9 48

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 48"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[21,21,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_48.

A1 Invariants.

Weight	Invariant
1	q3−2q + q−1 + 2q−5 + 2q−7−q−9 + q−11−2q−13
2	q10−2q8−2q6 + 6q4−6 + 5q−2 + 2q−4−7q−6 + 2q−8 + 4q−10−2q−12 + q−14 + 4q−16 + 4q−18−5q−20 + 6q−24−7q−26−4q−28 + 5q−30−3q−32−3q−34 + 3q−36 + q−38
3	q21−2q19−2q17 + 3q15 + 6q13−13q9−4q7 + 14q5 + 11q3−13q−19q−1 + 12q−3 + 26q−5−4q−7−27q−9−q−11 + 23q−13 + 7q−15−22q−17−7q−19 + 14q−21 + 11q−23−7q−25−9q−27 + 2q−29 + 14q−31 + 9q−33−15q−35−12q−37 + 12q−39 + 21q−41−15q−43−27q−45 + 4q−47 + 25q−49−2q−51−24q−53−6q−55 + 19q−57 + 9q−59−9q−61−7q−63 + 5q−65 + 8q−67−q−69−2q−71−2q−73
4	q36−2q34−2q32 + 3q30 + 3q28 + 6q26−7q24−13q22−4q20 + 5q18 + 32q16 + 8q14−25q12−34q10−19q8 + 52q6 + 51q4−63−81q−2 + 30q−4 + 93q−6 + 64q−8−44q−10−120q−12−27q−14 + 78q−16 + 104q−18 + 2q−20−106q−22−62q−24 + 33q−26 + 90q−28 + 31q−30−60q−32−57q−34−q−36 + 53q−38 + 38q−40−14q−42−43q−44−28q−46 + 20q−48 + 50q−50 + 35q−52−42q−54−62q−56−14q−58 + 60q−60 + 82q−62−29q−64−94q−66−63q−68 + 48q−70 + 120q−72 + 14q−74−85q−76−97q−78 + 5q−80 + 110q−82 + 57q−84−27q−86−84q−88−39q−90 + 52q−92 + 55q−94 + 21q−96−34q−98−40q−100 + q−102 + 17q−104 + 20q−106−q−108−13q−110−7q−112−3q−114 + 3q−116 + 3q−118 + q−120
5	q55−2q53−2q51 + 3q49 + 3q47 + 3q45−q43−7q41−13q39−4q37 + 14q35 + 23q33 + 20q31−4q29−37q27−57q25−18q23 + 48q21 + 88q19 + 69q17−24q15−128q13−149q11−29q9 + 144q7 + 229q5 + 139q3−105q−305q−1−269q−3 + 18q−5 + 328q−7 + 385q−9 + 115q−11−291q−13−471q−15−249q−17 + 206q−19 + 491q−21 + 361q−23−85q−25−460q−27−411q−29−18q−31 + 372q−33 + 424q−35 + 98q−37−283q−39−372q−41−141q−43 + 180q−45 + 312q−47 + 159q−49−111q−51−235q−53−155q−55 + 37q−57 + 179q−59 + 157q−61 + 10q−63−131q−65−166q−67−64q−69 + 96q−71 + 196q−73 + 133q−75−74q−77−229q−79−196q−81 + 25q−83 + 265q−85 + 295q−87 + 19q−89−300q−91−372q−93−115q−95 + 286q−97 + 454q−99 + 203q−101−245q−103−491q−105−306q−107 + 150q−109 + 483q−111 + 382q−113−30q−115−399q−117−420q−119−84q−121 + 288q−123 + 395q−125 + 182q−127−139q−129−317q−131−223q−133 + 23q−135 + 206q−137 + 204q−139 + 57q−141−98q−143−153q−145−90q−147 + 23q−149 + 79q−151 + 67q−153 + 19q−155−27q−157−45q−159−21q−161 + 3q−163 + 12q−165 + 17q−167 + 8q−169−q−171−4q−173−2q−175−2q−177

A2 Invariants.

Weight	Invariant
1,0	q4−q2−1 + q−2−q−4 + 2q−6 + q−8 + 2q−10 + 2q−12 + q−16−2q−18−2q−20
1,1	q12−4q10 + 10q8−20q6 + 30q4−42q2 + 58−62q−2 + 59q−4−48q−6 + 26q−8−2q−10−34q−12 + 68q−14−86q−16 + 112q−18−108q−20 + 114q−22−94q−24 + 70q−26−40q−28 + 2q−30 + 22q−32−46q−34 + 61q−36−68q−38 + 56q−40−48q−42 + 31q−44−22q−46 + 10q−48−2q−50 + 2q−52 + 2q−54
2,0	q12−q10−2q8 + 3q4 + 3q2−4−q−2 + 4q−4 + q−6−4q−8−2q−10 + 2q−12−q−14 + 2q−18 + 5q−20 + 2q−22 + 6q−24 + 4q−26−q−28 + q−30 + 3q−32−2q−34−7q−36−4q−38−q−40−4q−42−6q−44 + 4q−48 + 4q−50 + q−52

A3 Invariants.

Weight	Invariant
0,1,0	q8−2q6 + 3q2−4 + 3q−2 + 4q−4−6q−6 + q−10−4q−12 + 6q−16 + 6q−18 + 7q−20 + 4q−22 + 6q−24−3q−26−9q−28−q−30−6q−32−7q−34 + 4q−36 + q−38−q−40 + 3q−42
1,0,0	q5−q3−q−1 + q−3−q−5 + q−7 + q−9 + q−11 + 2q−13 + 2q−15 + 3q−17 + q−21−2q−23−2q−25−2q−27

A4 Invariants.

