10 156

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["10 156"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X12,4,13,3 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X2,12,3,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[-3:2:20]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_156.

A1 Invariants.

Weight	Invariant
1	−q13 + 2q11−2q9 + q7 + 2q−q−1 + 2q−3−q−5
2	−2q34 + 2q32 + 5q30−7q28−2q26 + 10q24−6q22−6q20 + 8q18−5q14 + q12 + 5q10−3q8−5q6 + 8q4 + 2q2−8 + 6q−2 + 6q−4−8q−6−q−8 + 6q−10−2q−12−2q−14 + q−16
3	q71−q67−5q65−q63 + 11q61 + 9q59−10q57−24q55 + 6q53 + 35q51 + 4q49−40q47−22q45 + 39q43 + 34q41−28q39−36q37 + 14q35 + 35q33−q31−28q29−11q27 + 20q25 + 17q23−13q21−27q19 + 7q17 + 33q15−39q11−5q9 + 42q7 + 18q5−37q3−29q + 30q−1 + 35q−3−12q−5−37q−7−3q−9 + 30q−11 + 15q−13−16q−15−18q−17 + 3q−19 + 14q−21 + q−23−6q−25−3q−27 + 2q−29 + 2q−31−q−33

A2 Invariants.

Weight	Invariant
1,0	−q18 + q16−q14 + q10−q8 + 2q6−q4 + 2q2 + 1 + q−4−q−6
1,1	q52−2q50−2q48 + 2q46 + 2q44−4q42 + 18q40−36q38 + 62q36−90q34 + 104q32−114q30 + 99q28−68q26 + 26q24 + 32q22−84q20 + 132q18−170q16 + 186q14−194q12 + 176q10−138q8 + 94q6−32q4−16q2 + 70−96q−2 + 107q−4−104q−6 + 84q−8−62q−10 + 40q−12−22q−14 + 10q−16−4q−18 + q−20
2,0	−q48 + q42 + 3q40−2q38−q36 + q34 + 2q32−2q30−5q28 + 3q26 + q24−3q22−q20 + 3q18−q16−q14 + 2q12 + q6 + 5q4−q2 + 5q−2 + q−4−4q−6−q−8 + 2q−10 + q−12−2q−14−q−16 + q−18

A3 Invariants.

Weight	Invariant
0,1,0	q42−q40−q38 + 4q36−3q34−4q32 + 7q30−4q28−4q26 + 6q24−3q22−2q20 + 2q18 + q16−q12 + 3q10 + 3q8−5q6 + 3q4 + 6q2−5 + 3q−2 + 4q−4−5q−6 + 2q−8−2q−12 + q−14
1,0,0	−q23 + q21−2q19 + q17−q15 + q13 + q9 + q7 + 2q3 + 2q−1−q−3 + q−5−q−7

A4 Invariants.

Weight	Invariant
0,1,0,0	2q52−q50−2q48 + 4q46 + q44−6q42 + 4q38−4q36−6q34 + 3q32 + 3q30−5q28 + 8q24−3q22−5q20 + 8q18−6q14 + 4q12 + 7q10−3q8−q6 + 6q4 + 4q2−3 + q−2 + 4q−4−2q−6−2q−8 + q−10−q−12−q−14 + q−16
1,0,0,0	−q28 + q26−2q24−q18 + q16 + 2q12 + 2q8 + 2q4 + 1 + q−2−q−4 + q−6−q−8

B2 Invariants.

Weight	Invariant
0,1	−q42 + q40−3q38 + 6q36−7q34 + 8q32−7q30 + 6q28−4q26 + 3q22−8q20 + 10q18−13q16 + 14q14−13q12 + 13q10−7q8 + 5q6 + q4−2q2 + 5−7q−2 + 8q−4−7q−6 + 6q−8−4q−10 + 2q−12−q−14
1,0	q68−q64−2q62 + 5q58 + 2q56−5q54−6q52 + 2q50 + 8q48 + q46−8q44−4q42 + 6q40 + 5q38−4q36−6q34 + 2q32 + 6q30−6q26−q24 + 5q22 + 3q20−3q18−4q16 + 4q14 + 5q12−2q10−7q8 + 2q6 + 8q4 + 4q2−6−6q−2 + 5q−4 + 8q−6−q−8−6q−10−2q−12 + 4q−14 + 2q−16−2q−18−2q−20 + q−24

D4 Invariants.

