10 147

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["10 147"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,4,11,3 X5,14,6,15 X15,20,16,1 X12,7,13,8 X8,18,9,17 X19,7,20,6 X16,12,17,11 X18,13,19,14 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,3,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_147.

A1 Invariants.

Weight	Invariant
1	q7−q5 + q3−q + q−1 + q−3 + q−7−2q−9 + q−11
2	q22−q20−2q18 + 3q16 + q14−4q12 + 2q10 + 4q8−3q6−2q4 + 3q2−1−3q−2 + 3q−4 + 3q−6−2q−8 + q−10 + 4q−12−q−14−5q−16 + 2q−18 + q−20−4q−22 + 2q−24 + 2q−26−q−28
3	q45−q43−2q41 + 4q37 + 3q35−5q33−6q31 + 3q29 + 9q27 + 2q25−10q23−8q21 + 6q19 + 12q17 + q15−13q13−7q11 + 11q9 + 13q7−6q5−15q3 + 3q + 14q−1 + q−3−15q−5−q−7 + 14q−9 + 3q−11−11q−13−3q−15 + 10q−17 + 7q−19−5q−21−11q−23−q−25 + 10q−27 + 8q−29−12q−31−14q−33 + 7q−35 + 19q−37−q−39−15q−41−3q−43 + 9q−45 + 6q−47−6q−49−4q−51 + q−53 + 2q−55 + q−57−q−59

A2 Invariants.

Weight	Invariant
1,0	q10 + q4−q2 + 2q−6 + q−10−q−12−q−14 + q−16
1,1	q28−2q26 + 6q24−12q22 + 19q20−28q18 + 36q16−38q14 + 33q12−24q10 + 12q8 + 14q6−31q4 + 48q2−60 + 62q−2−69q−4 + 56q−6−42q−8 + 32q−10−3q−12−6q−14 + 28q−16−38q−18 + 38q−20−40q−22 + 24q−24−18q−26 + 11q−28−2q−30−2q−32 + 2q−34 + 2q−36−2q−42 + q−44
2,0	q28−q24−q22 + q20 + 2q18−q16−q14 + q12 + 2q10 + q8−2q6 + q2−1−2q−2 + q−6 + q−8 + 2q−10 + 2q−12 + 2q−14 + 2q−16 + 2q−18−q−20−5q−22−q−24−2q−28−q−30 + 2q−32 + 3q−34−q−38

A3 Invariants.

Weight	Invariant
0,1,0	q20−q18 + q16 + q14−2q12 + 2q10−q6 + 3q4−1 + 2q−2−2q−6 + q−8 + q−12−q−14 + 2q−18−2q−20 + q−22 + 3q−24−2q−26 + 2q−30−2q−32−q−34 + q−36
1,0,0	q13 + q9 + q5−q3−q−1 + q−7 + 2q−9 + q−13−q−15−q−19 + q−21

A4 Invariants.

Weight	Invariant
0,1,0,0	q26 + 2q20 + q18−2q16 + q12−2q10−2q8 + 3q6 + 4q4−q2 + 2 + 6q−2−4q−6−q−10−5q−12−q−14 + 2q−16 + q−18−q−20 + 4q−22 + 3q−24−2q−26 + 3q−30−2q−34 + q−36 + q−38−q−40−2q−42 + q−46
1,0,0,0	q16 + q12 + q10 + q6−q4−1−q−2 + q−8 + q−10 + 2q−12 + q−16−q−18−q−24 + q−26

B2 Invariants.

Weight	Invariant
0,1	q20−q18 + 3q16−3q14 + 4q12−4q10 + 4q8−3q6 + q4−3 + 4q−2−6q−4 + 8q−6−7q−8 + 8q−10−5q−12 + 5q−14−2q−16 + 2q−20−3q−22 + 3q−24−4q−26 + 4q−28−4q−30 + 2q−32−q−34 + q−36
1,0	q34−q30−q28 + 2q26 + 2q24−2q22−3q20 + q18 + 4q16 + q14−4q12−2q10 + 4q8 + 4q6−q4−4q2 + 3q−2 + q−4−2q−6−2q−8 + 2q−10 + 2q−12−q−14−3q−16 + 2q−18 + 4q−20−4q−24−q−26 + 3q−28 + 2q−30−3q−32−3q−34 + 2q−36 + 4q−38−3q−42−q−44 + 2q−46 + 3q−48−q−50−2q−52−q−54 + q−58

D4 Invariants.

Weight	Invariant
1,0,0,0	q26−q24 + 2q22−2q20 + 4q18−3q16 + 3q14−3q12 + 4q10−2q8 + q6 + 3−3q−2 + 4q−4−5q−6 + 5q−8−6q−10 + 6q−12−6q−14 + 5q−16−4q−18 + 3q−20−q−22 + q−24 + q−26−q−28 + 3q−30−2q−32 + 4q−34−3q−36 + 2q−38−3q−40 + 3q−42−2q−44−q−48 + q−50

G2 Invariants.

