10 4

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 4"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 12 14 20 18 16 4 2 10 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[613]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_4.

A1 Invariants.

Weight	Invariant
1	q11−q9 + q7 + q3−q−1−q−5 + q−7 + q−11
2	q30−q28 + 2q24−q22−q16 + q6−q4 + 1 + q−6 + 2q−8 + q−14−q−16−2q−18 + q−20−2q−24 + q−26 + q−28−q−30 + q−34
3	q57−q55 + q51 + q49−q47−2q45 + q43 + q41−q39−2q37 + q35 + 3q33−q31−3q29−q27 + 4q25 + q23−3q21−q19 + 2q17 + 3q15−q11−2q9 + q7 + 3q5 + 2q3−3q−q−1 + 2q−3 + q−5−2q−7−q−9 + q−11−q−15−q−17 + q−19−q−25 + q−29 + 2q−31 + q−33−q−37 + q−39 + 2q−41−3q−45−2q−47 + 2q−49 + 2q−51−q−53−3q−55 + 2q−59 + q−61−q−63−q−65 + q−69
4	q92−q90 + q86 + q82−3q80 + q76 + q72−4q70 + 2q68 + 3q66−3q62−6q60 + 4q58 + 6q56 + 2q54−5q52−7q50 + 4q48 + 8q46 + 3q44−5q42−8q40 + 2q38 + 7q36 + 5q34−2q32−9q30−3q28 + 2q26 + 6q24 + 5q22−2q20−6q18−5q16 + q14 + 7q12 + 4q10−3q8−6q6−4q4 + 4q2 + 6−3q−4−2q−6 + 3q−8 + 4q−10−2q−12−3q−14−q−16 + 5q−18 + 6q−20−2q−22−5q−24−3q−26 + 4q−28 + 6q−30−4q−34−5q−36−q−38 + 5q−40 + 2q−42−3q−46−4q−48 + q−50 + q−52 + 2q−54 + q−56−3q−58−q−60−2q−62 + 4q−66 + 2q−68 + 3q−70−2q−72−4q−74−q−76 + q−78 + 6q−80 + 2q−82−3q−84−4q−86−4q−88 + 3q−90 + 4q−92 + 2q−94−q−96−5q−98−q−100 + q−102 + 2q−104 + 2q−106−q−108−q−110−q−112 + q−116
5	q135−q133 + q129−q123−q121 + q117−q113 + q109 + 2q107−q105−4q103−4q101 + 2q99 + 7q97 + 5q95−q93−8q91−6q89 + 2q87 + 10q85 + 6q83−3q81−8q79−4q77 + 3q75 + 7q73 + 2q71−6q69−8q67 + q65 + 10q63 + 8q61−11q57−13q55−2q53 + 11q51 + 13q49 + 4q47−8q45−12q43−9q41 + q39 + 10q37 + 11q35 + 6q33−q31−10q29−13q27−5q25 + 7q23 + 14q21 + 11q19−q17−12q15−12q13−2q11 + 8q9 + 11q7 + 3q5−3q3−4q−q−1 + 4q−3 + 4q−5−2q−7−5q−9−4q−11 + 3q−13 + 7q−15 + 6q−17−2q−19−11q−21−10q−23−q−25 + 9q−27 + 12q−29 + 5q−31−9q−33−15q−35−9q−37 + 5q−39 + 15q−41 + 14q−43−14q−47−15q−49−2q−51 + 11q−53 + 17q−55 + 8q−57−6q−59−15q−61−11q−63 + q−65 + 11q−67 + 12q−69 + 5q−71−5q−73−10q−75−8q−77 + 6q−81 + 6q−83 + 4q−85−5q−89−5q−91−2q−93−q−95 + q−97 + 3q−99 + 3q−101 + q−103−3q−107−5q−109−4q−111 + q−113 + 5q−115 + 7q−117 + 5q−119−q−121−7q−123−8q−125−3q−127 + 4q−129 + 8q−131 + 7q−133 + q−135−5q−137−8q−139−5q−141 + q−143 + 5q−145 + 6q−147 + 3q−149−2q−151−5q−153−3q−155−q−157 + q−159 + 3q−161 + 2q−163−q−167−q−169−q−171 + q−175

A2 Invariants.

Weight	Invariant
1,0	q16 + q10 + q6−q−2−q−4−q−6−q−8 + q−10 + q−12 + q−14 + q−16
1,1	q44−2q42 + 2q40−2q38 + 5q36−4q34 + 4q32−4q30 + 5q28−4q26 + 4q24−2q22 + 4q20−4q18−q12−4q8−4q4 + 2q2−4 + 10q−2−7q−4 + 20q−6−10q−8 + 18q−10−14q−12 + 12q−14−12q−16 + 2q−18−5q−20−2q−22 + 6q−24−10q−26 + 11q−28−12q−30 + 12q−32−10q−34 + 6q−36−4q−38 + 4q−40 + q−44
2,0	q40 + q34 + q32 + q26−2q22−2q16−q14 + q12−q8−q6 + 1 + 2q−2 + q−4 + q−6 + 2q−8 + 2q−10 + q−12 + 2q−14 + q−16 + q−18−q−20−2q−22−2q−24−2q−26−2q−28−2q−30 + q−34 + q−36 + q−40 + q−42 + q−44

A3 Invariants.

Weight	Invariant
0,1,0	q36−q34−q32 + 2q30−q26 + 2q24 + q22 + q18 + q16−q14−q6 + q2−1−q−2 + 2q−10 + q−14−2q−18−q−20 + q−26 + 2q−28 + q−32
1,0,0	q21 + q17 + q13 + q9−q−3−q−5−2q−7−q−9−q−11 + q−13 + q−15 + 2q−17 + q−19 + q−21

B2 Invariants.

