9 42

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["9 42"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 -14 2 -16 -18 -6 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,3,2-]

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(4,{1,1,1,−2,−1,−1,3,−2,3})

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]=

{ 4, 9, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	q7 + q−7
2	q22−q18 + q6 + 1 + q−6−q−18 + q−22
3	q45−q41−q39 + q35 + q23 + q21−q19−q17 + q13−q9 + q5 + q3 + q−3 + q−5−q−13−q−15 + q−21 + 2q−23−q−27 + 2q−31−3q−35−q−37 + q−39 + q−41
4	q76−q72−q70−q68 + q66 + q64 + q62−q58 + q48 + q46−2q42−2q40 + q36 + 2q34−q30−q28 + 2q24 + q22 + q20−q18−2q16 + q12 + 2q10−2q6−q4 + 2 + q−2−q−4 + q−8 + 2q−10 + q−12−q−14 + q−20−q−24−q−34−2q−36 + q−40 + 2q−42 + 2q−44−q−46−q−48−2q−50 + q−52 + 3q−54−2q−60−q−62 + q−68
5	q115−q111−q109−q107 + q103 + 2q101 + q99−q95−q93−q91 + q87 + q81 + q79−q75−3q73−2q71 + 2q67 + 3q65 + 2q63−2q59−2q57−q55 + q53 + 2q51 + 2q49 + q47−q45−2q43−3q41−2q39 + q37 + 3q35 + 3q33 + q31−2q29−4q27−2q25 + q23 + 3q21 + 4q19 + q17−2q15−3q13−q11 + 2q9 + 3q7 + 2q5−q3−3q−2q−1 + q−3 + 3q−5 + 2q−7−2q−11−q−13 + q−15 + 2q−17 + q−19−q−21−q−23 + q−27−q−31−q−33 + q−37 + q−39 + q−55 + 2q−57 + q−59−2q−61−4q−63−4q−65 + 5q−69 + 6q−71 + q−73−5q−75−6q−77−3q−79 + 3q−81 + 6q−83 + 5q−85−3q−89−3q−91−2q−93 + q−97 + q−99

A2 Invariants.

Weight	Invariant
1,0	q10 + q8 + q6−q2−1−q−2 + q−6 + q−8 + q−10
1,1	q28 + 2q24 + 2q20−2q18−2q16−2q14−2q12 + 2q6 + 4q4 + 4q2 + 2 + 2q−2−2q−4−4q−8−2q−12 + 2q−14 + 2q−18 + 2q−24−2q−26 + q−28−2q−30 + 2q−32
2,0	q28 + q26 + q24−q18−q16−2q14−2q12−q10 + q8 + 3q6 + 2q4 + 3q2 + 2 + q−2−q−4−q−6−q−8−q−10 + q−16 + q−20−q−22−q−24 + q−28 + q−30

A3 Invariants.

Weight	Invariant
0,1,0	q20 + q16 + q14 + q12−q4−q−4 + q−12 + q−14 + q−16 + q−20
1,0,0	q13 + q11 + 2q9 + q7−q3−2q−2q−1−q−3 + q−7 + 2q−9 + q−11 + q−13

A4 Invariants.

Weight	Invariant
0,1,0,0	q26 + q24 + 2q22 + 2q20 + 2q18 + q16−2q12−3q10−3q8−2q6−q4 + q2 + 4 + 4q−2 + 3q−4 + 2q−6−3q−10−3q−12−3q−14−2q−16 + 2q−20 + 3q−22 + 2q−24 + 2q−26 + q−28−q−32
1,0,0,0	q16 + q14 + 2q12 + 2q10 + q8−q4−2q2−3−2q−2−q−4 + q−8 + 2q−10 + 2q−12 + q−14 + q−16

B2 Invariants.

Weight	Invariant
0,1	q20 + q16 + q14 + q12−q4−2−q−4 + q−12 + q−14 + q−16 + q−20
1,0	q34 + q26 + q18−q14 + 1−q−14 + q−18 + q−26 + q−34

D4 Invariants.

