10 140

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["10 140"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,19,12,18 X14,5,15,6 X6,17,7,18 X16,7,17,8 X8,15,9,16 X13,1,14,20 X19,13,20,12 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4,3,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_140.

A1 Invariants.

Weight	Invariant
1	−q15 + q9 + q7 + q−1
2	q44−q40−q34−q32 + q28 + q26 + q22−q18−q14−q12 + q10 + q8 + q6 + q2 + 2−q−4
3	−q87 + q83 + q81−q77 + q73 + q71−q67−2q65−q63 + q59−q55 + 2q51 + 2q49−q45 + q41−2q37−2q35−q29 + q25 + q23 + q17 + q15−q11−q9 + 2q7 + 3q5−2q + 3q−3 + q−5−q−7−q−9

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	q46 + q42−q38−2q36−2q34−2q32−3q30 + 2q24 + 2q22 + 4q20 + 3q18 + 2q16 + 2q14−q10−2q8−q6−q4 + 2 + q−2 + q−4
1,0,0	−q29−q27−2q25−q23 + 2q19 + 3q17 + 3q15 + 2q13−q9−2q7−q5 + q + q−1 + q−3

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	−q46−q42−q38 + q30 + 2q26 + 2q22 + q18−q10−q6 + q4 + q−2 + q−4
1,0	q76 + q68−q64−q62−q58−2q56−q54−q48 + q42 + q38 + q34 + q32 + q30 + q26 + 2q24 + q22 + q16−q12−q10−q4 + 1 + q−2 + q−6

D4 Invariants.

Weight	Invariant
1,0,0,0	q62 + q58 + q54−q52−q50−2q48−3q46−3q44−4q42−2q40−2q38 + q36 + q34 + 4q32 + 4q30 + 6q28 + 4q26 + 4q24 + 2q22 + q20−2q16−2q14−3q12−q10−2q8 + q2 + 2 + q−2 + q−4 + q−6

G2 Invariants.

Weight	Invariant
1,0	q108 + q104−q102−q96 + q94−q92−q90−q88−q86−q82−4q80−4q70 + q68 + 3q66−q62−q60 + 3q58 + 5q56 + 2q54−q52 + q50 + 3q48 + 4q46−2q42 + q40 + 3q38 + q36−q34−2q30 + q28−3q24−q20−q18 + q16−q14−q12−q8 + q6 + 2 + q−4 + q−6 + q−10

.

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["10 140"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−2t + 3−2t−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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

1−q−1 + q−2−q−3 + 2q−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 + 4a4−z2a2−2a2 + 1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a6z8 + a4z8 + a7z7 + 2a5z7 + a3z7−6a6z6−6a4z6−6a7z5−11a5z5−5a3z5 + 11a6z4 + 12a4z4 + a2z4 + 10a7z3 + 16a5z3 + 6a3z3−8a6z2−12a4z2−4a2z2−4a7z−6a5z−2a3z + 2a6 + 4a4 + 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["10 140"];

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 + 3−2t−1 + t−2, 1−q−1 + q−2−q−3 + 2q−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]=

{8_20, K11n73, K11n74,}

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	−q + 1 + 2q−1−2q−2 + 3q−4−2q−5 + q−7−2q−8 + q−9−q−11 + 2q−12−q−13 + 2q−15−2q−16−q−17 + 2q−18−q−19−q−20 + q−21
3	−q3 + 2q + 2−4q−1−2q−2 + 4q−3 + 5q−4−5q−5−5q−6 + 4q−7 + 6q−8−4q−9−5q−10 + 3q−11 + 6q−12−3q−13−5q−14 + 2q−15 + 5q−16−2q−17−5q−18 + 5q−20−4q−22−q−23 + 4q−24 + q−25−2q−26−q−27 + 2q−28−q−30 + q−32−q−33−2q−34 + q−35 + 2q−36−2q−38 + q−40 + q−41−q−42
4	q3−2q2−2q + 4q−1 + 7q−2−5q−3−6q−4−4q−5 + 6q−6 + 12q−7−4q−8−7q−9−8q−10 + 4q−11 + 14q−12−3q−13−6q−14−7q−15 + 3q−16 + 11q−17−4q−18−4q−19−5q−20 + 3q−21 + 9q−22−3q−23−2q−24−5q−25 + q−26 + 7q−27−q−28−5q−30−2q−31 + 4q−32 + 3q−34−2q−35−4q−36 + 5q−39 + 2q−40−3q−41−3q−42−2q−43 + 3q−44 + 5q−45−3q−47−3q−48−q−49 + 4q−50−q−53−2q−54 + 3q−55−2q−56 + 4q−60−2q−61−q−62−q−63−q−64 + 3q−65−q−68−q−69 + q−70