Weight	Invariant
1,0,0,0	q58−q56 + q54−2q52 + 5q50−6q48 + 4q46−6q44 + 7q42−6q40 + 3q38−4q36 + q34 + 2q32−4q30 + 4q28−8q26 + 10q24−9q22 + 11q20−10q18 + 11q16−7q14 + 9q12−5q10 + 3q8 + q6 + 3q2−3 + 7q−2−6q−4 + 6q−6−6q−8 + 5q−10−4q−12 + 2q−14−2q−16 + q−18

G2 Invariants.

Weight

Invariant

1,0

q100−q98 + q96−2q90 + 2q88 + 2q86−6q84 + 11q82−16q80 + 10q78−3q76−13q74 + 29q72−35q70 + 27q68−8q66−16q64 + 35q62−36q60 + 23q58−20q54 + 28q52−21q50 + 2q48 + 21q46−34q44 + 32q42−18q40−5q38 + 27q36−42q34 + 42q32−31q30 + 11q28 + 15q26−35q24 + 44q22−33q20 + 15q18 + 10q16−28q14 + 31q12−16q10−3q8 + 25q6−32q4 + 24q2−1−22q−2 + 36q−4−35q−6 + 22q−8−3q−10−16q−12 + 25q−14−22q−16 + 16q−18−5q−20−3q−22 + 5q−24−7q−26 + 4q−28−2q−30 + q−32

.

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["10 156"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−4t2 + 8t−9 + 8t−1−4t−2 + t−3

In[5]:=

Conway[K][z]

Out[5]=

z6 + 2z4 + z2 + 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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q2 + 3q−4 + 6q−1−6q−2 + 6q−3−5q−4 + 3q−5−q−6

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

a2z6−a4z4 + 4a2z4−z4−2a4z2 + 5a2z2−2z2−a4 + 2a2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a4z8 + a2z8 + a5z7 + 4a3z7 + 3az7−a4z6 + 2a2z6 + 3z6−a5z5−9a3z5−7az5 + z5a−1 + 3a6z4 + 2a4z4−9a2z4−8z4 + a7z3 + 4a5z3 + 8a3z3 + 3az3−2z3a−1−2a6z2 + a4z2 + 7a2z2 + 4z2−a7z−2a5z−2a3z−az−a4−2a2

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["10 156"];

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]=

{ t3−4t2 + 8t−9 + 8t−1−4t−2 + t−3, −q2 + 3q−4 + 6q−1−6q−2 + 6q−3−5q−4 + 3q−5−q−6 }

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_16, K11n15, K11n56, K11n58,}

In[6]:=

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

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

Out[6]=

{8_16,}

n	Jn
2	q7−3q6 + 9q4−10q3−7q2 + 23q−10−21q−1 + 33q−2−4q−3−34q−4 + 35q−5 + 4q−6−38q−7 + 29q−8 + 9q−9−30q−10 + 15q−11 + 9q−12−14q−13 + 3q−14 + 4q−15−2q−16
3	−q15 + 3q14−5q12−4q11 + 10q10 + 13q9−16q8−25q7 + 12q6 + 44q5−q4−58q3−22q2 + 69q + 46−63q−1−81q−2 + 61q−3 + 101q−4−39q−5−128q−6 + 27q−7 + 140q−8−6q−9−154q−10−7q−11 + 154q−12 + 24q−13−151q−14−38q−15 + 137q−16 + 51q−17−115q−18−59q−19 + 87q−20 + 59q−21−53q−22−54q−23 + 26q−24 + 41q−25−9q−26−23q−27−3q−28 + 11q−29 + 5q−30−4q−31−q−32−q−33 + q−34
4	q26−3q25 + 5q23 + 4q21−17q20−6q19 + 17q18 + 11q17 + 31q16−46q15−47q14 + 5q13 + 27q12 + 121q11−25q10−96q9−89q8−50q7 + 222q6 + 108q5−26q4−189q3−268q2 + 183q + 253 + 208q−1−148q−2−510q−3−21q−4 + 273q−5 + 477q−6 + 34q−7−646q−8−267q−9 + 180q−10 + 667q−11 + 238q−12−677q−13−453q−14 + 66q−15 + 763q−16 + 390q−17−646q−18−572q−19−38q−20 + 782q−21 + 498q−22−549q−23−624q−24−159q−25 + 689q−26 + 566q−27−348q−28−572q−29−289q−30 + 458q−31 + 533q−32−89q−33−378q−34−332q−35 + 163q−36 + 357q−37 + 83q−38−132q−39−228q−40−22q−41 + 135q−42 + 86q−43 + 8q−44−78q−45−44q−46 + 16q−47 + 25q−48 + 19q−49−7q−50−11q−51−2q−52 + 3q−54 + q−55−q−56