Weight	Invariant
1,0,0,0	q26 + q22 + q20 + 2q18 + q16 + q14 + q12−q6−q4−2q2−2q−2−q−4−q−6 + q−12 + q−14 + q−16 + 2q−18 + q−20 + q−22 + q−26

G2 Invariants.

Weight	Invariant
1,0	q46 + q42 + 2q32 + q26 + q24 + q22 + q20−q18 + q16 + q14−q12 + q10−q8−q4−2q2−1−q−2−q−4−2q−6−q−8−q−10 + q−12−q−14−q−16 + q−20 + q−22 + q−24 + q−26 + 3q−30 + q−34 + q−36 + q−40 + q−46−q−50−q−54 + q−56−q−60 + q−62

.

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["9 42"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t2 + 2t−1 + 2t−1−t−2

In[5]:=

Conway[K][z]

Out[5]=

−z4−2z2 + 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, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q3−q2 + q−1 + q−1−q−2 + q−3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4 + a2z2 + z2a−2−4z2 + 2a2 + 2a−2−3

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

az7 + z7a−1 + a2z6 + z6a−2 + 2z6−5az5−5z5a−1−5a2z4−5z4a−2−10z4 + 6az3 + 6z3a−1 + 6a2z2 + 6z2a−2 + 12z2−2az−2za−1−2a2−2a−2−3

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["9 42"];

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 + 2t−1 + 2t−1−t−2, q3−q2 + q−1 + q−1−q−2 + q−3 }

