10 132

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["10 132"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[23,3,2-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_132.

A1 Invariants.

Weight	Invariant
1	−q15 + q7 + q5 + q3
2	q44−q40−q30−q24 + q16 + q14 + q10 + q6 + 2−q−4
3	−q87 + q83 + q81−q77 + q67−q63 + q59 + q57−q55−q53−q45−q43 + q39−q35 + q31−q27−q25 + q17 + 2q15 + 2q13 + q7 + 2q5 + q3−q−q−1

A2 Invariants.

Weight	Invariant
1,0	−q22−q20−q18 + q14 + q12 + 2q10 + q8 + q6
1,1	q60 + 2q56−2q50−2q48−2q44 + 2q36−2q34−2q30−q28−2q24 + 2q22−q20 + 4q18 + 6q14 + q12 + 4q10 + 2q8−2q6 + 2q4−2q2 + 2−2q−2
2,0	q58 + q56 + q54−q48−q46−2q44−2q42−2q40−q38 + q36−q34 + q28 + q24 + 2q22 + 2q20 + q18 + q16 + q14 + q10 + q8 + q4 + q2 + 1−q−2−q−4

A3 Invariants.

Weight	Invariant
0,1,0	q46 + q42 + q40−2q36−2q34−4q32−5q30−2q28−q26 + 2q24 + 3q22 + 6q20 + 4q18 + 3q16 + 2q14 + q12−q10−q8−q4 + 1
1,0,0	−q29−q27−2q25−q23 + q19 + 2q17 + 2q15 + 2q13 + q11 + q9

A4 Invariants.

Weight	Invariant
0,1,0,0	q60 + q58 + 2q56 + 2q54 + 2q52 + q50−q48−4q46−6q44−8q42−9q40−8q38−5q36 + 4q32 + 8q30 + 11q28 + 11q26 + 8q24 + 6q22 + q20−q18−3q16−2q14−2q12−q10 + q6 + 1
1,0,0,0	−q36−q34−2q32−2q30−q28 + q24 + 2q22 + 3q20 + 2q18 + 2q16 + q14 + q12

B2 Invariants.

Weight	Invariant
0,1	−q46−q42−q40 + q30 + q26 + q22 + q16 + q12 + q10 + q8 + q4−1
1,0	q76 + q68−q58−2q56−q54−q52−2q50−2q48−q46−q44 + q38 + q36 + 2q34 + 2q32 + 2q30 + q28 + 2q26 + 2q24 + q22 + q18−q12−q4 + 1

D4 Invariants.

Weight	Invariant
1,0,0,0	q62 + q58 + q56 + q54−q50−2q48−4q46−4q44−5q42−4q40−3q38 + q34 + 4q32 + 5q30 + 6q28 + 5q26 + 4q24 + 3q22 + q20 + q18−q16−q14−q12−q8 + 1

G2 Invariants.

Weight	Invariant
1,0	q108 + q104−q100−q92−q90−q86−q84−q82−2q80−q78−q76−2q74−q68−q64 + q62 + q60 + q58 + q56 + q54 + 2q52 + 3q50 + q48 + q46 + 2q44 + q42 + 2q40 + q38 + q34 + q32−q28 + q26−q24−q18 + q16−q12 + q4−q2 + 1 + q−6−q−8

.

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["10 132"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−t + 1−t−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]=

{1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q−2 + q−4−q−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 + 4z2a4 + 3a4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a6z8 + a4z8 + a7z7 + 2a5z7 + a3z7−6a6z6−6a4z6−6a7z5−12a5z5−6a3z5 + 10a6z4 + 10a4z4 + 10a7z3 + 19a5z3 + 9a3z3−6a6z2−7a4z2−a2z2−5a7z−8a5z−4a3z−az + 2a6 + 3a4

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["10 132"];

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−t + 1−t−1 + t−2, q−2 + q−4−q−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]=

{5_1,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{5_1,}

n	Jn
2	−q + 1 + 2q−1−3q−2 + q−3 + 3q−4−4q−5 + 2q−6 + 2q−7−3q−8 + 2q−9 + q−10−3q−11 + 2q−12−2q−14 + 2q−15−q−16−q−17 + 2q−18−q−19−q−20 + q−21
3	−q−1 + 2q−3 + q−4−2q−5−q−6 + 2q−7 + 3q−8−2q−9−2q−10 + q−11 + 3q−12−2q−13−3q−14 + q−15 + 4q−16−q−17−4q−18 + 5q−20−5q−22−q−23 + 5q−24 + q−25−5q−26−q−27 + 4q−28 + q−29−3q−30−q−31 + 3q−32−2q−34 + 2q−36−2q−38 + q−40 + q−41−q−42
4	q5−q4−q3−q2 + 5−q−2−6q−3−2q−4 + 9q−5 + 2q−6−9q−8−3q−9 + 10q−10 + 3q−11 + q−12−10q−13−3q−14 + 11q−15 + 3q−16−q−17−10q−18−3q−19 + 11q−20 + 2q−21−2q−22−9q−23−2q−24 + 10q−25 + 2q−26−2q−27−7q−28−2q−29 + 8q−30 + 2q−31−2q−32−5q−33−q−34 + 6q−35 + q−36−3q−37−3q−38 + q−39 + 5q−40−5q−42−2q−43 + 2q−44 + 6q−45−6q−47−2q−48 + q−49 + 6q−50 + q−51−4q−52−2q−53−q−54 + 5q−55−2q−57−q−58−q−59 + 4q−60−q−61−q−62−q−63−q−64 + 3q−65−q−68−q−69 + q−70