Weight	Invariant
0,1,0,0	q10−q8−2q6 + 2q4 + q2−2 + 5q−4 + q−6−5q−8 + 3q−12−5q−14−7q−16 + 3q−18 + q−20 + q−22 + 9q−24 + 14q−26 + 9q−28 + 9q−30 + 10q−32 + q−34−10q−36−8q−38−7q−40−13q−42−10q−44−q−46 + 2q−48 + 2q−52 + 4q−54 + 3q−56
1,0,0,0	q6−q4−q−2 + q−4−q−6 + q−8 + q−12 + q−14 + 2q−16 + 2q−18 + 3q−20 + 3q−22 + q−26−2q−28−2q−30−2q−32−2q−34

B2 Invariants.

Weight	Invariant
0,1	q8−2q6 + 4q4−5q2 + 4−5q−2 + 4q−4−2q−6 + 3q−10−4q−12 + 8q−14−8q−16 + 10q−18−7q−20 + 8q−22−4q−24 + 3q−26 + q−28−3q−30 + 4q−32−5q−34 + 4q−36−5q−38 + 3q−40−3q−42
1,0	q14−2q10−2q8 + 2q6 + 4q4−q2−4−q−2 + 6q−4 + 4q−6−4q−8−5q−10 + 3q−14−q−16−4q−18−2q−20 + 4q−22 + 3q−24 + 6q−30 + 7q−32 + 3q−34−2q−36 + 2q−38 + 4q−40−7q−44−5q−46 + q−48 + q−50−5q−52−7q−54−q−56 + 4q−58 + 3q−60−2q−62−2q−64 + q−66 + 3q−68

D4 Invariants.

Weight	Invariant
1,0,0,0	q10−2q8 + 2q6−3q4 + 4q2−4 + 4q−2−3q−4 + 4q−6−2q−8−q−10−2q−14 + 2q−16−6q−18 + 6q−20−3q−22 + 12q−24−q−26 + 12q−28 + 10q−32−q−34−5q−38−6q−40−2q−42−8q−44−q−46−6q−48 + 4q−50−2q−52 + 4q−54−2q−56 + 3q−58

G2 Invariants.

Weight

Invariant

1,0

q18−2q16 + 4q14−6q12 + 3q10 + q8−6q6 + 14q4−14q2 + 15−8q−2−7q−4 + 16q−6−23q−8 + 18q−10−9q−12−5q−14 + 13q−16−13q−18 + 10q−20 + q−22−13q−24 + 16q−26−13q−28 + q−30 + 15q−32−24q−34 + 27q−36−14q−38 + 11q−40 + 7q−42−22q−44 + 29q−46−22q−48 + 19q−50−2q−52−14q−54 + 21q−56−8q−58 + 6q−60 + q−62−15q−64 + 14q−66−5q−68−8q−70 + 14q−72−24q−74 + 21q−76−5q−78−10q−80 + 11q−82−18q−84 + 16q−86−9q−88−2q−90 + 3q−92−7q−94 + 7q−96−2q−98 + q−100 + q−102

.

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 48"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t2 + 7t−11 + 7t−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]=

{3,t + 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]=

−2q6 + 3q5−4q4 + 6q3−4q2 + 4q−3 + q−1

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4a−2−z2a−2 + 3z2a−4 + z2 + 3a−4−2a−6

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z7a−3 + z7a−5 + 3z6a−2 + 4z6a−4 + z6a−6 + 3z5a−1 + 2z5a−3−z5a−5−5z4a−2−6z4a−4 + z4−5z3a−1−3z3a−3 + 5z3a−5 + 3z3a−7 + 2z2a−2 + 2z2a−4−z2a−6−z2−za−3−5za−5−4za−7 + 3a−4 + 2a−6

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 48"];

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 + 7t−11 + 7t−1−t−2, −2q6 + 3q5−4q4 + 6q3−4q2 + 4q−3 + q−1 }

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]=

{K11n1,}

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	q18 + 2q17−6q16 + q15 + 10q14−15q13−2q12 + 23q11−21q10−7q9 + 32q8−21q7−10q6 + 29q5−15q4−12q3 + 20q2−6q−9 + 9q−1−3q−3 + q−4
3	−2q35 + q33 + 9q32−5q31−12q30−q29 + 27q28 + 5q27−37q26−19q25 + 49q24 + 32q23−58q22−50q21 + 61q20 + 68q19−67q18−74q17 + 58q16 + 92q15−62q14−86q13 + 47q12 + 94q11−44q10−83q9 + 26q8 + 79q7−15q6−67q5 + 2q4 + 53q3 + 8q2−37q−12 + 22q−1 + 14q−2−13q−3−9q−4 + 4q−5 + 5q−6−3q−8 + q−9
4	q58 + 2q57−6q55−4q54−5q53 + 14q52 + 21q51−9q50−20q49−46q48 + 20q47 + 76q46 + 25q45−23q44−137q43−25q42 + 133q41 + 109q40 + 30q39−242q38−127q37 + 145q36 + 208q35 + 136q34−314q33−238q32 + 114q31 + 273q30 + 247q29−336q28−312q27 + 66q26 + 293q25 + 324q24−321q23−342q22 + 18q21 + 278q20 + 353q19−269q18−327q17−36q16 + 222q15 + 350q14−178q13−268q12−93q11 + 127q10 + 306q9−70q8−166q7−119q6 + 22q5 + 213q4 + 6q3−58q2−90q−41 + 102q−1 + 24q−2 + 5q−3−39q−4−40q−5 + 31q−6 + 9q−7 + 14q−8−6q−9−16q−10 + 4q−11 + 5q−13−3q−15 + q−16