10 101

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["10 101"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X12,20,13,19 X18,8,19,7 X6,14,7,13 X8,18,9,17 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 18 2 16 6 20 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[21:2:2]

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(5,{1,1,1,2,−1,3,−2,1,3,2,2,4,−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, 14, 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[{3, 9}, {2, 5}, {1, 3}, {10, 7}, {8, 6}, {7, 4}, {9, 11}, {5, 10}, {12, 8}, {11, 2}, {4, 12}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_101.

A1 Invariants.

Weight	Invariant
1	q−3−2q−5 + 4q−7−3q−9 + 4q−11−q−15 + 2q−17−4q−19 + 3q−21−3q−23 + q−25
2	q−6−2q−8 + q−10 + 6q−12−9q−14 + 3q−16 + 17q−18−24q−20 + 35q−24−26q−26−15q−28 + 34q−30−7q−32−22q−34 + 11q−36 + 16q−38−16q−40−15q−42 + 27q−44−2q−46−33q−48 + 25q−50 + 16q−52−35q−54 + 9q−56 + 23q−58−18q−60−5q−62 + 12q−64−2q−66−3q−68 + q−70
3	q−9−2q−11 + q−13 + 3q−15−5q−19 + 3q−21 + 11q−23−11q−25−19q−27 + 30q−29 + 42q−31−44q−33−90q−35 + 59q−37 + 150q−39−41q−41−214q−43−7q−45 + 255q−47 + 78q−49−255q−51−147q−53 + 201q−55 + 200q−57−121q−59−219q−61 + 32q−63 + 201q−65 + 58q−67−176q−69−125q−71 + 130q−73 + 181q−75−95q−77−218q−79 + 44q−81 + 250q−83 + 12q−85−260q−87−79q−89 + 248q−91 + 146q−93−198q−95−201q−97 + 124q−99 + 226q−101−41q−103−202q−105−36q−107 + 151q−109 + 78q−111−86q−113−82q−115 + 29q−117 + 59q−119 + 4q−121−34q−123−9q−125 + 11q−127 + 7q−129−2q−131−3q−133 + q−135

A2 Invariants.

Weight	Invariant
1,0	q−6−2q−8 + 2q−10 + q−12−2q−14 + 4q−16 + 2q−20 + 2q−22−q−24 + 2q−26−4q−28−q−30−3q−34 + q−36 + q−38
2,0	q−12−2q−14−q−16 + 7q−18−q−20−11q−22 + 6q−24 + 16q−26−5q−28−18q−30 + 9q−32 + 23q−34−10q−36−16q−38 + 14q−40 + 13q−42−5q−44−5q−46 + 8q−48−2q−50−3q−52 + 4q−54−6q−56−14q−58 + 2q−60 + 11q−62−15q−64−15q−66 + 9q−68 + 15q−70−9q−72−14q−74 + 13q−76 + 14q−78−3q−80−8q−82 + 3q−84 + 8q−86−2q−88−5q−90−2q−92 + q−94 + q−96

A3 Invariants.

Weight	Invariant
0,1,0	q−12−2q−14 + 6q−18−7q−20−3q−22 + 18q−24−10q−26−11q−28 + 28q−30−8q−32−16q−34 + 31q−36−3q−38−11q−40 + 14q−42 + 2q−44−9q−46−13q−48 + 4q−50 + q−52−25q−54 + 7q−56 + 19q−58−23q−60 + 8q−62 + 20q−64−20q−66 + 6q−68 + 10q−70−12q−72 + 4q−74 + 2q−76−3q−78 + q−80
1,0,0	q−9−2q−11 + 2q−13−q−15 + 2q−17−2q−19 + 4q−21 + 2q−25 + 2q−27 + 2q−29 + 2q−31−q−33 + 2q−35−4q−37−q−39−4q−41−3q−45 + q−47 + q−49 + q−51

B2 Invariants.

Weight	Invariant
0,1	q−12−2q−14 + 4q−16−8q−18 + 13q−20−17q−22 + 24q−24−28q−26 + 33q−28−32q−30 + 28q−32−16q−34 + 5q−36 + 13q−38−27q−40 + 44q−42−56q−44 + 63q−46−65q−48 + 58q−50−49q−52 + 33q−54−17q−56−q−58 + 15q−60−26q−62 + 32q−64−34q−66 + 32q−68−28q−70 + 20q−72−14q−74 + 8q−76−3q−78 + q−80
1,0	q−18−2q−22−2q−24 + 2q−26 + 7q−28 + 2q−30−10q−32−10q−34 + 6q−36 + 21q−38 + 7q−40−20q−42−22q−44 + 11q−46 + 34q−48 + 7q−50−31q−52−18q−54 + 23q−56 + 31q−58−9q−60−28q−62 + q−64 + 29q−66 + 7q−68−24q−70−13q−72 + 16q−74 + 13q−76−17q−78−20q−80 + 9q−82 + 20q−84−9q−86−30q−88−4q−90 + 32q−92 + 18q−94−26q−96−31q−98 + 14q−100 + 37q−102 + 6q−104−29q−106−18q−108 + 18q−110 + 22q−112−5q−114−16q−116−4q−118 + 9q−120 + 5q−122−3q−124−3q−126 + q−130

G2 Invariants.