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	q10−q9−q8 + 2q7−q6−q5 + 2q4−q3 + q−1 + q−1−q−3 + 2q−4−q−5−q−6 + 2q−7−q−8−q−9 + q−10
3	q19−2q17−2q16 + 4q15 + 2q14−4q13−3q12 + 5q11 + 4q10−5q9−4q8 + 5q7 + 3q6−5q5−3q4 + 5q3 + 3q2−4q−3 + 4q−1 + 3q−2−3q−3−3q−4 + 3q−5 + 2q−6−2q−7−2q−8 + 2q−9 + q−10−2q−11 + 2q−13−2q−15 + 2q−17−q−19−q−20 + q−21
4	q32−q31−q29−q28 + 3q27−q26 + 3q25−3q24−4q23 + 4q22−q21 + 6q20−3q19−5q18 + 3q17−3q16 + 7q15−2q14−5q13 + 3q12−3q11 + 6q10−q9−4q8 + 2q7−3q6 + 5q5 + q4−3q3 + q2−4q + 4 + 3q−1−2q−2−q−3−5q−4 + 3q−5 + 5q−6−2q−8−6q−9 + q−10 + 6q−11 + 2q−12−2q−13−5q−14−q−15 + 5q−16 + 2q−17−q−18−3q−19−2q−20 + 4q−21−q−23−q−24−q−25 + 4q−26−q−27−q−28−q−29−q−30 + 3q−31−q−34−q−35 + q−36
5	q47−q45−2q44−q43 + 4q41 + 5q40−5q38−7q37−3q36 + 5q35 + 11q34 + 5q33−6q32−12q31−7q30 + 5q29 + 13q28 + 8q27−5q26−13q25−8q24 + 5q23 + 13q22 + 8q21−5q20−13q19−8q18 + 6q17 + 13q16 + 7q15−6q14−13q13−7q12 + 7q11 + 12q10 + 6q9−6q8−11q7−6q6 + 6q5 + 10q4 + 5q3−4q2−9q−5 + 4q−1 + 7q−2 + 4q−3−2q−4−6q−5−4q−6 + 3q−7 + 4q−8 + 2q−9−q−10−3q−11−q−12 + 2q−13 + 2q−14−q−15−3q−16−q−17 + 2q−18 + 4q−19 + 2q−20−3q−21−6q−22−2q−23 + 3q−24 + 5q−25 + 4q−26−2q−27−6q−28−3q−29 + q−30 + 4q−31 + 4q−32−4q−34−2q−35 + 2q−37 + 2q−38−q−39−2q−40−q−41 + q−42 + 2q−43 + q−44−q−45−q−46−2q−47 + 2q−49 + q−50−q−53−q−54 + q−55
6	q66−q65−q62−q61−q60 + 3q59 + q58 + 4q57 + q56−2q55−7q54−6q53 + 2q51 + 14q50 + 7q49 + 2q48−13q47−14q46−7q45−q44 + 20q43 + 13q42 + 9q41−12q40−16q39−13q38−5q37 + 20q36 + 14q35 + 13q34−11q33−14q32−15q31−7q30 + 19q29 + 13q28 + 14q27−11q26−14q25−15q24−6q23 + 19q22 + 13q21 + 14q20−10q19−14q18−15q17−5q16 + 17q15 + 11q14 + 14q13−8q12−12q11−15q10−5q9 + 13q8 + 9q7 + 16q6−5q5−9q4−16q3−6q2 + 8q + 7 + 18q−1−5q−3−17q−4−8q−5 + 2q−6 + 4q−7 + 19q−8 + 5q−9−16q−11−8q−12−4q−13−q−14 + 17q−15 + 8q−16 + 4q−17−12q−18−5q−19−7q−20−5q−21 + 12q−22 + 7q−23 + 5q−24−8q−25−5q−27−5q−28 + 8q−29 + 3q−30 + q−31−7q−32 + 2q−33−q−34−q−35 + 8q−36 + 2q−37−3q−38−8q−39−q−40−q−41 + q−42 + 9q−43 + 4q−44−2q−45−6q−46−3q−47−3q−48−q−49 + 8q−50 + 3q−51−2q−53−2q−54−2q−55−2q−56 + 6q−57−q−59−q−60−q−61−q−62−q−63 + 5q−64−q−67−q−68−2q−69−q−70 + 3q−71 + q−73−q−76−q−77 + q−78
7	q87−q85−q84−q83−q82 + q80 + 5q79 + 4q78 + q77−q76−5q75−7q74−9q73−4q72 + 7q71 + 15q70 + 11q69 + 9q68−q67−15q66−22q65−20q64 + q63 + 18q62 + 23q61 + 24q60 + 8q59−15q58−27q57−30q56−10q55 + 15q54 + 25q53 + 30q52 + 13q51−12q50−24q49−31q48−13q47 + 12q46 + 23q45 + 29q44 + 13q43−13q42−22q41−29q40−12q39 + 13q38 + 22q37 + 29q36 + 12q35−13q34−23q33−29q32−11q31 + 14q30 + 23q29 + 28q28 + 11q27−14q26−24q25−27q24−9q23 + 15q22 + 22q21 + 25q20 + 9q19−14q18−22q17−24q16−7q15 + 13q14 + 19q13 + 21q12 + 8q11−11q10−17q9−19q8−7q7 + 9q6 + 13q5 + 16q4 + 9q3−6q2−10q−14−7q−1 + 3q−2 + 5q−3 + 10q−4 + 9q−5−3q−7−6q−8−6q−9−2q−10−3q−11 + q−12 + 6q−13 + 3q−14 + 5q−15 + 3q−16−q−17−2q−18−8q−19−9q−20−2q−21 + q−22 + 8q−23 + 11q−24 + 7q−25 + 4q−26−7q−27−14q−28−10q−29−6q−30 + 3q−31 + 12q−32 + 13q−33 + 10q−34−11q−36−11q−37−11q−38−4q−39 + 7q−40 + 10q−41 + 10q−42 + 4q−43−4q−44−6q−45−7q−46−5q−47 + 3q−48 + 5q−49 + 4q−50 + q−51−3q−52−3q−53−2q−54 + 4q−56 + 6q−57 + 2q−58−2q−59−6q−60−6q−61−2q−62 + 4q−64 + 8q−65 + 5q−66−4q−68−7q−69−3q−70−2q−71 + 6q−73 + 4q−74 + 2q−75−q−76−4q−77−q−78−q−79−q−80 + 4q−81 + q−82−3q−85−q−86−q−87 + 4q−89 + q−90 + q−92−2q−93−q−94−2q−95−q−96 + 2q−97 + q−98 + q−100−q−103−q−104 + q−105