Weight	Invariant
0,1	q36−q34 + q32−2q30 + 2q28−q26 + 2q24−q22 + 2q20−q18 + q16 + q14 + 2q10−2q8 + 3q6−4q4 + 3q2−5 + 3q−2−4q−4 + 2q−6−2q−8−q−14 + 2q−16−2q−18 + 3q−20−2q−22 + 2q−24−q−26 + 2q−28 + q−32
1,0	q58−q54−q52 + 2q48 + q46−q44−q42 + 2q38 + q36−q32 + q28 + q26−q24−q22 + 2q18−q14−q12 + q8−q4−q2 + 1−q−2−q−4 + 2q−8 + q−10−q−12−q−14 + 2q−16 + 3q−18 + q−20−2q−22−q−24 + q−26 + q−28−q−30−3q−32−q−34 + q−36 + q−38−q−40−q−42 + q−44 + 2q−46 + q−54

G2 Invariants.

Weight	Invariant
1,0	q86−q84 + q82−q80−q74 + 3q72−2q70 + 2q68−q66 + q62−2q60 + 3q58−2q56 + 2q48−q46 + 2q44−q42 + q40 + 2q38−q36 + q34 + q32 + q30 + q26−q24 + q22−q20−q14−q10 + 2q4−4q2 + 2−3q−4 + 4q−6−5q−8 + 2q−10−q−12−q−14 + 2q−16−3q−18 + 2q−20−2q−22−q−26−q−28 + 2q−36−2q−38 + q−40 + q−42−2q−44 + 5q−46−5q−48 + 3q−50 + q−52−2q−54 + 5q−56−4q−58 + 3q−60 + q−66−2q−68 + 2q−70 + q−74

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−3t2 + 7t−7 + 7t−1−3t−2

In[5]:=

Conway[K][z]

Out[5]=

−3z4−5z2 + 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]=

{ 27, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q5−q4 + 2q3−3q2 + 3q−4 + 4q−1−3q−2 + 3q−3−2q−4 + q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a4 + a4−z4a2−2z2a2−z4−2z2−z4a−2−3z2a−2−2a−2 + z2a−4 + 2a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z9a−1 + z9a−3 + 3z8a−2 + z8a−4 + 2z8 + 3az7−3z7a−1−6z7a−3 + 3a2z6−17z6a−2−7z6a−4−7z6 + 3a3z5−10az5−2z5a−1 + 11z5a−3 + 3a4z4−6a2z4 + 29z4a−2 + 16z4a−4 + 4z4 + 2a5z3−4a3z3 + 8az3 + 7z3a−1−7z3a−3 + a6z2−3a4z2−16z2a−2−13z2a−4 + z2−3az−za−1 + 2za−3 + a4 + 2a−2 + 2a−4

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 4"];

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]=

{ −3t2 + 7t−7 + 7t−1−3t−2, q5−q4 + 2q3−3q2 + 3q−4 + 4q−1−3q−2 + 3q−3−2q−4 + q−5 }

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	q16−q15−q14 + 3q13−q12−4q11 + 5q10−7q8 + 6q7 + 2q6−8q5 + 6q4 + 4q3−9q2 + 5q + 4−8q−1 + 4q−2 + 3q−3−6q−4 + 3q−5 + 3q−6−6q−7 + 3q−8 + 2q−9−5q−10 + 3q−11 + q−12−2q−13 + q−14
3	q33−q32−q31 + 3q29−3q27−3q26 + 5q25 + 3q24−3q23−7q22 + 4q21 + 6q20−q19−8q18 + 2q17 + 7q16−7q14 + q13 + 6q12−q11−6q10 + q9 + 7q8−3q7−6q6 + 2q5 + 8q4−5q3−7q2 + 5q + 9−8q−1−9q−2 + 10q−3 + 10q−4−10q−5−12q−6 + 11q−7 + 11q−8−7q−9−13q−10 + 8q−11 + 9q−12−3q−13−10q−14 + 3q−15 + 7q−16−q−17−6q−18 + q−19 + 4q−20−4q−22 + q−23 + q−24 + q−25−2q−26 + q−27
4	q56−q55−q54 + 4q51−q50−2q49−2q48−4q47 + 8q46 + 2q45−3q43−11q42 + 8q41 + 3q40 + 5q39 + q38−16q37 + 6q36 + 7q34 + 6q33−17q32 + 8q31−4q30 + 5q29 + 7q28−19q27 + 12q26−3q25 + 4q24 + 7q23−24q22 + 13q21 + 6q19 + 10q18−30q17 + 9q16 + q15 + 10q14 + 16q13−32q12 + 2q11−q10 + 13q9 + 24q8−33q7−4q6−3q5 + 14q4 + 30q3−34q2−9q−4 + 17q−1 + 36q−2−36q−3−17q−4−6q−5 + 20q−6 + 43q−7−33q−8−23q−9−12q−10 + 19q−11 + 47q−12−26q−13−22q−14−16q−15 + 14q−16 + 41q−17−19q−18−15q−19−14q−20 + 9q−21 + 31q−22−16q−23−7q−24−9q−25 + 5q−26 + 20q−27−14q−28−5q−30 + 3q−31 + 10q−32−11q−33 + 3q−34−2q−35 + 2q−36 + 4q−37−6q−38 + 2q−39−q−40 + q−41 + q−42−2q−43 + q−44