Weight

Invariant

1,0

q−30−2q−32 + 4q−34−6q−36 + 6q−38−5q−40 + 11q−44−21q−46 + 33q−48−38q−50 + 31q−52−14q−54−15q−56 + 52q−58−84q−60 + 107q−62−105q−64 + 68q−66 + 3q−68−87q−70 + 169q−72−206q−74 + 183q−76−94q−78−40q−80 + 166q−82−226q−84 + 203q−86−85q−88−58q−90 + 169q−92−188q−94 + 107q−96 + 45q−98−192q−100 + 265q−102−220q−104 + 73q−106 + 125q−108−289q−110 + 359q−112−307q−114 + 147q−116 + 50q−118−229q−120 + 322q−122−304q−124 + 183q−126−14q−128−146q−130 + 224q−132−204q−134 + 85q−136 + 65q−138−188q−140 + 214q−142−138q−144−16q−146 + 177q−148−270q−150 + 259q−152−149q−154−14q−156 + 158q−158−232q−160 + 224q−162−139q−164 + 32q−166 + 59q−168−106q−170 + 105q−172−71q−174 + 33q−176 + q−178−19q−180 + 20q−182−16q−184 + 8q−186−3q−188 + q−190

.

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["10 101"];

In[4]:=

Alexander[K][t]

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

Out[4]=

7t2−21t + 29−21t−1 + 7t−2

In[5]:=

Conway[K][z]

Out[5]=

7z4 + 7z2 + 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]=

{ 85, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q12−4q11 + 7q10−11q9 + 13q8−14q7 + 14q6−10q5 + 7q4−3q3 + q2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z4a−4 + 3z4a−6 + 3z4a−8 + z2a−4 + 5z2a−6 + 5z2a−8−4z2a−10 + 2a−6 + 2a−8−4a−10 + a−12

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z9a−9 + 2z9a−11 + 6z8a−8 + 11z8a−10 + 5z8a−12 + 7z7a−7 + 10z7a−9 + 7z7a−11 + 4z7a−13 + 6z6a−6−6z6a−8−24z6a−10−11z6a−12 + z6a−14 + 3z5a−5−8z5a−7−31z5a−9−31z5a−11−11z5a−13 + z4a−4−8z4a−6 + z4a−8 + 15z4a−10 + 3z4a−12−2z4a−14−2z3a−5 + 4z3a−7 + 26z3a−9 + 28z3a−11 + 8z3a−13−z2a−4 + 7z2a−6−z2a−8−9z2a−10 + z2a−12 + z2a−14−8za−9−9za−11−za−13−2a−6 + 2a−8 + 4a−10 + a−12

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["10 101"];

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]=

{ 7t2−21t + 29−21t−1 + 7t−2, q12−4q11 + 7q10−11q9 + 13q8−14q7 + 14q6−10q5 + 7q4−3q3 + q2 }

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]=

{K11a200,}

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	q34−4q33 + q32 + 15q31−21q30−12q29 + 56q28−35q27−56q26 + 107q25−26q24−114q23 + 138q22 + 3q21−156q20 + 137q19 + 35q18−161q17 + 104q16 + 50q15−120q14 + 55q13 + 39q12−59q11 + 20q10 + 15q9−18q8 + 6q7 + 3q6−3q5 + q4
3	q66−4q65 + q64 + 9q63 + 5q62−24q61−24q60 + 47q59 + 60q58−54q57−135q56 + 43q55 + 224q54 + 19q53−322q52−123q51 + 385q50 + 286q49−424q48−448q47 + 388q46 + 630q45−322q44−775q43 + 207q42 + 902q41−84q40−981q39−55q38 + 1025q37 + 192q36−1032q35−310q34 + 974q33 + 426q32−889q31−479q30 + 723q29 + 524q28−568q27−478q26 + 375q25 + 416q24−235q23−301q22 + 113q21 + 209q20−62q19−110q18 + 22q17 + 60q16−16q15−24q14 + 10q13 + 11q12−8q11−2q10 + 2q9 + 3q8−3q7 + q6