20
3,0	−q132−q130−2q128−2q126−2q124−q122 + 2q118 + 4q116 + 4q114 + 5q112 + 4q110 + 4q108 + 2q106 + q104−q94−2q92−3q90−4q88−5q86−5q84−5q82−4q80−3q78−2q76−q74 + q54 + q52 + 2q50 + 2q48 + 3q46 + 3q44 + 4q42 + 3q40 + 3q38 + 2q36 + 2q34 + q32 + q30

A3 Invariants.

Weight	Invariant
0,1,0	q70−q48−2q46−3q44−3q42−3q40−2q38 + q34 + 3q32 + 3q30 + 4q28 + 3q26 + 3q24 + q22 + q20
1,0,0	−q39−q37−2q35−q33−q31 + q27 + q25 + 2q23 + 2q21 + 2q19 + q17 + q15
1,0,1	q112 + q78 + q76 + 3q74 + 3q72 + 4q70 + 3q68 + q66−3q64−7q62−10q60−14q58−14q56−13q54−8q52−3q50 + 3q48 + 8q46 + 11q44 + 12q42 + 11q40 + 10q38 + 7q36 + 5q34 + 2q32 + q30

A4 Invariants.

Weight	Invariant
0,1,0,0	q88 + q86 + q84 + q82 + q80−q66−2q64−4q62−5q60−7q58−7q56−6q54−4q52−q50 + 2q48 + 5q46 + 6q44 + 8q42 + 6q40 + 6q38 + 4q36 + 3q34 + q32 + q30
1,0,0,0	−q48−q46−2q44−2q42−2q40−q38 + q34 + 2q32 + 2q30 + 3q28 + 2q26 + 2q24 + q22 + q20

B2 Invariants.

Weight	Invariant
0,1	−q70−q48−q44−q42−q40 + q34 + q32 + q30 + 2q28 + q26 + q24 + q22 + q20
1,0	q112−q78−q76−q74−q72−2q70−q68−q66−q64−q62 + q54 + q50 + q48 + 2q46 + q44 + 2q42 + q40 + 2q38 + q36 + q34 + q30

D4 Invariants.

Weight	Invariant
0,1,0,0	q168−q136−q134−3q132−3q130−4q128−4q126−q124 + 3q120 + 8q118 + 11q116 + 12q114 + 15q112 + 12q110 + 12q108 + 9q106 + 6q104 + 2q102−6q98−10q96−16q94−22q92−27q90−32q88−33q86−32q84−27q82−19q80−8q78 + 12q74 + 19q72 + 24q70 + 26q68 + 27q66 + 22q64 + 20q62 + 14q60 + 10q58 + 6q56 + 4q54 + q52 + q50
1,0,0,0	q98−q66−q64−3q62−3q60−4q58−4q56−3q54−2q52−q50 + 2q48 + 3q46 + 4q44 + 5q42 + 4q40 + 4q38 + 3q36 + 2q34 + q32 + q30

G2 Invariants.

Weight	Invariant
1,0	q168−q136−q134−q128−q126−q124−q118−q116−q102−q96−q94−q92 + q88−2q84 + 2q80 + q78 + q72 + 3q70 + 2q68 + q64 + 2q62 + 2q60 + q58 + q54 + q52 + q50

.

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["7 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−t2 + t−1 + t−1−t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + 5z4 + 6z2 + 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]=

{ 7, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−3 + q−5−q−6 + q−7−q−8 + q−9−q−10

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4a8−4z2a8−3a8 + z6a6 + 6z4a6 + 10z2a6 + 4a6

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

za13 + z2a12 + z3a11−za11 + z4a10−2z2a10 + z5a9−3z3a9 + za9 + z6a8−5z4a8 + 7z2a8−3a8 + z5a7−4z3a7 + 3za7 + z6a6−6z4a6 + 10z2a6−4a6

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["7 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]=

{ t3−t2 + t−1 + t−1−t−2 + t−3, q−3 + q−5−q−6 + q−7−q−8 + q−9−q−10 }

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]=

{}

n	Jn
2	q−6 + q−9−q−11 + q−12−q−14 + q−15−q−17 + q−18−q−20−q−23 + q−24−q−26 + q−27
3	q−9 + q−13−q−16 + q−17−q−20 + q−21−q−24 + q−25−q−28 + q−29−q−31−q−32 + q−33−q−35 + q−37−q−39 + q−41−q−43 + q−45 + q−46−q−47 + q−50−q−51
4	q−12 + q−17−q−21 + q−22−q−26 + q−27−q−31 + q−32−q−36 + q−37−2q−41 + q−42−2q−46 + q−47 + q−48−2q−51 + q−52 + q−53−2q−56 + q−57 + q−58−2q−61 + q−62 + 2q−63−2q−66 + q−67 + q−68−2q−71 + q−72 + q−73−2q−76 + q−77−q−81 + q−82
5	q−15 + q−21−q−26 + q−27−q−32 + q−33−q−38 + q−39−q−44 + q−45−q−50−q−56 + q−60−q−62 + q−66−q−68 + q−72−q−74 + q−78 + q−84−q−87 + q−90−q−93 + q−96−q−99−q−105 + q−107−q−111 + q−113 + q−119−q−120
6	q−18 + q−25−q−31 + q−32−q−38 + q−39−q−45 + q−46−q−52 + q−53−q−59 + q−60−q−61−q−66 + q−67−q−68 + q−72−q−73 + q−74−q−75 + q−79−q−80 + q−81−q−82 + q−86−q−87 + q−88−q−89 + q−93−q−94 + q−95−q−96 + q−97 + q−100−q−101 + q−102−q−103 + q−104−q−106 + q−107−q−108 + q−109−q−110 + q−111−q−113 + q−114−q−115 + q−116−q−117 + q−118−q−120 + q−121−q−122 + q−123−q−124 + q−125−q−126−q−127 + q−128−q−129 + q−130−q−131 + q−132−q−134 + q−135−q−136 + q−137−q−138 + q−139−q−141 + q−142−q−143 + q−144−q−145 + q−146 + q−149−q−150 + q−151−q−152 + q−156−q−157 + q−158−q−159−q−164 + q−165
7	q−21 + q−29−q−36 + q−37−q−44 + q−45−q−52 + q−53−q−60 + q−61−q−68 + q−69−q−71−q−76 + q−77−q−79 + q−85−q−87 + q−93−q−95 + q−101−q−103 + q−109−q−111 + q−114 + q−117−q−119 + q−122−q−127 + q−130−q−135 + q−138−q−143 + q−146−q−150−q−151 + q−154−q−158 + q−162−q−166 + q−170−q−174 + q−178 + q−179−q−182 + q−187−q−190 + q−195−q−198−q−201 + q−203−q−209 + q−211 + q−216−q−217