10 130

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["10 130"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X13,6,14,7 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 -14 2 -16 -18 -6 -20 -12 -10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[311,3,2-]

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,−2,−2,−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, 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[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_130.

A1 Invariants.

Weight	Invariant
1	−q15−q11 + q9 + q7 + q5 + q3 + q−1−q−3
2	q44−q40 + q38 + q36−2q34−q32−q28−q26 + q22−q20 + 2q16 + 3q10 + q8 + q2−q−2
3	−q87 + q83 + q81−q79−2q77 + 3q73 + 2q71−q69−2q67 + 2q63 + 2q61−3q57−2q55 + q53 + 3q51−2q49−4q47−q45 + 3q43−3q39−2q37 + 2q35 + q33−2q31−q29 + 4q27 + 2q25−2q23−q21 + 2q19 + 3q17 + q15−q13−2q11 + 2q9 + 5q7 + 2q5−6q3−3q + 4q−1 + 4q−3−3q−5−2q−7 + q−9 + q−11−q−13−q−15 + q−17

A2 Invariants.

Weight	Invariant
1,0	−q22−q20−q18−q16 + q14 + q12 + 2q10 + 2q8 + q6 + q4−q−4
1,1	q60 + 4q56−4q54 + 6q52−8q50 + 6q48−8q46 + q44−4q40 + 6q38−7q36 + 8q34−10q32 + 6q30−11q28 + 4q26−6q24 + 4q22 + 3q20 + 2q18 + 8q16 + 4q14 + 5q12 + 2q10 + 2q8−2q6 + 2q4−4q2 + 2−4q−2 + q−4−2q−6 + q−12
2,0	q58 + q56 + q54 + q50 + q48−3q44−3q42−3q40−2q38−2q36−3q34−q32−q30 + q28 + 3q24 + 4q22 + 5q20 + 3q18 + 3q16 + 2q14 + q12 + q10−q8−q6−1−2q−2 + q−6

A3 Invariants.

Weight	Invariant
0,1,0	q46 + 2q42 + q40−q36−3q34−5q32−6q30−4q28−2q26 + 2q24 + 3q22 + 6q20 + 5q18 + 4q16 + 3q14 + 2q12 + q10 + q8−q2−1−q−2−q−4−q−6 + q−10
1,0,0	−q29−q27−2q25−q23−q21 + q19 + q17 + 2q15 + 2q13 + 2q11 + 2q9 + q7 + q5−q−1−q−5

A4 Invariants.

Weight	Invariant
0,1,0,0	q60 + q58 + 2q56 + 3q54 + 3q52 + 2q50 + q48−3q46−7q44−10q42−11q40−12q38−10q36−3q34 + 3q32 + 6q30 + 10q28 + 14q26 + 11q24 + 9q22 + 6q20 + 4q18−q10−2q8−q4−3q2−2−q−2−q−4−q−6 + q−10 + q−12
1,0,0,0	−q36−q34−2q32−2q30−q28−q26 + q24 + q22 + 2q20 + 2q18 + 2q16 + 2q14 + 2q12 + 2q10 + q8 + q6−1−q−2−q−6

B2 Invariants.

Weight	Invariant
0,1	−q46−2q42 + q40−2q38 + q36−q34 + q32 + 2q26−2q24 + 3q22−2q20 + 3q18−2q16 + 3q14 + q10 + q8 + 2q4−q2 + 1−q−2 + q−4−q−6−q−10
1,0	q76 + 2q68 + q66−q64−q62 + q60−2q56−3q54−2q52−q50−2q48−3q46−3q44 + q40 + q38 + 2q34 + 3q32 + 3q30 + q28 + 2q26 + 3q24 + 3q22 + q16 + 2q14−q10−q8 + q6−q2−1−q−8−q−10 + q−16

D4 Invariants.