Weight

Invariant

1,0

q46−q44 + 3q42−4q40 + 3q38−q36−3q34 + 10q32−11q30 + 12q28−6q26−5q24 + 12q22−14q20 + 11q18−5q16−4q14 + 12q12−9q10 + 3q8 + 5q6−13q4 + 14q2−7−5q−2 + 10q−4−14q−6 + 18q−8−11q−10 + 4q−12 + 4q−14−13q−16 + 15q−18−13q−20 + 7q−22−7q−26 + 11q−28−6q−30 + 3q−32 + 6q−34−14q−36 + 11q−38−q−40−7q−42 + 13q−44−15q−46 + 11q−48 + q−50−6q−52 + 7q−54−11q−56 + 7q−58−3q−62 + 2q−64−2q−66 + q−68 + q−70 + 2q−72−q−74−q−78−q−84 + q−86

.

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["10 147"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−2t2 + 7t−9 + 7t−1−2t−2

In[5]:=

Conway[K][z]

Out[5]=

−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]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

q5−3q4 + 4q3−4q2 + 5q−4 + 3q−1−2q−2 + q−3

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z4a−2−z4 + a2z2−z2a−2 + z2a−4−2z2 + a2 + a−2−1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−2 + z8 + 2az7 + 4z7a−1 + 2z7a−3 + a2z6−z6a−2 + z6a−4−z6−8az5−14z5a−1−6z5a−3−4a2z4−2z4a−2−6z4 + 8az3 + 13z3a−1 + 8z3a−3 + 3z3a−5 + 4a2z2 + z2a−2 + z2a−6 + 6z2−2az−4za−1−3za−3−za−5−a2−a−2−1

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["10 147"];

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 + 7t−9 + 7t−1−2t−2, q5−3q4 + 4q3−4q2 + 5q−4 + 3q−1−2q−2 + q−3 }

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_11, K11n122,}

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	−q13 + 3q12−7q10 + 8q9 + q8−14q7 + 12q6 + 6q5−17q4 + 9q3 + 11q2−17q + 3 + 13q−1−13q−2−2q−3 + 12q−4−6q−5−4q−6 + 6q−7−q−8−2q−9 + q−10
3	−q28 + 2q27 + q26−q25−6q24 + 13q22 + 2q21−18q20−12q19 + 27q18 + 22q17−30q16−33q15 + 29q14 + 42q13−28q12−44q11 + 19q10 + 48q9−16q8−41q7 + 6q6 + 40q5−2q4−30q3−9q2 + 26q + 14−17q−1−20q−2 + 8q−3 + 23q−4 + 2q−5−22q−6−10q−7 + 17q−8 + 16q−9−11q−10−16q−11 + 3q−12 + 14q−13 + q−14−9q−15−3q−16 + 5q−17 + 2q−18−q−19−2q−20 + q−21
4	−q46 + 2q45 + 2q44−4q43−3q42−4q41 + 12q40 + 16q39−11q38−23q37−30q36 + 30q35 + 70q34 + 5q33−61q32−105q31 + 21q30 + 151q29 + 72q28−74q27−201q26−36q25 + 202q24 + 153q23−43q22−255q21−106q20 + 200q19 + 193q18 + 2q17−251q16−141q15 + 172q14 + 181q13 + 37q12−213q11−149q10 + 135q9 + 150q8 + 64q7−163q6−150q5 + 87q4 + 109q3 + 93q2−94q−140 + 27q−1 + 52q−2 + 107q−3−15q−4−99q−5−16q−6−18q−7 + 79q−8 + 38q−9−33q−10−10q−11−62q−12 + 20q−13 + 35q−14 + 14q−15 + 26q−16−52q−17−17q−18 + q−19 + 12q−20 + 41q−21−15q−22−13q−23−15q−24−5q−25 + 24q−26 + q−27−7q−29−7q−30 + 6q−31 + q−32 + 2q−33−q−34−2q−35 + q−36
5	q66−2q65−3q64 + 3q63 + 7q62 + 5q61−2q60−20q59−27q58 + 7q57 + 47q56 + 60q55 + 10q54−81q53−133q52−58q51 + 119q50 + 240q49 + 144q48−130q47−359q46−295q45 + 102q44 + 492q43 + 464q42−27q41−574q40−655q39−99q38 + 620q37 + 819q36 + 238q35−609q34−927q33−377q32 + 552q31 + 992q30 + 488q29−490q28−995q27−553q26 + 407q25 + 972q24 + 594q23−357q22−920q21−597q20 + 290q19 + 874q18 + 598q17−249q16−808q15−589q14 + 175q13 + 751q12 + 599q11−114q10−671q9−596q8 + 12q7 + 583q6 + 603q5 + 79q4−470q3−574q2−190q + 336 + 536q−1 + 269q−2−189q−3−448q−4−326q−5 + 40q−6 + 337q−7 + 336q−8 + 78q−9−200q−10−293q−11−159q−12 + 68q−13 + 210q−14 + 185q−15 + 35q−16−110q−17−153q−18−92q−19 + 11q−20 + 93q−21 + 101q−22 + 47q−23−22q−24−66q−25−71q−26−30q−27 + 22q−28 + 56q−29 + 52q−30 + 15q−31−25q−32−45q−33−36q−34−3q−35 + 30q−36 + 33q−37 + 14q−38−6q−39−23q−40−20q−41−q−42 + 13q−43 + 10q−44 + 5q−45−9q−47−5q−48 + 2q−49 + 2q−50 + q−51 + 2q−52−q−53−2q−54 + q−55