Weight	Invariant
1,0,0,0	q62 + 2q58 + 3q54−q52−3q48−3q46−6q44−6q42−5q40−4q38−q34 + 5q32 + 3q30 + 8q28 + 3q26 + 6q24 + q22 + 4q20 + q18 + 2q16 + q14 + q12 + 2q10 + q6−2q4−2−q−2−2q−4−q−8 + q−14

G2 Invariants.

Weight	Invariant
1,0	q108 + 2q104−2q102 + q100−2q96 + 3q94−4q92 + 2q90−q88−3q86 + 2q84−4q82−q80 + q78−4q76−q74−4q70 + 3q68−3q66 + q62−2q60 + 4q58−2q56 + 4q54 + 3q50 + 2q48 + 4q44−q42 + 3q40 + 2q38−q36 + 2q34 + 3q32−3q30 + 4q28−q26−q24 + 4q22−4q20 + 3q18−q16 + q14 + q12−q10 + 1−2q−2 + q−4−q−6−q−12−q−16−q−20 + q−24

.

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["10 130"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t2−4t + 5−4t−1 + 2t−2

In[5]:=

Conway[K][z]

Out[5]=

2z4 + 4z2 + 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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q + 2−2q−1 + 3q−2−2q−3 + 3q−4−2q−5 + q−6−q−7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a6−2a6 + z4a4 + 3z2a4 + 2a4 + z4a2 + 3z2a2 + 2a2−z2−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a6z8 + a4z8 + a7z7 + 3a5z7 + 2a3z7−5a6z6−3a4z6 + 2a2z6−6a7z5−15a5z5−8a3z5 + az5 + 7a6z4−7a2z4 + 11a7z3 + 21a5z3 + 8a3z3−2az3−4a6z2 + 6a2z2 + 2z2−6a7z−9a5z−3a3z + az + za−1 + 2a6 + 2a4−2a2−1

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.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 130"];

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]=

{ 2t2−4t + 5−4t−1 + 2t−2, −q + 2−2q−1 + 3q−2−2q−3 + 3q−4−2q−5 + q−6−q−7 }

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]=

{7_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]=

{K11n61,}

n	Jn
2	−1 + q−1 + q−2−2q−3 + q−4 + 2q−5−2q−7 + 2q−8 + 2q−9−4q−10 + q−11 + 4q−12−5q−13 + 4q−15−4q−16−q−17 + 3q−18−q−19−q−20 + q−21
3	q7−2q6 + 2q4 + q3−5q2−q + 9 + q−1−12q−2−4q−3 + 17q−4 + 4q−5−15q−6−8q−7 + 18q−8 + 6q−9−13q−10−9q−11 + 15q−12 + 5q−13−9q−14−7q−15 + 10q−16 + 4q−17−6q−18−6q−19 + 6q−20 + 3q−21−3q−22−3q−23 + 2q−24−q−26 + 2q−27−3q−29−2q−30 + 5q−31 + 2q−32−3q−33−4q−34 + 3q−35 + 3q−36−3q−38 + q−40 + q−41−q−42
4	−q12 + 2q11−2q9 + 2q8−2q7 + 4q6−5q5−5q4 + 10q3 + q2 + 7q−17−15q−1 + 19q−2 + 13q−3 + 16q−4−28q−5−31q−6 + 18q−7 + 22q−8 + 30q−9−27q−10−41q−11 + 11q−12 + 19q−13 + 35q−14−17q−15−38q−16 + 4q−17 + 11q−18 + 32q−19−8q−20−33q−21 + q−22 + 5q−23 + 29q−24−29q−26−5q−27 + 27q−29 + 10q−30−24q−31−11q−32−7q−33 + 21q−34 + 20q−35−15q−36−12q−37−13q−38 + 9q−39 + 22q−40−5q−41−5q−42−11q−43−3q−44 + 14q−45−2q−46 + 3q−47−2q−48−5q−49 + 5q−50−7q−51 + 3q−52 + 4q−53 + 5q−55−9q−56−2q−57 + q−58 + 2q−59 + 7q−60−4q−61−2q−62−2q−63−q−64 + 4q−65−q−68−q−69 + q−70